% Encoding: UTF-8
@Article{aabdls,
author = {Fairweather, G. and Gladwell, I.},
title = {Algorithms for almost block diagonal linear systems},
journal = {SIAM Rev.},
year = {2004},
volume = {46},
pages = {49--58}
}
@Article{AACI,
author = {Artuso, R. and Aurell, E. and Cvitanovi\'c, P.},
title = {Recycling of strange sets: {I. Cycle} expansions},
journal = {Nonlinearity},
year = {1990},
volume = {3},
pages = {325--359},
doi = {10.1088/0951-7715/3/2/005}
}
@Article{AACII,
author = {Artuso, R. and Aurell, E. and Cvitanovi\'c, P.},
title = {Recycling of strange sets: {II. Applications}},
journal = {Nonlinearity},
year = {1990},
volume = {3},
pages = {361--386},
doi = {10.1088/0951-7715/3/2/006}
}
@Unpublished{ABBBBB08,
Title = {Universal intermittent properties of particle trajectories in highly turbulent flows},
Author = {A. Arneodo and R. Benzi and J. Berg and L. Biferale and E. Bodenschatz and A. Busse and E. Calzavarini and B. Castaing and M. Cencini and L. Chevillard and R. Fisher and R. Grauer and H. Homann and D. Lamb and A.S. Lanotte and E. Leveque and B. Luethi and J. Mann and N. Mordant and W.-C. Mueller and S. Ott and N.T. Ouellette and J.-F. Pinton and S. B. Pope and S.G. Roux and F. Toschi and H. Xu and P.K. Yeung},
Note = {\arXiv{0802.3776}},
Year = {1989}
}
@Article{ABBR12,
Title = {Algorithm 924: {TIDES}, a {Taylor} series integrator for differential equations},
Author = {Abad, A. and Barrio, R. and Blesa, F. and Rodr\'{\i}guez, M.},
Journal = {ACM Trans. Math. Softw.},
Year = {2012},
Pages = {5:1--5:28},
Volume = {39},
DOI = {10.1145/2382585.2382590},
Numpages = {28}
}
@Article{AbBrKe91,
author = {H. D. I. Abarbanel and R. Brown and M. B. Kennel},
title = {{Lyapunov} exponents in chaotic systems: their importance and their evaluation using observed data},
journal = {Int. J. Mod. Phys. B},
year = {1991},
volume = {5},
pages = {1347--1375}
}
@Article{AbBrKe91a,
Title = {Variation of {Lyapunov} exponents on a strange attractor},
Author = {H. D. I. Abarbanel and R. Brown and M. B. Kennel},
Journal = {J. Nonlin. Sci.},
Year = {1991},
Pages = {175--199},
Volume = {1}
}
@Book{ablowbook,
Title = {Solitions, Nonlinear Evolution Equations and Inverse Scattering},
Author = {M. J. Ablowitz and P. A. Clarkson},
Publisher = {Cambridge Univ. Press},
Year = {1992},
Address = {Cambridge}
}
@Article{Abraham95,
Title = {Structural similarities and differences among attractors and their intensity maps in the laser-{Lorenz} model},
Author = {N. B. Abraham and U. A. Allen and E. Peterson and A. Vicens and R. Vilaseca and V. Espinosa and G. L. Lippi},
Journal = {Optics Commun.},
Year = {1995},
Pages = {367--384},
Volume = {117},
Abstract = {Numerical studies of the laser-{Lorenz} model using parameters reasonably
accessible for recent experiments with a single mode homogeneously
broadened laser demonstrate that the form of the return map of successive
peak values of the intensity changes from a sharply cusped map in
resonance to a map with a smoothly rounded maximum as the laser is
detuned into the period doubling regime. This transformation appears
to be related to the disappearance (with detuning) of the heteroclinic
structural basis for the stable manifold which exists in resonance.
This is in contrast to the evidence reported by Tang and Weiss (Phys.
Rev. A 49 (1994) 1296) of a cusped map for both the period doubling
chaos found with detuning and the spiral chaos found in resonance
for seemingly Lorenz-like behavior of the far-infrared ammonia laser
from which it was concluded that there existed a ?unique chaotic
attractor for a single-mode laser?. }
}
@Article{Abramov09,
author = {R. V. Abramov},
title = {Short-time linear response with reduced-rank tangent map},
journal = {Chin. Ann. Math.},
year = {2009},
volume = {30},
pages = {447--462},
doi = {10.1007/s11401-009-0088-3}
}
@Article{Abramov10,
author = {R. V. Abramov},
title = {Approximate linear response for slow variables of dynamics with explicit time scale separation},
journal = {J. Comput. Phys.},
year = {2010},
volume = {229},
pages = {7739--7746},
doi = {10.1016/j.jcp.2010.06.029},
abstract = {Many real-world numerical models are notorious for the time
scale separation of different subsets of variables and the inclusion of
random processes. The existing algorithms of linear response to external
forcing are vulnerable to the time scale separation due to increased
response errors at fast scales. Here we develop the approximate linear
response algorithm for slow variables in a two-scale dynamical system
with explicit separation of slow and fast variables, which has improved
numerical stability and reduced computational expense. }
}
@Article{Abramov12,
author = {Abramov, R. V.},
title = {Improved linear response for stochastically driven systems},
journal = {Front. Math. China},
year = {2012},
volume = {7},
pages = {199--216},
doi = {10.1007/s11464-012-0192-7}
}
@Article{Abramov12a,
author = {Abramov, R. V.},
title = {A simple linear response closure approximation for slow dynamics of a multiscale system with linear coupling},
journal = {Multiscale Modeling \&\ Simulation},
year = {2012},
volume = {10},
pages = {28--47},
doi = {10.1137/110844696}
}
@Unpublished{Abramov14,
Title = {Linear response of the {Lyapunov} exponent to small external perturbations},
Author = {Abramov, R. V.},
Note = {\arXiv{1403.7116}},
Year = {2014}
}
@Unpublished{AbrKje13,
Title = {The response of reduced models of multiscale dynamics to small external perturbations},
Author = {{Abramov}, R. V. and {Kjerland}, M. P.},
Note = {\arXiv{1305.0862}},
Year = {2013}
}
@Article{AbrMaj08,
Title = {New approximations and tests of linear fluctuation-response for chaotic nonlinear forced-dissipative dynamical systems},
Author = {Abramov, R. V. and Majda, A. J.},
Journal = {J. Nonlinear Sci.},
Year = {2008},
Pages = {303--341},
Volume = {18},
Abstract = {We develop and test two novel computational approaches for predicting
the mean linear response of a chaotic dynamical system to small change
in external forcing via the fluctuation-dissipation theorem. Unlike
the earlier work in developing fluctuation-dissipation theorem-type
computational strategies for chaotic nonlinear systems with forcing
and dissipation, the new methods are based on the theory of Sinai-Ruelle-Bowen
probability measures, which commonly describe the equilibrium state
of such dynamical systems. The new methods take into account the
fact that the dynamics of chaotic nonlinear forced-dissipative systems
often reside on chaotic fractal attractors, where the classical quasi-Gaussian
formula of the fluctuation-dissipation theorem often fails to produce
satisfactory response prediction, especially in dynamical regimes
with weak and moderate degrees of chaos. A simple new low-dimensional
chaotic nonlinear forced-dissipative model is used to study the response
of both linear and nonlinear functions to small external forcing
in a range of dynamical regimes with an adjustable degree of chaos.
We demonstrate that the two new methods are remarkably superior to
the classical fluctuation-dissipation formula with quasi-Gaussian
approximation in weakly and moderately chaotic dynamical regimes,
for both linear and nonlinear response functions. One straightforward
algorithm gives excellent results for short-time response while the
other algorithm, based on systematic rational approximation, improves
the intermediate and long time response predictions.},
DOI = {10.1007/s00332-007-9011-9}
}
@Book{AbrMars78,
Title = {Foundations of Mechanics},
Author = {R. Abraham and J. E. Marsden},
Publisher = {Benjamin-Cummings},
Year = {1978},
Address = {Reading, Mass.}
}
@Book{AbSh92,
Title = {Dynamics - The Geometry of Behavior},
Author = {R. H. Abraham and C. D. Shaw},
Publisher = {Wesley},
Year = {1992},
Address = {Reading, MA}
}
@Article{AbSm70,
Title = {Nongenericity of {$\Omega$}-stability},
Author = {R. Abraham and S. Smale},
Journal = {Proc. Symp. Pure Math.},
Year = {1970},
Pages = {5--8},
Volume = {14}
}
@Article{ACFK09,
author = {H. D. I. Abarbanel and D. R. Creveling and R. Farsian and M. Kostuk},
title = {Dynamical state and parameter estimation},
journal = {SIAM J. Appl. Math.},
year = {2009},
volume = {8},
pages = {1341--1381}
}
@Article{ACHKW11,
author = {Willis, A. P. and Cvitanovi{\'c}, P. and Avila, M.},
title = {Revealing the state space of turbulent pipe flow by symmetry reduction},
journal = {J. Fluid Mech.},
year = {2013},
volume = {721},
pages = {514--540},
note = {\arXiv{1203.3701}},
doi = {10.1017/jfm.2013.75}
}
@Article{ACK89,
Title = {Phase transitions on strange irrational sets},
Author = {Artuso, R. and Cvitanovi\'{c}, P. and Kenny, B. G.},
Journal = {Phys. Rev. A},
Year = {1989},
Pages = {268--281},
Volume = {39},
DOI = {10.1103/PhysRevA.39.268}
}
@Book{Adams:2003wi,
Title = {Sobolev spaces},
Author = {Adams, R. A. and Fournier, J. J. F.},
Publisher = {Academic},
Year = {2003},
Address = {New York}
}
@Article{AdChDj12,
author = {Aderogba, A. A. and Chapwanya, M. and Djoko, J. K.},
title = {Travelling wave solution of the {Kuramoto-Sivashinsky} equation: A computational study},
journal = {AIP Conf. Proc.},
year = {2012},
volume = {1479},
pages = {777--780},
doi = {10.1063/1.4756252}
}
@Article{AEOGS10,
Title = {Observation of periodic orbits on curved two-dimensional geometries},
Author = {Avlund, M. and Ellegaard, C. and Oxborrow, M. and Guhr, T. and S\o{}ndergaard, N.},
Journal = {Phys. Rev. Lett.},
Year = {2010},
Pages = {164101},
Volume = {104},
DOI = {10.1103/PhysRevLett.104.164101},
Issue = {16}
}
@Article{afind,
Title = {Characterization of unstable periodic orbits in chaotic attractors and repellers},
Author = {O. Biham and W. Wenzel},
Journal = {Phys. Rev. Lett.},
Year = {1989},
Pages = {819},
Volume = {63}
}
@Article{AGHks89,
Title = {{Kuramoto-Sivashinsky} dynamics on the center-unstable manifold},
Author = {D. Armbruster and J. Guckenheimer and P. Holmes},
Journal = {SIAM J. Appl. Math.},
Year = {1989},
Pages = {676--691},
Volume = {49}
}
@Article{AGHO288,
Title = {Heteroclinic cycles and modulated travelling waves in systems with {O(2)} symmetry},
Author = {D. Armbruster and J. Guckenheimer and P. Holmes},
Journal = {Physica D},
Year = {1988},
Pages = {257--282},
Volume = {29}
}
@Article{AGIP90,
Title = {On the topology of the {H\'enon} map},
Author = {G. D'Alessandro, P. Grassberger, S. Isola and A. Politi},
Journal = {J. Phys. A},
Year = {1990},
Pages = {5285},
Volume = {23}
}
@Article{ahuja_template-based_2007,
Title = {Template-based stabilization of relative equilibria in systems with continuous symmetry},
Author = {S. Ahuja and {I.G.} Kevrekidis and {C.W.} Rowley},
Journal = {J. Nonlin. Sci.},
Year = {2007},
Pages = {109--143},
Volume = {17},
Abstract = {We present an approach to the design of feedback control laws that
stabilize relative equilibria of general nonlinear systems with continuous
symmetry. Using a template-based method, we factor out the dynamics
associated with the symmetry variables and obtain evolution equations
in a reduced frame that evolves in the symmetry direction. The relative
equilibria of the original systems are fixed points of these reduced
equations. Our controller design methodology is based on the linearization
of the reduced equations about such fixed points. We present two
different approaches of control design. The first approach assumes
that the closed loop system is affine in the control and that the
actuation is equivariant. We derive feedback laws for the reduced
system that minimize a quadratic cost function. The second approach
is more general; here the actuation need not be equivariant, but
the actuators can be translated in the symmetry direction. The controller
resulting from this approach leaves the dynamics associated with
the symmetry variable unchanged. Both approaches are simple to implement,
as they use standard tools available from linear control theory.
We illustrate the approaches on three examples: a rotationally invariant
planar {ODE,} an inverted pendulum on a cart, and the {Kuramoto-Sivashinsky}
equation with periodic boundary conditions.}
}
@Article{AKcgl02,
Title = {The world of the complex {Ginzburg-Landau} equation},
Author = {I. S. Aranson and L. Kramer},
Journal = {Rev. Mod. Phys.},
Year = {2002},
Pages = {99--143},
Volume = {74}
}
@Article{Akhmediev04,
Title = {Strongly asymmetric soliton explosions},
Author = {Akhmediev, N. and Soto-Crespo, J. M.},
Journal = {Phys. Rev. E},
Year = {2004},
Volume = {70}
}
@Article{AlIsPo91,
Title = {Geometric properties of the pruning front},
Author = {D'Alessandro, G. and Isola, S. and Politi, A.},
Journal = {Progr. Theor. Phys.},
Year = {1991},
Pages = {1149--1157},
Volume = {86},
Mrnumber = {93a:58052}
}
@Misc{AllBar12,
Title = {Chaotic dynamics in multidimensional transition states},
Author = {A. Allahem and T. Bartsch},
Year = {2012}
}
@Article{AllgGeorg88,
Title = {Numerically stable homotopy methods without an extra dimension},
Author = {E. L. Allgower and K. Georg},
Journal = {Lect. Appl. Math.},
Year = {1990},
Pages = {1--13},
Volume = {26},
URL = {http://www.math.colostate.edu/emeriti/georg/homExtra.pdf}
}
@Book{almeida88,
Title = {Hamiltonian systems: Chaos and Quantization},
Author = {Ozorio de Almeida, A. M.},
Publisher = {Cambridge Univ. Press},
Year = {1988},
Address = {Cambridge}
}
@Article{AltTel08,
Title = {Poincar\'e recurrences and transient chaos in systems with leaks},
Author = {Altmann, E. G. and T\'el, T.},
Journal = {Phys. Rev. E},
Year = {2009},
Note = {\arXiv{0808.3785}},
Pages = {016204},
Volume = {79},
Abstract = {Simulating observational and experimental situations, we consider
a leak in the phase space of a chaotic dynamical system. We obtain
an expression for the escape rate of the survival probability applying
the theory of transient chaos. This expression improves previous
estimates based on the properties of the closed system and explains
dependencies on the position and size of the leak and on the initial
ensemble. With a subtle choice of the initial ensemble, we obtain
an equivalence to the classical problem of Poincare recurrences in
closed systems, which is treated in the same framework. Finally,
we show how our results apply to weakly chaotic systems and justify
a split of the invariant saddle in hyperbolic and non-hyperbolic
components, related, respectively, to the intermediate exponential
and asymptotic power-law decays of the survival probability.},
Numpages = {12}
}
@Article{alvarez_monodromy_2005,
Title = {Monodromy and stability for nilpotent critical points},
Author = {M. Alvarez},
Journal = {Int. J. Bifur. Chaos},
Year = {2005},
Pages = {1253--1266},
Volume = {15},
ISSN = {0218-1274}
}
@Misc{Alves06,
author = {Alves, J. F.},
title = {Chaotic dynamics: physical measures and statistical features},
year = {2006},
url = {http://cmup.fc.up.pt/cmup/jfalves/CIM.pdf}
}
@Incollection{AlvSou13,
author = {Alves, J. F. and Soufi, M.},
title = {Statistical stability in chaotic dynamics},
booktitle = {Progress and Challenges in Dynamical Systems},
publisher = {Springer},
year = {2013},
editor = {Ib\'a\~nez, S. and P\'erez del Río, J. S. and Pumari\~no, A. and Rodríguez, J. \'A.},
volume = {54},
pages = {7--24},
address = {New York},
doi = {10.1007/978-3-642-38830-9_2}
}
@Article{AmLeAg06,
Title = {Piecewise affine models of chaotic attractors: {The R\"ossler} and {Lorenz} systems},
Author = {G. F. V. Amaral and C. Letellier and L. A. Aguirre},
Journal = {Chaos},
Year = {2006},
Pages = {013115},
Volume = {16},
DOI = {10.1063/1.2149527},
Numpages = {14}
}
@Incollection{ampatt,
Title = {Amplitude equations for pattern forming systems},
Author = {M. van Hecke and P. C. Hohenberg and van Saarloos, W.},
Booktitle = {Fundamental Problems in Statistical Mechanics},
Publisher = {North-Holland},
Year = {1994},
Address = {Amsterdam},
Editor = {H. van Beijeren and M. H. Ernst},
Volume = {VIII}
}
@Article{AmZaSa10,
author = {J. M. Amig{\'o} and S. Zambrano and M. A. F. Sanju{\'a}n},
title = {Permutation complexity of spatiotemporal dynamics},
journal = {Europhys. Lett.},
year = {2010},
volume = {90},
pages = {10007},
doi = {10.1209/0295-5075/90/10007}
}
@Book{AnAr88,
Title = {Dynamical Systems {I}: {Ordinary} Differential Equations and Smooth Dynamical Systems},
Author = {D. V. Anosov and V. I. Arnol'd},
Publisher = {Springer},
Year = {1988},
Address = {New York}
}
@Article{AnBoAi07,
Title = {Automatic control and tracking of periodic orbits in chaotic systems},
Author = {Ando, H. and Boccaletti, S. and Aihara, K.},
Journal = {Phys. Rev. E},
Year = {2007},
Pages = {066211},
Volume = {75}
}
@Article{angen88,
Title = {The periodic orbits of an area preserving twist-map},
Author = {S. B. Angenent},
Journal = {Commun. Math. Phys.},
Year = {1988},
Pages = {353--374},
Volume = {115}
}
@Article{anosov67,
Title = {Geodesic flows on compact {Riemannian} manifolds of negative curvature},
Author = {D. V. Anosov},
Journal = {Proc.\ Steklov.\ Inst.\ of Math.},
Year = {1967},
Volume = {90},
Annote = {His famous paper about Anosov systems?}
}
@Article{Arai07a,
Title = {On hyperbolic plateaus of the {H\'enon} map},
Author = {Arai, Z.},
Journal = {Experimental Math.},
Year = {2007},
Pages = {181--188},
Volume = {16},
Abstract = { We propose a rigorous computational method to prove the uniform hyperbolicity
of discrete dynamical systems. Applying the method to the real {H\'enon}
family, we prove the existence of many regions of hyperbolic parameters
in the parameter plane of the family. },
DOI = {10.1080/10586458.2007.10128992}
}
@Unpublished{Arai07b,
Title = {On loops in the hyperbolic locus of the complex {H\'enon} map and their monodromies},
Author = {Arai, Z.},
Note = {\arXiv{0704.2978}},
Year = {2007}
}
@Article{ArCho91,
Title = {Heteroclinic orbits in a spherically invariant system},
Author = {D. Armbruster and P. Chossat},
Journal = {Physica D},
Year = {1991},
Pages = {155--176},
Volume = {50},
Abstract = {The existence and stability of structurally stable heteroclinic cycles
are discussed in a codimension-2 bifurcation problem with O(3)-symmetry,
when the critical spherical modes l = 1 and l = 2 occur simultaneously.
Several types of heteroclinic cycles are found which may explain
aperiodic attractors found in numerical simulations for the onset
of convection in a self-gravitating fluid in a spherical shell.},
DOI = {10.1016/0167-2789(91)90173-7}
}
@Book{ArKoNe88,
Title = {Mathematical Aspects of Classical and Celestial Mechanics},
Author = {V. I. Arnol'd and V. V. Kozlov and A. I. Neishtadt},
Publisher = {Springer},
Year = {1988},
Address = {New York}
}
@Article{ArMi06,
author = {Arai, Z. and Mischaikow, K.},
title = {Rigorous computations of homoclinic tangencies},
journal = {SIAM J. Appl. Dyn. Syst.},
year = {2006},
volume = {5},
pages = {280--292},
doi = {10.1137/050626429}
}
@Book{arnold82,
Title = {Geometrical Methods in the Theory of Ordinary Differential Equations},
Author = {V. I. Arnolʹd},
Publisher = {Springer},
Year = {1982},
Address = {Berlin}
}
@Book{arnold89,
Title = {Mathematical Methods for Classical Mechanics},
Author = {V. I. Arnol'd},
Publisher = {Springer},
Year = {1989},
Address = {New York}
}
@Article{arnold91k,
Title = {Kolmogorov's hydrodynamic attractors},
Author = {V. I. Arnol'd},
Journal = {Proc. R. Soc. Lond. A},
Year = {1991},
Pages = {19--22},
Volume = {434},
Abstract = {A review of Kolmogorov's efforts relating the Navier-Stokes equation
to the theory of dynamical system. Several interesting questions
regarding the connection are exposed.}
}
@Book{arnold92,
Title = {Ordinary Differential Equations},
Author = {V. I. Arnol'd},
Publisher = {Springer},
Year = {1992},
Address = {New York}
}
@Article{art03int,
Title = {Cycle expansions for intermittent maps},
Author = {R. Artuso and P. Cvitanovi\'{c} and G. Tanner},
Journal = {Proc. Theo. Phys. Supp.},
Year = {2003},
Pages = {1--21},
Volume = {150}
}
@Article{art91,
author = {R. Artuso},
title = {Diffusive dynamics and periodic orbits of dynamic systems},
journal = {Phys. Lett. A},
year = {1991},
volume = {160},
pages = {528--530},
doi = {10.1016/0375-9601(91)91062-I}
}
@Article{ArtMaz65,
author = {Artin, M. and Mazur, B.},
title = {On periodic points},
journal = {Ann. Math.},
year = {1965},
volume = {81},
pages = {82--99},
url = {http://www.jstor.org/stable/1970384}
}
@Article{Artuso94,
author = {R. Artuso},
title = {Recycling deterministic diffusion},
journal = {Physica D},
year = {1994},
volume = {76},
pages = {1--7},
doi = {10.1016/0167-2789(94)90245-3}
}
@Book{ArWe05,
Title = {Mathematical Methods for Physicists: A Comprehensive Guide},
Author = {G. B. Arfken and H. J. Weber},
Publisher = {Academic},
Year = {2005},
Address = {New York},
Edition = {6}
}
@Article{AS87,
Title = {A study of hairpin vortices in a laminar boundary layer},
Author = {M. S. Acarlar and C. R. Smith},
Journal = {J. Fluid Mech.},
Year = {1987},
Pages = {1--41 and 45--83},
Volume = {175}
}
@Article{Asami06,
Title = {Statistical properties of periodic orbits in a 4-disk billiard system ---{The} pruning-proof property---},
Author = {Asamizuya, T.},
Journal = {Progr. Theor. Phys.},
Year = {2006},
Pages = {247--271},
Volume = {116},
Abstract = {We investigate the statistical properties of the actual periodic orbits
in a 4-disk billiard system in the context of pruning. For served
value of the system parameter, we numerically obtain approximately
1,000,000 periodic orbits that have no more than 20 collisions. We
also compute some statistical quantities of the resultant periodic
orbits. In these statistics, we observe a periodic peak structure
that survives pruning; that is, it possesses a "pruning-proof property".}
}
@Article{Ascher95,
Title = {Implicit-explicit methods for time-dependent partial differential equations},
Author = {U. Ascher and S. Ruuth and B. Wetton},
Journal = {SIAM J. Numer. Anal.},
Year = {1995},
Month = jun,
Number = {3},
Pages = {797--823},
Volume = {32}
}
@Article{AschKnauf97,
author = {Asch, J. and Knauf, A.},
title = {Motion in periodic potentials},
journal = {Nonlinearity},
year = {1998},
volume = {11},
pages = {175–200},
addendum = {\arXiv{cond-mat/9710169}},
doi = {10.1088/0951-7715/11/1/011},
abstract = {We consider motion in a periodic potential in a classical, quantum, and semiclassical context. Various results on the distribution of asymptotic velocities are proven.}
}
@Article{AshBoMe96,
Title = {Forced symmetry breaking of homoclinic cycles in a {PDE} with {O(2)} symmetry},
Author = {Ashwin, P. and B\"ohmer, K. and Mei, Z.},
Journal = {J. Comput. Appl. Math.},
Year = {1996},
Pages = {297--310},
Volume = {70},
URL = {http://wrap.warwick.ac.uk/18622/}
}
@Article{AshDang05,
Title = {Reduced dynamics and symmetric solutions for globally coupled weakly dissipative oscillators},
Author = {Ashwin, P. and Dangelmayr, G.},
Journal = {Dyn. Sys.},
Year = {2005},
Pages = {333--367},
Volume = {20}
}
@Article{AshMe97,
Title = {Noncompact drift for relative equilibria and relative periodic orbits},
Author = {P. Ashwin and I. Melbourne},
Journal = {Nonlinearity},
Year = {1997},
Pages = {595},
Volume = {10}
}
@Article{AshMeNi99,
Title = {Drift bifurcations of relative equilibria and transitions of spiral waves},
Author = {P. Ashwin and I. Melbourne and M. Nicol},
Journal = {Nonlinearity},
Year = {1999},
Pages = {741},
Volume = {12},
Abstract = {We consider dynamical systems that are equivariant under a noncompact
Lie group of symmetries and the drift of relative equilibria in such
systems. In particular, we investigate how the drift for a parametrized
family of normally hyperbolic relative equilibria can change character
at what we call a `drift bifurcation'. To do this, we use results
of Arnold to analyse parametrized families of elements in the Lie
algebra of the symmetry group. We examine effects in physical space
of such drift bifurcations for planar reaction-diffusion systems
and note that these effects can explain certain aspects of the transition
from rigidly rotating spirals to rigidly propagating `retracting
waves'. This is a bifurcation observed in numerical simulations of
excitable media where the rotation rate of a family of spirals slows
down and gives way to a semi-infinite translating wavefront.}
}
@Article{ASTIK09,
author = {Asakura, J. and Sakurai, T. and Tadano, H. and Ikegami, T. and Kimura, K.},
title = {A numerical method for nonlinear eigenvalue problems using contour integrals},
journal = {JSIAM Lett.},
year = {2009},
volume = {1},
pages = {52--55},
doi = {10.14495/jsiaml.1.52}
}
@Article{AstMelb06,
author = {P. Aston and I. Melbourne},
title = {{Lyapunov} exponents of symmetric attractors},
journal = {Nonlinearity},
year = {2006},
volume = {19},
pages = {2455}
}
@Article{Astor10,
author = {Astorino, M.},
title = {Kauffman knot invariant from {$\mathrm{SO}(N)$ or $\mathrm{Sp}(N)$ Chern-Simons} theory and the {Potts} model},
journal = {Phys. Rev. D},
year = {2010},
volume = {81},
pages = {125026},
doi = {10.1103/PhysRevD.81.125026}
}
@Article{AtJaJo10,
author = {F. M. Atay and S. Jalan and J. Jost},
title = {Symbolic dynamics and synchronization of coupled map networks with multiple delays},
journal = {Phys. Lett. A },
year = {2010},
volume = {375},
pages = {130--135},
doi = {10.1016/j.physleta.2010.10.044}
}
@Article{atlas12,
Title = {Cartography of high-dimensional flows: {A} visual guide to sections and slices},
Author = {Cvitanovi\'c, P. and Borrero-Echeverry, D. and Carroll, K. and Robbins, B. and Siminos, E.},
Journal = {Chaos},
Year = {2012},
Pages = {047506},
Volume = {22},
DOI = {10.1063/1.4758309}
}
@Article{AuAb90,
Title = {Chaotic trajectories in the standard map. {The} concept of anti-integrability},
Author = {Aubry, S. and Abramovici, G.},
Journal = {Physica D},
Year = {1990},
Pages = {199--219},
Volume = {43},
Abstract = {A rigorous proof is given in the standard map for the existence of
chaotic trajectories with unbounded momenta for large enough coupling
constant k. The obtained chaotic trajectories correspond to stationary
configurations of the Frenkel-Kontorowa model with a finite (non-zero)
photon gap (called gap parameter in dimensionless units). This property
implies that the trajectory (or the configuration {ui}) can be uniquely
continued as a uniformly continuous function of the model parameter
k in some neighborhood of the initial configuration. A non-zero gap
parameter implies that the Lyapunov coefficient is strictly positive
(when it is defined). In addition, the existence of dilating and
contracting manifolds is proven for these chaotic trajectories. ``Exotic''?
trajectories such as ballistic trajectories are also proven to exist
as a consequence of these theorems. The concept of anti-integrability
emerges from these theorems. In the anti-integrable limit which can
be only defined for a discrete time dynamical system, the coordinates
of the trajectory at time i do not depend on the coordinates at time
i - 1. Thus, at this singular limit, the existence of chaotic trajectories
is trivial and the dynamical system reduces to a Bernoulli shift.
It appears that the chaotic trajectories of dynamical systems originate
by continuity from those which exists at the anti-integrable limits.},
DOI = {10.1016/0167-2789(90)90133-A}
}
@Article{aub95ant,
Title = {Anti-integrability in dynamical and variational problems},
Author = {S. Aubry},
Journal = {Physica D},
Year = {1995},
Pages = {284--296},
Volume = {86}
}
@Article{Aubry88,
Title = {The dynamics of coherent structures in the wall region of turbulent boundary layer},
Author = {N. Aubry and P. Holmes and J. L. Lumley and E. Stone},
Journal = {J. Fluid Mech.},
Year = {1988},
Pages = {115--173},
Volume = {192}
}
@Article{AuDae83,
Title = {The discrete {Frenkel-Kontorova} model and its extensions. {I}. {Exact} results for the ground-states},
Author = {Aubry, S. and Le Daeron, P. Y.},
Journal = {Phys. D},
Year = {1983},
Pages = {381--422},
Volume = {8},
Annote = {Original paper about Aubry-Mather sets}
}
@Unpublished{AuKrTr13,
Title = {Numerical algorithms based on analytic function values at roots of unity},
Author = {Austin, A. P and Kravanja, P. and Trefethen, L. N.},
Note = {SIAM J. Numer. Anal., to appear.},
Year = {2013}
}
@Book{ausloos_logistic_2005,
Title = {The Logistic Map and the Route to Chaos: From the Beginnings to Modern Applications},
Author = {M. Ausloos and M. Dirickx},
Publisher = {Springer},
Year = {2005},
Address = {New York}
}
@Manual{auto,
Title = {AUTO 97: Continuation and Bifurcation Software for Ordinary Differential Equations (with {HomCont})},
Author = {Doedel, E. J. and Champneys, A. R. and Fairgrieve, T. F. and Kuznetsov, Y. A. and Sandstede, B. and Wang, X.},
Year = {1998}
}
@Incollection{autod89,
Title = {On Automatic Differentiation},
Author = {A. Griewank},
Booktitle = {Mathematical Programming: Recent Developments and Applications},
Publisher = {Kluwer},
Year = {1989},
Address = {Dordrecht},
Editor = {M. Iri and K. Tanabe},
Pages = {83--108}
}
@Article{ave99,
Title = {Average patterns of spatiotemporal chaos: {A} boundary effect},
Author = {V. M. Egu\'{i}luz and P. Alstr{\o}m and E. Hern\'{a}ndez-Garc\'{\i}a and O. Piro},
Journal = {Phys. Rev. E},
Year = {1999},
Number = {3},
Pages = {2822},
Volume = {59}
}
@Incollection{aver_dasbuch,
Author = {P. Cvitanovi\'{c}},
Booktitle = {{Chaos: Classical and Quantum}},
Publisher = {Niels Bohr Inst.},
Year = {2015},
Address = {Copenhagen},
Chapter = {{Averaging}},
URL = {http://ChaosBook.org/paper.shtml#average}
}
@Article{AvMeRoHo13,
Title = {Streamwise-localized solutions at the onset of turbulence in pipe flow},
Author = {M. Avila and F. Mellibovsky and N. Roland and B. Hof},
Journal = {Phys. Rev. Lett.},
Year = {2013},
Pages = {224502},
Volume = {110},
DOI = {10.1103/PhysRevLett.110.224502}
}
@Unpublished{Axen11,
Title = {Non {Hamiltonian} chaos from {Nambu} dynamics of surfaces},
Author = {Axenides, M.},
Note = {\arXiv{1109.0470}},
Year = {2011}
}
@Article{AxFl08,
author = {Axenides, M. and Floratos, E.},
title = {{Nambu-Lie} 3-algebras on fuzzy 3\mbox{-}{m}anifolds},
journal = {JHEP},
year = {2009},
volume = {902},
pages = {39},
note = {\arXiv{0809.3493}},
doi = {10.1088/1126-6708/2009/02/039}
}
@Article{AxFl09,
Title = {Strange attractors in dissipative {Nambu} mechanics: {Classical} and quantum aspects},
Author = {Axenides, M. and Floratos, E.},
Journal = {JHEP},
Year = {2010},
Note = {\arXiv{0910.3881}},
Pages = {036},
Volume = {1004},
DOI = {10.1007/JHEP04(2010)036}
}
@Article{AxFlNi09,
Title = {{Nambu} quantum mechanics on discrete 3-tori},
Author = {Axenides, M. and Floratos, E.G. and Nicolis, S.},
Journal = {J. Phys. A},
Year = {2009},
Note = {\arXiv{0901.2638}},
Pages = {275201},
Volume = {42},
DOI = {10.1088/1751-8113/42/27/275201}
}
@Article{B84,
Title = {The evolution of disturbances in shear flows at high {Reynolds} numbers},
Author = {D. J. Benney},
Journal = {Stud. Appl. Math.},
Year = {1984},
Pages = {1--19},
Volume = {70}
}
@Article{BaBaBo12,
author = {D.H. Bailey and R. Barrio and J.M. Borwein},
title = {High-precision computation: {Mathematical} physics and dynamics},
journal = {Appl. Math. Computation },
year = {2012},
volume = {218},
pages = {10106--10121},
doi = {10.1016/j.amc.2012.03.087},
abstract = {At the present time, \{IEEE\} 64-bit floating-point
arithmetic is sufficiently accurate for most scientific applications.
However, for a rapidly growing body of important scientific computing
applications, a higher level of numeric precision is required. Such
calculations are facilitated by high-precision software packages that
include high-level language translation modules to minimize the
conversion effort. This paper presents an overview of recent applications
of these techniques and provides some analysis of their numerical
requirements. We conclude that high-precision arithmetic facilities are
now an indispensable component of a modern large-scale scientific
computing environment. }
}
@Article{BaCseGaHa08,
author = {B. B\'anhelyi and T. Csendes and B. M. Garay and L. Hatvani},
title = {A computer-assisted proof of {$\Sigma_3$}-chaos in the forced damped pendulum equation},
journal = {SIAM J. Appl. Dyn. Syst.},
year = {2008},
volume = {7},
pages = {843--867},
doi = {10.1137/070695599}
}
@Article{BaEvCo93,
author = {Baranyai, A. and Evans, D. J. and Cohen, E. G. D.},
title = {Field-dependent conductivity and diffusion in a two-dimensional {Lorentz} gas},
journal = {J. Stat. Phys.},
year = {1993},
volume = {70},
pages = {1085--1098},
doi = {10.1007/BF01049423}
}
@Unpublished{Bagheri12,
Title = {Spectral analysis of nonlinear flows},
Author = {Bagheri, S.},
Note = {Early version of \refref{Bagheri13}.},
Year = {2012}
}
@Unpublished{Bagheri13,
Title = {Spectral analysis of nonlinear flows},
Author = {Bagheri, S.},
Note = {Submitted to J. Fluid Mech.},
Year = {2013}
}
@Article{Bai06,
Title = {Multifractal analysis of the spectral measure of the {Thue-Morse} sequence: a periodic orbit approach},
Author = {Bai, Z.-Q.},
Journal = {J. Phys. A},
Year = {2006},
Pages = {10959},
Volume = {39},
Abstract = {The Fourier spectral density of the {Thue-Morse} sequence is reinterpreted
as the invariant measure of a stochastic dynamical system. Based
on this fact, its generalized (Renyi) dimension and $f(\alphs)$ statistics
are calculated with high precision by cycle expansions of spectral
determinant and dynamical zeta function. ? q at integer values of
q are also computed in an operator scheme and the asymptotic result
in the large q limit is derived.},
DOI = {10.1088/0305-4470/39/35/002}
}
@Article{Bai07,
Title = {On the cycle expansion for the {Lyapunov} exponent of a product of random matrices},
Author = {Bai, Z.-Q.},
Journal = {J. Phys. A},
Year = {2007},
Pages = {8315},
Volume = {40},
Abstract = {The cycle expansion of the thermodynamical zeta function for the Lyapunov
exponent of a product of random matrices typically converges exponentially
with the maximal cycle length (Mainieri 1992 Phys. Rev. Lett. 68
1965). In this paper we show that the convergent exponents are given
by the spectrum of a properly defined evolution operator, which describes
how a steady distribution of vector direction is established under
the action of random matrices. The exponential decay terms are automatically
eliminated in the cycle expansion of the spectral determinant, which
greatly accelerates the convergence provided all matrix elements
are positive numbers. As a marginal case, the random Fibonacci series
is studied in detail, and it is shown that this method is helpful.},
DOI = {10.1088/1751-8113/40/29/008}
}
@Article{Bai09,
Title = {An infinite transfer matrix approach to the product of random {$2\times2$} positive matrices},
Author = {Bai, Z.-Q.},
Journal = {J. Phys. A},
Year = {2009},
Pages = {015003},
Volume = {42},
Abstract = {... efficient and precise determination of the Laypunov exponent (and
other statistical properties) of a product of random $2\times 2$
matrices. By considering the ensemble average of an infinite series
of regular functions and its iteration, we construct a transfer matrix,
which is shown to be a trace class operator in a {Hilbert} space given
that the positiveness of the random matrices is assumed. This fact
gives a theoretical explanation of the superior convergence of the
cycle expansion of the Lyapunov exponent (Bai 2007 J. Phys. A: Math.
Theor. 40 8315). A numerical method based on the infinite transfer
matrix is applied to a one-dimensional Ising model with a random
field and a generalized Fibonacci sequence. It is found that, in
the presence of continuous distribution of a disorder or degenerated
random matrix, the transfer matrix approach is more efficient than
the cycle expansion method.},
DOI = {10.1088/1751-8113/42/1/015003}
}
@Article{Bai11,
Title = {A transfer operator approach to random {Fibonacci} sequences},
Author = {Bai, Z.-Q.},
Journal = {J. Phys. A},
Year = {2011},
Pages = {115002},
Volume = {44},
DOI = {10.1088/1751-8113/44/11/115002}
}
@Misc{BaJePo13,
Title = {Periodic points, escape rates and escape measures},
Author = {O. F. Bandtlow and O. Jenkinson and M. Pollicott},
Year = {2013},
Abstract = {For piecewise real analytic expanding Markov maps, with Markov hole,
it is shown that the escape rate and corresponding escape measure
can be rapidly approximated using periodic points.},
URL = {homepages.warwick.ac.uk/~masdbl/periodic-escape6.pdf}
}
@Article{BakasovAbraham93,
Title = {Laser second threshold: {Its} exact analytical dependence on detuning and relaxation rates},
Author = {Bakasov, A. A. and Abraham, N. B.},
Journal = {Phys. Rev. A},
Year = {1993},
Pages = {1633--1660},
Volume = {48}
}
@Article{BaKnTu90,
Title = {Spiral wave dynamics in a simple model of excitable media: {Transition} from simple to compound rotation},
Author = {D. Barkley and M. Kness and L. S. Tuckerman},
Journal = {Phys. Rev. A},
Year = {1990},
Pages = {2489--2492},
Volume = {42}
}
@Book{Baladi00,
Title = {Positive transfer operators and decay of correlations},
Author = {Baladi, V.},
Publisher = {World Scientific},
Year = {2000},
Address = {Singapore}
}
@Article{Baladi07,
author = {Baladi, V.},
title = {On the susceptibility function of piecewise expanding interval maps},
journal = {Comm. Math. Phys.},
year = {2007},
volume = {275},
pages = {839--859},
doi = {10.1007/s00220-007-0320-5}
}
@Article{Baladi08,
author = {V. Baladi},
title = {Linear response despite critical points},
journal = {Nonlinearity},
year = {2008},
volume = {21},
pages = {81--90},
doi = {10.1088/0951-7715/21/6/T01},
abstract = {Recent results and open problems about the differentiability
(or lack thereof) of SRB measures as functions of the dynamics are
discussed in this paper.}
}
@Unpublished{Baladi14,
Title = {Linear response, or else},
Author = {V. Baladi},
Note = {\arXiv{1408.2937}},
Year = {2014}
}
@Article{Baladi98,
Title = {Periodic orbits and dynamical spectra},
Author = {Baladi, V.},
Journal = {Ergod. Theor. Dynam. Syst.},
Year = {1998},
Pages = {255--292},
Volume = {18}
}
@Article{Baldwin88,
author = {Baldwin, P. R.},
title = {Soft billiard systems},
journal = {Physica D},
year = {1988},
volume = {29},
pages = {321--342},
doi = {10.1016/0167-2789(88)90034-6}
}
@Article{BALL92,
Title = {Existence of stable periodic orbits in the $x^2 y^2$ potential: a semiclassical approach},
Author = {D. Biswas and M. Azam and Q. V. Lawande and S. V Lawande},
Journal = {J. Phys. A},
Year = {1992},
Pages = {L297},
Volume = {25},
Abstract = {... the semiclassical periodic orbit theory to identify the recently
discovered one-parameter family of stable periodic orbits in the
$x^2 y^2$ potential occupying an area of 0.005\% on the surface of
section. They also indicate the presence of another stable family
of periodic orbits of higher length. The sensitivity of the method
provides hope for ruling out ergodicity in other systems.},
DOI = {10.1088/0305-4470/25/7/003}
}
@Article{BalSma08,
Title = {Linear response formula for piecewise expanding unimodal maps},
Author = {V. Baladi and D. Smania},
Journal = {Nonlinearity},
Year = {2008},
Note = {\arXiv{0705.3383}},
Pages = {677},
Volume = {21},
DOI = {10.1088/0951-7715/21/4/003}
}
@Unpublished{BalTod15,
Title = {Linear response for intermittent maps},
Author = {{Baladi}, V. and {Todd}, M.},
Note = {\arXiv{1508.02700}},
Year = {2015}
}
@Misc{BaMaReSe11,
Title = {Ensemble dynamics and bred vectors},
Author = {{Balci}, N. and {Mazzucato}, A. L. and {Restrepo}, J. M. and {Sell}, G. R.},
Note = {\arXiv{1108.4918}},
Year = {2011}
}
@Article{Bandelow1998,
Title = {Frequency regions for forced locking of self-pulsating multi-section {DFB} lasers},
Author = {U. Bandelow, L. Recke and B. Sandstede},
Journal = {Opt. Commun.},
Year = {1998},
Pages = {212--218},
Volume = {147}
}
@Article{baranger88,
author = {M. Baranger and K. T. R. Davies and J. H. Mahoney},
title = {The calculation of periodic trajectories},
journal = {Ann. Phys.},
year = {1988},
volume = {186},
pages = {95--110},
doi = {10.1016/S0003-4916(88)80018-6}
}
@Article{BarBle09,
author = {R. Barrio and F. Blesa},
title = {Systematic search of symmetric periodic orbits in {2DOF Hamiltonian} systems},
journal = {Chaos Solit. Fract.},
year = {2009},
volume = {41},
pages = {560--582},
doi = {10.1016/j.chaos.2008.02.032},
abstract = {study in detail the grid search numerical method to locate
symmetric periodic orbits in Hamiltonian systems of two degrees of
freedom. The method is based on the classical search method but combining
up-to-date numerical algorithms in the search and in the integration
process. Instead of using Newton methods that requires to differentiate
the Poincar\' map we use the Brent?s method and in the integration process
a Taylor series method that permits us to compute the orbits using
extended precision, something highly interesting in the case of unstable
periodic orbits. These facts have permitted us to obtain many more
periodic orbits than other researchers. Once the families of periodic
orbits have been found we study the bifurcations just by comparing with
the stability index and the classical generic bifurcations for
Hamiltonian systems with and without symmetries. We illustrate the method
with four important classical Hamiltonian problems. }
}
@Article{BarChor98,
Title = {New perspectives in turbulence: {Scaling} laws, asymptotics, and intermittency},
Author = {G. I. Barenblatt and A. J. Chorin},
Journal = {SIAM Review},
Year = {1998},
Pages = {265--291},
Volume = {40}
}
@Article{BarKev94,
Title = {A dynamical systems approach to spiral wave dynamics},
Author = {Barkley, D. and Kevrekidis, I. G.},
Journal = {Chaos},
Year = {1994},
Pages = {453--460},
Volume = {4},
DOI = {10.1063/1.166023}
}
@Article{Barkley92,
author = {Barkley, D.},
title = {Linear stability analysis of rotating spiral waves in excitable media},
journal = {Phys. Rev. Lett.},
year = {1992},
volume = {68},
pages = {2090--2093},
month = mar,
doi = {10.1103/PhysRevLett.68.2090}
}
@Article{Barkley94,
author = {Barkley, D.},
title = {{Euclid}ean symmetry and the dynamics of rotating spiral waves},
journal = {Phys. Rev. Lett.},
year = {1994},
volume = {72},
pages = {164--167}
}
@Article{Barreira15,
author = {L. Barreira},
title = {Dimension theory of flows: {A} survey},
journal = {Discrete Contin. Dyn. Syst.},
year = {2015},
volume = {20},
pages = {3345--3362},
doi = {10.3934/dcdsb.2015.20.3345},
abstract = {This is a survey on recent developments of the
dimension theory of flows, with emphasis on hyperbolic flows. In
particular, we describe various results of the dimension theory and
multifractal analysis of flows, including the dimension of
hyperbolic sets, the dimension of invariant measures, the
multifractal analysis of equilibrium measures, conditional
variational principles, multidimensional spectra, and dimension
spectra taking both into account past and future. The dimension
theory and the multifractal analysis of dynamical systems grew out
exponentially during the last two decades, but for various reasons
flows have initially been given less attention than maps.}
}
@Article{BarRod14,
author = {R. Barrio and M. Rodr{\'\i}guez},
title = {Systematic computer assisted proofs of periodic orbits of {Hamiltonian systems}},
journal = {Commun. Nonlinear Sci. Numer. Simul.},
year = {2014},
doi = {10.1016/j.cnsns.2013.12.025},
abstract = {The numerical study of Dynamical Systems leads to obtain
invariant objects of the systems such as periodic orbits, invariant tori,
attractors and so on, that helps to the global understanding of the
problem. In this paper we focus on the rigorous computation of periodic
orbits and their distribution on the phase space, which configures the so
called skeleton of the system. We use Computer Assisted Proof techniques
to make a rigorous proof of the existence and the stability of families
of periodic orbits in two-degrees of freedom Hamiltonian systems, which
provide rigorous skeletons of periodic orbits. To that goal we show how
to prove the existence and stability of a huge set of discrete initial
conditions of periodic orbits, and later, how to prove the existence and
stability of continuous families of periodic orbits. We illustrate the
approach with two paradigmatic problems: the H{\'{e}}non-Heiles Hamiltonian and
the Diamagnetic Kepler problem. }
}
@Incollection{Basu07,
Author = {A. Basu},
Booktitle = {{ChaosBook.org/projects}},
Publisher = {Georgia Inst. of Technology},
Year = {2007},
Chapter = {Construction of {Poincar\'e} return maps for {R\"ossler} flow},
URL = {http://ChaosBook.org/projects/index.shtml#Basu}
}
@Article{BaTo03,
author = {B\'alint, P. and T\'oth, I. P.},
title = {Correlation decay in certain soft billiards},
journal = {Comm. Math. Phys.},
year = {2003},
volume = {243},
pages = {55--91},
doi = {10.1007/s00220-003-0954-x}
}
@Article{BaTo04,
author = {B\'alint, P. and T\'oth, I. P.},
title = {Mixing and its rate in `soft' and `hard' billiards motivated by the {Lorentz} process},
journal = {Physica D},
year = {2004},
volume = {187},
pages = {128--135},
doi = {10.1016/j.physd.2003.09.004}
}
@Article{bauer1995,
Title = {Relativistic Ponderomotive Force, Uphill Acceleration, and Transition to Chaos},
Author = {Bauer, D. and Mulser, P. and Steeb, W. -H.},
Journal = {Phys. Rev. Lett.},
Year = {1995},
Pages = {4622--4625},
Volume = {75},
Abstract = {Starting from a covariant cycle-averaged Lagrangian the relativistic
oscillation center equation of motion of a point charge is deduced,
and analytical formulas for the ponderomotive force in a traveling
wave of arbitrary strength are presented. It is further shown that
the pondermotive forces for transverse and longitudinal waves are
different; in the latter, uphill acceleration can occur. In a standing
wave there exists a threshold intensity above which, owing to transition
to chaos, the secular motion can no longer be described by a regular
ponderomotive force.},
DOI = {10.1103/PhysRevLett.75.4622}
}
@Article{bauer2007,
Title = {Vacuum heating versus skin layer absorption of intense femtosecond laser pulses},
Author = {Bauer, D. and Mulser, P.},
Journal = {Phys. Plasmas},
Year = {2007},
Pages = {023301--023301-11},
Volume = {14},
Abstract = {The crossing of the narrow skin layer in solid targets by electrons
in a time shorter than a laser cycle represents one of the numerous
collisionless absorption mechanisms of intense laser-matter interaction.
This kinetic effect is studied at normal and oblique laser beam incidence
and particle injection by a test particle approach in an energy interval
extending into the relativistic domain. Three main results obtained
are the strong dependence of the energy gain by the single particle
on the instant of injection relative to the phase of the light wave,
the reflection of the particles primarily contributing to absorption
well in front of the target rather than in the Debye layer, and the
low degree of absorption hardly exceeding the 10\% limit. The simulation
results offer a more unambiguous interpretation of the absorption
mechanism often referred to as ?vacuum heating.? In particular, it
is clearly revealed that the absorption in the vacuum region prevails
on that originating from the skin layer. Relativistic ponderomotive
effects are also tested, however their contribution to absorption
is not significant.},
DOI = {10.1063/1.2435326}
}
@Article{BaVo86,
Title = {Chaos on the pseudosphere},
Author = {N. Balasz and A. Voros},
Journal = {Phys. Rep.},
Year = {1986},
Pages = {109--240},
Volume = {143}
}
@Article{BayMor88,
author = {D. Bayer and I. Morrison},
title = {Standard bases and geometric invariant theory {I. Initial} ideals and state polytopes},
journal = {J. Symbolic Comp.},
year = {1988},
volume = {6},
pages = {209--217},
doi = {10.1016/S0747-7171(88)80043-9}
}
@InProceedings{BayOrt04,
author = {E. Bayro-Corrochano and J. Ortegon-Aguilar},
title = {Lie algebra template tracking},
booktitle = {International Conf. on Pattern Recognition},
year = {2004},
volume = {2},
pages = {56--59},
doi = {10.1109/ICPR.2004.1334036}
}
@Article{BaySti92,
author = {D. Bayer and M. Stillman},
title = {Computation of {Hilbert} functions},
journal = {J. Symbolic Comp.},
year = {1992},
volume = {14},
pages = {31--50},
doi = {10.1016/0747-7171(92)90024-X}
}
@Article{BBLTFC11,
author = {Boghosian, B. M. and Brown, Aa. and L\"att, J. and Tang, H. and Fazendeiro, L. M. and Coveney, P. V.},
title = {Unstable periodic orbits in the {Lorenz} attractor},
journal = {Philos. Trans. Royal Soc. A},
year = {2011},
volume = {369},
pages = {2345--2353}
}
@Article{BC89,
Title = {A mean flow first harmonic theory for hydrodynamic instabilities},
Author = {D. J. Benney and K. A. Chow},
Journal = {Stud. Appl. Math.},
Year = {1989},
Pages = {37--73},
Volume = {80}
}
@Article{BCFLR08,
Title = {Abundance of regular orbits and out-of-equilibrium phase transitions in the thermodynamic limit for long-range systems},
Author = {R. Bachelard and C. Chandre and D. Fanelli and X. Leoncini and S. Ruffo},
Journal = {Phys. Rev. Lett.},
Year = {2008},
Pages = {260603},
Volume = {101}
}
@Article{BCISV93,
Title = {Advection of vector fields by chaotic flows},
Author = {Balmforth, N. J. and Cvitanovi{\'c}, P. and Ierley, G. R. and Spiegel, E. A. and Vattay, G.},
Journal = {Ann. New York Acad. Sci.},
Year = {1993},
Note = {Volume entitled {\em Stochastic Processes in Astrophysics}},
Pages = {148--160},
Volume = {706}
}
@Article{BCOR11,
Title = {Bifurcations in the {Lozi} map},
Author = {Botella-Soler, V. and Castelo, J. M. and Oteo, J. A. and Ros, J.},
Journal = {J. Phys. A},
Year = {2011},
Note = {\arXiv{1102.0034}},
Pages = {D5101},
Volume = {44}
}
@Misc{BCS12,
Title = {2-dimensional {Landau} turbulence sliced},
Author = {Borrero-Echeverry, D. and Schatz, M. and Cvitanovi\'c, P.},
Note = {In preparation},
Year = {2012}
}
@Article{BeBoRa13,
author = {Benzi, M. and Boito, P. and Razouk, N.},
title = {Decay properties of spectral projectors with applications to electronic structure},
journal = {SIAM Review},
year = {2013},
volume = {55},
pages = {3--64},
doi = {10.1137/100814019}
}
@Article{BeCa80,
Title = {Phase transitions on strange sets: the {Ising} quasicrystal},
Author = {V. G. Benza and V. Callegaro},
Journal = {J. Phys. A},
Year = {1990},
Pages = {L841},
Volume = {23},
Abstract = {A quantum Ising spin chain with nearest-neighbour couplings arranged
in a quasiperiodic sequence is considered. The Cantor set structure
of the energy spectrum is analysed in terms of the thermodynamic
description of multifractals. Evidence is given that the spectrum
of scales develops a singular behaviour: this is associated with
a first-order phase transition of a new type. It is argued that this
effect involves, not only quantum spins, but the whole class of phonon-like
propagation problems on quasiperiodic chains.}
}
@Article{BeKoSt03,
Title = {On self-propulsion of micro-machines at low {Reynolds} number: {Purcell's} three-link swimmer},
Author = {Becker, L. E. and Koehler, S. A. and Stone, H. A.},
Journal = {J. Fluid Mech.},
Year = {2003},
Pages = {15--35},
Volume = {490}
}
@Article{Belot03,
author = {G. Belot},
title = {Symmetry and gauge freedom},
journal = {Studies in History and Philosophy of Modern Physics},
year = {2003},
volume = {34},
pages = {189--225},
doi = {10.1016/S1355-2198(03)00004-2},
abstract = {The classical field theories that underlie the quantum
treatments of the electromagnetic, weak, and strong forces share a
peculiar feature: specifying the initial state of the field determines
the evolution of some degrees of freedom of the theory while leaving the
evolution of some others wholly arbitrary. This strongly suggests that
some of the variables of the standard state space lack physical
content?intuitively, the space of states of such a theory is of higher
dimension than the corresponding space of genuine physical possibilities.
The structure of such theories can helpfully be characterized in terms of
the action of symmetry groups on their space of states; and the
conceptual problems surrounding their strange behavior can be sharpened
in light of the observation that it is usually possible to eliminate the
redundant variables associated with these symmetries?which turn out to be
precisely those variables whose evolution is unconstrained by the
dynamical laws of the theory. This paper discusses this approach, uses it
to frame questions about the interpretation of classical gauge theories,
and to reflect (pessimistically) on our prospects of reaching
satisfactory answers to these questions. }
}
@Article{bene80a,
author = {Benettin, G. and Galgani, L. and Giorgilli, A. and Strelcyn, J. M.},
title = {{Lyapunov} characteristic exponents for smooth dynamical systems; a method for computing all of them. {P}art 1: theory},
journal = {Meccanica},
year = {1980},
volume = {15},
pages = {9--20}
}
@Article{bene80b,
author = {Benettin, G. and Galgani, L. and Giogilli, A. and Strelcyn, J. M.},
title = {{Lyapunov} characteristic exponents for smooth dynamical systems; a method for computing all of them. {P}art 2: numerical application},
journal = {Meccanica},
year = {1980},
volume = {15},
pages = {21--30}
}
@Article{benisti2010,
Title = {Self-Organization and Threshold of Stimulated {Raman} Scattering},
Author = {B\'enisti, D. and Morice, O. and Gremillet, L. and Siminos, E. and Strozzi, D. J.},
Journal = {Phys. Rev. Lett.},
Year = {2010},
Pages = {015001},
Volume = {105},
Abstract = {We derive, both theoretically and using an envelope code, threshold
intensities for stimulated Raman scattering, which compare well with
results from Vlasov simulations. To do so, we account for the nonlinear
decrease of Landau damping and for the detuning induced by both the
nonlinear wave number shift ?kp and the frequency shift ??p of the
plasma wave. In particular, we show that the effect of ?kp may cancel
out that of ??p, but only in that plasma region where the laser intensity
decreases along the direction of propagation of the scattered wave.
Elsewhere, ?kp enhances the detuning effect of ??p.},
DOI = {10.1103/PhysRevLett.105.015001}
}
@Article{benisti2010-1,
Title = {Nonlinear group velocity of an electron plasma wave},
Author = {B\'enisti, D. and Morice, O. and Gremillet, L. and Siminos, E. and Strozzi, D. J.},
Year = {2010},
Pages = {082301},
Volume = {17},
Abstract = {The nonlinear group velocity of an electron plasma wave is investigated
numerically using a Vlasov code, and is found to assume values which
agree very well with those predicted by a recently published theory
[ D. B\'enisti et al., Phys. Rev. Lett. 103, 155002 (2009) ], which
we further detail here. In particular we show that, once Landau damping
has been substantially reduced due to trapping, the group velocity
of an electron plasma wave is not the derivative of its frequency
with respect to its wave number. This result is moreover discussed
physically, together with its implications in the saturation of stimulated
Raman scattering.},
DOI = {10.1063/1.3464467}
}
@Article{benisti2010-2,
Title = {Nonlinear kinetic description of {Raman} growth using an envelope code, and comparisons with {Vlasov} simulations},
Author = {B\'enisti, D. and Morice, O. and Gremillet, L. and Siminos, E. and Strozzi, D. J.},
Year = {2010},
Month = oct,
Pages = {102311},
Volume = {17},
Abstract = {In this paper, we present our nonlinear kinetic modeling of stimulated
Raman scattering in a uniform and collisionless plasma using envelope
equations. We recall the derivation of these equations, as well as
our theoretical predictions for each of the nonlinear kinetic terms,
the precision of which having been carefully checked against Vlasov
simulations. We particularly focus here on the numerical resolution
of these equations, which requires the additional concept of ?self-optimization?
that we explain, and we describe the envelope code {BRAMA} that we
used. As an application of our modeling, we present one-dimensional
{BRAMA} simulations of stimulated Raman scattering which predict
threshold intensities, as well as time scales for Raman growth above
threshold, in very good agreement with those inferred from Vlasov
simulations. Finally, we discuss the differences between our modeling
and other published ones.},
DOI = {10.1063/1.3494223}
}
@Article{BenKS66,
Title = {Long waves on liquid films},
Author = {Benney, D. J.},
Journal = {J. Math. Phys.},
Year = {1966},
Pages = {150},
Volume = {45}
}
@Incollection{BeOtRo14,
author = {Beyn, W.-J. and Otten, D. and Rottmann-Matthes, J.},
title = {Stability and computation of dynamic patterns in {PDEs}},
booktitle = {Current Challenges in Stability Issues for Numerical Differential Equations},
publisher = {Springer},
year = {2014},
series = {Lect. Notes Math.},
pages = {89--172},
doi = {10.1007/978-3-319-01300-8_3},
editors = {Dieci, L. and Guglielmi, N.}
}
@Article{Berkooz94,
Title = {Observations regarding ``{C}oherence and chaos in a model of turbulent boundary layer'' by {X}. {Z}hou and {L}. {S}irovich},
Author = {G. Berkooz and P. Holmes and J. L. Lumley and N. Aubry and E. Stone},
Journal = {Phys. Fluids},
Year = {1994},
Pages = {1574--1578},
Volume = {6}
}
@Article{Bernoff88,
author = {A. J. Bernoff},
title = {Slowly varying fully nonlinear wavetrains in the {Ginzburg-Landau} equation},
journal = {Physica D},
year = {1988},
volume = {30},
pages = {363--381},
doi = {10.1016/0167-2789(88)90026-7},
abstract = {Following the ideas of Howard and Kopell [9] a perturbation
theory is developed for slowly varying fully nonlinear wavetrains (i.e.
solutions which appear locally as travelling waves, but with frequencies
and wavelengths which may vary widely on long length and time scales).
This perturbation theory is applied to the Ginzburg-Landau equation. The
motion and stability of slowly varying wavetrains is shown to be governed
by a simple wave equation which can develop shocks corresponding to rapid
changes in wavenumber. Numerical results supporting this theory are
presented. A shock structure is proposed and numerically verified. These
results together with a winding invariant valid in the limit of slow
variation suggest that over a large range of parameters many initial
conditions relax to uniform wavetrains. The evolution of a marginally
diffusively stable wavetrain is also examined; it is argued that the
evolution is governed by a perturbed Korteweg-de Vries equation. }
}
@Article{Berry12,
Title = {{Martin Gutzwiller} and his periodic orbits},
Author = {M. V. Berry},
Journal = {Commun. Swiss Phys. Soc.},
Year = {2012},
Number = {23--30},
Pages = {4839--4849},
Volume = {37}
}
@Article{BerstelPocchiola94,
Title = {Average cost of {D}uval's algorithm for generating {L}yndon words},
Author = {J. Berstel and M. Pocchiola},
Journal = {Theoret. Comput. Sci.},
Year = {1994},
Pages = {415--425},
Volume = {132}
}
@Article{BerTab76,
Title = {Closed orbits and the regular bound spectrum},
Author = {M. V. Berry and M. Tabor},
Journal = {Proc. Roy. Soc. Ser A},
Year = {1976},
Pages = {101--123},
Volume = {349},
DOI = {10.1098/rspa.1976.0062}
}
@Article{BeTh04,
Title = {Freezing solutions of equivariant evolution equations},
Author = {W.-J. Beyn and V. Th\"ummler},
Journal = {SIAM J. Appl. Dyn. Syst.},
Year = {2004},
Pages = {85--116},
Volume = {3}
}
@Article{Beyn12,
author = {W.-J. Beyn},
title = {An integral method for solving nonlinear eigenvalue problems},
journal = {Linear Algebra Appl.},
year = {2012},
volume = {436},
pages = {3839--3863},
doi = {10.1016/j.laa.2011.03.030},
abstract = {We propose a numerical method for computing all eigenvalues
(and the corresponding eigenvectors) of a nonlinear holomorphic
eigenvalue problem that lie within a given contour in the complex plane.
No initial approximations of eigenvalues and eigenvectors are needed. The
method is particularly suitable for moderately large eigenvalue problems
where k is much smaller than the matrix dimension.}
}
@Article{BFLTC11,
author = {Boghosian, B. M. and Fazendeiro, L. M. and L\"att, J. and Tang, H. and Coveney, P. V.},
title = {New variational principles for locating periodic orbits of differential equations},
journal = {Philos. Trans. Royal Soc. A},
year = {2011},
volume = {369},
pages = {2211--2218},
doi = {10.1098/rsta.2011.0066}
}
@Article{BG81,
author = {D. J. {Benney} and L. H. {Gustavsson}},
title = {A new mechanism for linear and nonlinear hydrodynamic instability},
journal = {Studies in Appl. Math.},
year = {1981},
volume = {64},
pages = {185--209}
}
@Book{BH86,
Title = {Asymptotic Expansions of Integrals},
Author = {N. Bleistein and R. A. Handelsman},
Publisher = {Dover},
Year = {1986},
Address = {New York}
}
@Book{BHLV03,
Title = {Bifurcations in {Hamiltonian} Systems: Computing Singularities by Gr\"obner Bases},
Author = {Broer, H. W. and Hoveijn, I. and Lunter, G. A. and Vegter, G.},
Publisher = {Springer},
Year = {2003},
Address = {New York}
}
@Article{BiBiFo10,
Title = {Computation of the drift velocity of spiral waves using response functions},
Author = {I. V. Biktasheva and V. N. Biktashev and A. J. Foulkes},
Journal = {Phys. Rev. E},
Year = {2010},
Note = {\arXiv{0909.5372}},
Volume = {81},
DOI = {10.1103/PhysRevE.81.066202}
}
@Article{biham90c,
Title = {Unstable periodic orbits and the symbolic dynamics of the complex {H\'enon} map},
Author = {O. Biham and W. Wenzel},
Journal = {Phys. Rev. E},
Year = {1990},
Pages = {4639},
Volume = {42},
Abstract = {Previous work is extended to the complex plane so that 2^n complex
orbits are found for symbol sequences of length n.}
}
@Article{BiHo95,
author = {V. N. Biktashev and A. V. Holden},
title = {Resonant drift of autowave vortices in two dimensions and the effects of boundaries and inhomogeneities},
journal = {Chaos Solit. Fract.},
year = {1995},
volume = {5},
pages = {575--622},
doi = {10.1016/0960-0779(93)E0044-C}
}
@Article{BiHoNi96,
Title = {Spiral wave meander and symmetry of the plane},
Author = {V. N. Biktashev and A. V. Holden and E. V. Nikolaev},
Journal = {Int. J. Bifur. Chaos},
Year = {1996},
Pages = {2433--2440},
Volume = {6}
}
@Article{BiKoTi13,
Title = {Stalling chaos control accelerates convergence},
Author = {C. Bick and C. Kolodziejski and M. Timme},
Journal = {New J. Phys.},
Year = {2013},
Pages = {063038},
Volume = {15},
DOI = {10.1088/1367-2630/15/6/063038}
}
@Article{biktasheva2003wave,
Title = {Wave-particle dualism of spiral waves dynamics},
Author = {Biktasheva, I. V. and Biktashev, V. N.},
Journal = {Phys. Rev. E},
Year = {2003},
Pages = {026221},
Volume = {67}
}
@Article{Birkhoff31,
author = {Birkhoff, G. D.},
title = {Proof of the {Ergodic Theorem}},
journal = {Proc. Natl. Acad. Sci. USA},
year = {1931},
volume = {17},
pages = {656--660},
doi = {10.1073/pnas.17.2.656}
}
@Article{BisPal98,
Title = {Symmetry reduction and semiclassical analysis of axially symmetric systems},
Author = {S. Pal and D. Biswas},
Journal = {Phys. Rev. E},
Year = {1998},
Pages = {1475--1484},
Volume = {57},
Abstract = {semiclassical trace formula for a symmetry-reduced part of the spectrum
in axially symmetric systems. The classical orbits that contribute
are closed in (?,z,p?,pz) and have $p_\phi=m\hbar$, where m is the
azimuthal quantum number. For $m \neq 0$, these orbits vary with
energy and almost never lie on periodic trajectories in the full
phase space in contrast to the case of discrete symmetries.}
}
@Phdthesis{BittihnThesis,
Title = {Complex Structure and Dynamics of the Heart},
Author = {Bittihn, P.},
School = {Georg-August Universit\"at},
Year = {2013},
Address = { G\"{o}ttingen}
}
@Article{BJNRZ12,
author = {B. Barker and M. A. Johnson and P. Noble and L. M. Rodrigues and K. Zumbrun},
title = {Stability of periodic { Kuramoto-Sivashinsky} waves},
journal = {Appl. Math. Lett.},
year = {2012},
volume = {25},
pages = {824--829},
doi = {10.1016/j.aml.2011.10.026},
}
@Article{BjoGol73,
author = {Bj{\"o}rck, \AA. and Golub, G. H.},
title = {Numerical methods for computing angles between linear subspaces},
journal = {Math. Comput.},
year = {1973},
volume = {27},
pages = {579--594},
doi = {10.1090/S0025-5718-1973-0348991-3}
}
@Article{BK90,
Title = {A rule for quantizing chaos},
Author = {M. V. Berry and J. P. Keating},
Journal = {J. Phys. A},
Year = {1990},
Pages = {4839--4849},
Volume = {23}
}
@Article{bknk,
Title = {Formations of spatial patterns and holes in the generalized {Ginzburg-Landau} equation},
Author = {N. Bekki and K. Nozaki},
Journal = {Phys. Lett.},
Year = {1985},
Pages = {133--135},
Volume = {110A}
}
@Article{bknk83,
Title = {Pattern selection and spatiotemporal transition to chaos in the {Ginzburg-Landau} equation},
Author = {K. Nozaki and N. Bekki},
Journal = {Phys. Rev. Lett.},
Year = {1983},
Pages = {2171},
Volume = {51}
}
@Article{bknk84,
Title = {Exact solutions of the generalized {Ginzburg-Landau} equation},
Author = {K. Nozaki and N. Bekki},
Journal = {J. Phys. Soc. Japan},
Year = {1984},
Pages = {1581--1582},
Volume = {53}
}
@Book{bl,
Title = {Introduction to Numerical Analysis},
Author = {J. Stoer and R. Bulirsch},
Publisher = {Springer},
Year = {1983},
Address = {New York}
}
@Article{BlGaPa05,
author = {Blomgren, P. and Gasner, S. and Palacios, A.},
title = {Hopping behavior in the {Kuramoto–Sivashinsky} equation},
journal = {Chaos},
year = {2005},
volume = {15},
pages = {013706},
doi = {10.1063/1.1848311}
}
@Article{BlMaLo05,
author = {H. M. Blackburn and F. Marques and Lopez,J. M.},
title = {Symmetry breaking of two-dimensional time-periodic wakes},
journal = {J. Fluid Mech.},
year = {2005},
volume = {522},
pages = {395--411},
doi = {10.1017/S0022112004002095},
issn = {1469-7645},
numpages = {17}
}
@Article{BlMaZe05,
Title = {Nonholonomic dynamics},
Author = {Bloch, A. M. and Marsden, J. E. and Zenkov, D. V.},
Journal = {Notices Amer. Math. Soc.},
Year = {2005},
Pages = {324--333},
Volume = {52}
}
@Unpublished{BloWan13,
Title = {{Multigrid-in-time for sensitivity analysis of chaotic dynamical systems}},
Author = {{Blonigan}, P. and {Wang}, Q.},
Note = {\arXiv{1305.6878}},
Year = {2013}
}
@Article{Bluman07,
Title = {Connections between symmetries and conservation laws},
Author = {G. Bluman},
Journal = {SIGMA},
Year = {2005},
Note = {\arXiv{math-ph/0511035}},
Pages = {011},
Volume = {1},
Abstract = {On connections between symmetries and conservation laws. After reviewing
Noether's theorem and its limitations, we present the Direct Construction
Method to show how to find directly the conservation laws for any
given system of differential equations. This method yields the multipliers
for conservation laws as well as an integral formula for corresponding
conserved densities. The action of a symmetry (discrete or continuous)
on a conservation law yields conservation laws. Conservation laws
yield non-locally related systems that, in turn, can yield nonlocal
symmetries and in addition be useful for the application of other
mathematical methods. From symmetries or multipliers for conservation
laws, one can determine whether or not a given system of differential
equations can be linearized by an invertible transformation.}
}
@Book{BlumanAnco02,
Title = {Symmetry and Integration Methods for Differential Equations},
Author = {G. W. Bluman and S. C. Anco},
Publisher = {Springer},
Year = {2002},
Address = {New York}
}
@Book{BlumanSDE89,
Title = {Symmetries and Differential Equations},
Author = {G. W. Bluman and S. Kumei},
Publisher = {Springer},
Year = {1989},
Address = {New York}
}
@Book{Bmack93,
Title = {Renormalisation in area-preserving maps},
Author = {MacKay, R. S.},
Publisher = {World Scientific},
Year = {1993},
Address = {Singapore}
}
@Article{BMS71,
Title = {Aperiodic behavior of a non-linear oscillator},
Author = {N. H. Baker and D. W. Moore and E. A. Spiegel},
Journal = {Quatr. J. Mech. and Appl. Math.},
Year = {1971},
Pages = {391},
Volume = {24}
}
@Book{Boccotti00,
Title = {Wave Mechanics for Ocean Engineering},
Author = {P. Boccotti},
Publisher = {Elsevier},
Year = {2000},
Address = {New York},
DOI = {10.1016/S0422-9894(00)80024-0}
}
@Article{Boccotti11,
author = {P. Boccotti},
title = {Field verification of quasi-determinism theory for wind waves in the space-time domain},
journal = {Ocean Eng.},
year = {2011},
volume = {38},
pages = {1503--1507},
doi = {10.1016/j.oceaneng.2011.07.015},
abstract = {Profiles of exceptionally large waves in wind seas were obtained using
a gauge array that was nearly aligned in the dominant wave direction;
the length of these profiles ranged from 1.5 to 3.0 times the dominant
wavelength. The profiles were also obtained through calculations
using the quasi-determinism theory from the datasets of the sea states.
The possibility to observe waves in the space-time domain enables
us to obtain a remarkable confirmation of the quasi-determinism theory.}
}
@InCollection{Bochi04,
author = {Bochi, J. and Viana, M.},
title = {How frequently are dynamical systems hyperbolic?},
booktitle = {Modern Dynamical Systems and Applications},
publisher = {Cambridge Univ. Press},
year = {2004},
editor = {Brin, M. and Hasselblatt, B. and Pesin, Y.},
pages = {271--297},
address = {Cambridge},
isbn = {9780521840736}
}
@Article{Bochi05,
author = {Bochi, J. and Viana, M.},
title = {The {Lyapunov} exponents of generic volume-preserving and symplectic maps},
journal = {Ann. Math.},
year = {2005},
volume = {161},
pages = {1423--1485},
url = {http://annals.princeton.edu/annals/2005/161-3/p06.xhtml}
}
@Book{BogMit61,
Title = {Asymptotic Methods in the Theory of Nonlinear Oscillations},
Author = {Bogoliubov, N. N. and Mitropolski, Y. A.},
Publisher = {Gordon and Breach},
Year = {1961},
Address = {New York}
}
@Article{BohHubOtt96,
Title = {The structure of spiral domain patterns},
Author = {Bohr, T. and Huber, G. and Ott, E.},
Journal = {Europhys. Lett.},
Year = {1996},
Pages = {589},
Volume = {33}
}
@Article{BohHubOtt97,
author = {Bohr, T. and Huber, G. and Ott, E.},
title = {The structure of spiral-domain patterns and shocks in the {2D} complex {Ginzburg-Landau} equation},
journal = {Physica D},
year = {1997},
volume = {106},
pages = {95--112}
}
@Book{bohr98tur,
Title = {Dynamical Systems Approach to Turbulence},
Author = {T. Bohr and M. H. Jensen and G. Paladin and A. Vulpiani},
Publisher = {Cambridge Univ. Press},
Year = {1998},
Address = {Cambridge}
}
@InProceedings{Bojanczyk92theperiodic,
Title = {The periodic {Schur} decomposition. {Algorithm}s and applications},
Author = {A. Bojanczyk and G. H. Golub and P. Van Dooren},
Booktitle = {Proc. SPIE Conference},
Year = {1992},
Pages = {31--42},
Volume = {1770}
}
@Article{bollt07,
Title = {Attractor modeling and empirical nonlinear model reduction of dissipative dynamical systems},
Author = {Bollt, E.},
Journal = {Int. J. Bifur. Chaos},
Year = {2007},
Pages = {1199--1219},
Volume = {17}
}
@Unpublished{BoPo10,
Title = {Covariant {Lyapunov} vectors for rigid disk systems},
Author = {Bosetti, H. and Posch, H. A.},
Note = {\arXiv{1005.1172}},
Year = {2010}
}
@Article{Bor92,
Title = {{Yang-Mills} fields which are not self-dual},
Author = {G. Bor},
Journal = {Commun. Math. Phys.},
Year = {1992},
Pages = {393--410},
Volume = {145}
}
@Article{borghesi02,
Title = {Macroscopic Evidence of Soliton Formation in Multiterawatt Laser-Plasma Interaction},
Author = {Borghesi, M. and Bulanov, S. and {Campbell et. al.}, D. H.},
Journal = {Phys. Rev. Lett.},
Year = {2002},
Pages = {135002},
Volume = {88}
}
@Article{borot12,
Title = {Attosecond control of collective electron motion in plasmas},
Author = {Borot, Antonin and Malvache, Arnaud and Chen, Xiaowei and Jullien, Aur\'elie and Geindre, Jean-Paul and Audebert, Patrick and Mourou, G\'erard and Qu\'er\'e, Fabien and Lopez-Martens, Rodrigo},
Journal = {Nat. Phys.},
Year = {2012},
Number = {5},
Pages = {416--421},
Volume = {8},
Abstract = {Today, light fields of controlled and measured waveform can be used
to guide electron motion in atoms and molecules with attosecond precision.
Here, we demonstrate attosecond control of collective electron motion
in plasmas driven by extreme intensity (?1018 W cm?2) light fields.
Controlled few-cycle near-infrared waves are tightly focused at the
interface between vacuum and a solid-density plasma, where they launch
and guide subcycle motion of electrons from the plasma with characteristic
energies in the multi-kiloelectronvolt range?two orders of magnitude
more than has been achieved so far in atoms and molecules. The basic
spectroscopy of the coherent extreme ultraviolet radiation emerging
from the light?plasma interaction allows us to probe this collective
motion of charge with sub-200 as resolution. This is an important
step towards attosecond control of charge dynamics in laser-driven
plasma experiments.},
DOI = {10.1038/nphys2269},
Language = {en}
}
@Article{Bosetti2010a,
author = {Bosetti, H. and Posch, H. A. and Dellago, C. and Hoover, W. G.},
title = {Time-reversal symmetry and covariant {Lyapunov} vectors for simple particle models in and out of thermal equilibrium},
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year = {2010},
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pages = {1--10}
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@Article{BoSmWi36,
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title = {What Does Dynamical Systems Theory Teach Us about Fluids?},
journal = {Commun. Theor. Phys.},
year = {2014},
volume = {62},
pages = {451--468},
doi = {10.1088/0253-6102/62/4/03},
abstract = {We use molecular dynamics simulations to compute the Lyapunov
spectra of many-particle systems resembling simple fluids in thermal
equilibrium and in non-equilibrium stationary states. Here we review
some of the most interesting results and point to open questions.}
}
@Book{Botti89,
title = {The Kinematics of Mixing: Stretching, Chaos and Transport},
publisher = {Cambridge Univ. Press},
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Title = {Equilibrium States and the Ergodic Theory of {Anosov} Diffeomorphisms},
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author = {A. Bracco and J. Pedlosky},
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pages = {207--219}
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@Incollection{brand03,
Title = {Computational aspects of astrophysical {MHD} and turbulence},
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Booktitle = {Advances in Nonlinear Dynamos},
Publisher = {Taylor \& Francis},
Year = {2003},
Address = {London},
Editor = {A. Ferriz-Mas and M. N{\'{u}\~{n}ez}},
Note = {\arXiv{astro-ph/0109497}},
Pages = {269--344}
}
@Incollection{BrCv12,
Author = {S. Ortega Arango, S.},
Booktitle = {{ChaosBook.org/projects}},
Publisher = {Georgia Inst. of Technology},
Year = {2012},
Chapter = {Towards reducing continuous symmetry of baroclinic flows},
URL = {http://ChaosBook.org/projects/index.shtml#Ortega}
}
@Article{Bredon61,
Title = {On the structure of orbit spaces of generalized manifolds},
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Journal = {Trans. Amer. Math. Soc.},
Year = {1961},
Pages = {162--162},
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DOI = {10.1090/S0002-9947-1961-0126503-5}
}
@Book{Bredon72,
Title = {Introduction to Compact Transformation Groups},
Author = {G. Bredon},
Publisher = {Academic},
Year = {1972},
Address = {New York}
}
@Unpublished{BrEllGBAHo10,
Title = {{Un-reduction}},
Author = {{Bruveris}, M. and {Ellis}, D. C. P. and {Gay-Balmaz}, F. and {Holm}, D. D.},
Note = {\arXiv{1012.0076}},
Year = {2010}
}
@Misc{Bridges_priv,
Author = {T. J. Bridges},
Note = {private communication}
}
@Article{Bridges08,
Title = {Degenerate relative equilibria, curvature of the momentum map,and homoclinic bifurcation},
Author = {T. J. Bridges},
Journal = {J. Diff. Eqn.},
Year = {2008},
Number = {7},
Pages = {1629--1674},
Volume = {244},
Abstract = {A fundamental class of solutions of symmetric Hamiltonian systems
is relative equilibria. In this paper the nonlinear problem near
a degenerate relative equilibrium is considered. The degeneracy creates
a saddle-center and attendant homoclinic bifurcation in the reduced
system transverse to the group orbit. The surprising result is that
the curvature of the pullback of the momentum map to the Lie algebra
determines the normal form for the homoclinic bifurcation. There
is also an induced directional geometric phase in the homoclinic
bifurcation. The backbone of the analysis is the use of singularity
theory for smooth mappings between manifolds applied to the pullback
of the momentum map. The theory is constructive and generalities
are given for symmetric Hamiltonian systems on a vector space of
dimension (2n+2) with an n-dimensional abelian symmetry group. Examples
for n=1,2,3 are presented to illustrate application of the theory.}
}
@Article{Briggs1990,
Title = {An improved method for estimating {Liapunov} exponents of chaotic time series},
Author = {Briggs, K.},
Journal = {Phys. Lett. A},
Year = {1990},
Pages = {27?32},
Volume = {151}
}
@Incollection{BrKevr96,
author = {Brown, H. S. and Kevrekidis, I. G.},
title = {Modulated traveling waves for the {Kuramoto-Sivashinsky} equation},
booktitle = {Pattern Formation: Symmetry Methods and Applications},
publisher = {AMS},
year = {1996},
editor = {D. Benest and C. Froeschl\'{e}},
pages = {45--66},
address = {Providence, RI}
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@Article{bro94sur,
Title = {Nonlinear surface waves in a plasma with a diffuse boundary},
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Journal = {Phys. Plasmas},
Year = {1994},
Pages = {96},
Volume = {1}
}
@Article{broer_quasi-periodic_2007,
Title = {The quasi-periodic reversible {Hopf} bifurcation},
Author = {H. W. Broer and M. C. Ciocci and H. Hanssmann},
Journal = {Int. J. Bifur. Chaos},
Year = {2007},
Pages = {2605--2623},
Volume = {17}
}
@Article{broer97,
Title = {Algorithms for computing {Normally Hyperbolic Invariant Manifolds}},
Author = {H. W. Broer and H. M. Osinga and G. Vegter},
Journal = {Z. Angew. Math. Phys.},
Year = {1997},
Pages = {480},
Volume = {48}
}
@Article{BrOgYuRe05,
Title = {Uniform semiclassical trace formula for {$U(3) \to SO(3)$} symmetry breaking},
Author = {M. Brack and M. \"Ogren and Y. Yu and S. M. Reimann},
Journal = {J. Phys. A},
Year = {2005},
Pages = {9941},
Volume = {38}
}
@Article{bronski2005,
Title = {Uncertainty estimates and {$L_2$} bounds for the {Kuramoto-Sivashinsky} equation},
Author = {J. C. Bronski and T. N. Gambill},
Journal = {Nonlinearity},
Year = {2006},
Note = {\arXiv{math/0508481}},
Pages = {2023--2039},
Volume = {19}
}
@Article{Broucke75,
Title = {On relative periodic solutions of the planar general three-body problem},
Author = {R. Broucke},
Journal = {Celestial Mech. Dynam. Astronom.},
Year = {1975},
Pages = {439--462},
Volume = {12},
Abstract = {distinction between two types of periodic solutions: absolute or relative
periodic solutions. An algorithm for obtaining relative periodic
solutions using heliocentric coordinates is described. It is concluded
from the periodicity conditions that relative periodic solutions
must form families with a single parameter. Two such families have
been obtained numerically.}
}
@Article{Brouwer1,
author = {Brouwer, L. E. J.},
title = {\"Uber Abbildung von Mannigfaltigkeiten},
journal = {Math. Ann.},
year = {1911},
volume = {71},
pages = {97--115},
doi = {10.1007/BF01456931}
}
@Article{Brunet07,
Title = {Stabilized {Kuramoto-Sivashinsky} equation: {A} useful model for secondary instabilities and related dynamics of experimental one-dimensional cellular flows},
Author = {Brunet, P.},
Journal = {Phys. Rev. E},
Year = {2007},
Pages = {017204},
Volume = {76},
DOI = {10.1103/PhysRevE.76.017204}
}
@Article{BTBmaw03,
Title = {Nonlinear analysis of the {E}ckhaus instability: modulated amplitude waves and phase chaos with nonzero average phase gradient},
Author = {L. Brusch and A. Torcini and M. B{\"{a}}r},
Journal = {Physica D},
Year = {2003},
Pages = {152--167},
Volume = {174}
}
@Article{BTHmaw01,
author = {L. Brusch and A. Torcini and M. van Hecke and M. G. Zimmermann and M. B{\"{a}}r},
title = {Modulated amplitude waves and defect formation in the one-dimensional complex {Ginzburg-Landau} equation},
journal = {Physica D},
year = {2001},
volume = {160},
pages = {127--148}
}
@Article{BuBoCvSi14,
Title = {Periodic orbit analysis of a system with continuous symmetry - {A} tutorial},
Author = {Budanur, N. B. and Borrero-Echeverry, D. and Cvitanovi\'c, P.},
Journal = {Chaos},
Year = {2015},
Note = {\arXiv{1411.3303}},
Pages = {073112},
Volume = {25},
DOI = {10.1063/1.4923742}
}
@Unpublished{Bud15,
Title = {Invariant measurement of torsion within eigenframes of periodic orbits},
Author = {Budanur, N. B.},
Note = {In preparation},
Year = {2016}
}
@Phdthesis{BudanurThesis,
Title = {Exact Coherent Structures in Spatiotemporal Chaos: From Qualitative Description to Quantitative Predictions},
Author = {Budanur, N. B.},
School = {School of Physics, Georgia Inst. of Technology},
Year = {2015},
Address = {Atlanta},
URL = {http://ChaosBook.org/projects/theses.html}
}
@Article{BudCvi14,
Title = {Reduction of the {SO(2)} symmetry for spatially extended dynamical systems},
Author = {Budanur, N. B. and Cvitanovi\'c, P. and Davidchack, R. L. and Siminos, E.},
Journal = {Phys. Rev. Lett.},
Year = {2015},
Note = {\arXiv{1405.1096}},
Pages = {084102},
Volume = {114},
DOI = {10.1103/PhysRevLett.114.084102}
}
@Unpublished{BudCvi15,
Title = {Torus breakdown in the symmetry-reduced state space of the {Kuramoto-Sivashinsky} system},
Author = {Budanur, N. B. and Cvitanovi\'c, P.},
Note = {\arXiv{1509.08133}},
Year = {2015}
}
@Unpublished{BuDiCv15,
Title = {Turbulent {Kuramoto-Sivashinsky} system recycled},
Author = {Budanur, N. B. and Ding, D. and Cvitanovi\'c, P.},
Year = {2015}
}
@Article{BudMez12,
Title = {Geometry of the ergodic quotient reveals coherent structures in flows},
Author = {M. Budi\v{s}i\'c and I. Mezi\'c},
Journal = {Physica D},
Year = {2012},
Note = {\arXiv{1204.2050}},
Pages = {1255--1269},
Volume = {241},
Abstract = {The method we present in this paper uses trajectory averages of scalar
functions on the state space to: (a) identify invariant sets in the
state space, and (b) to form coherent structures by aggregating invariant
sets that are similar across multiple spatial scales. First, we construct
the ergodic quotient, the object obtained by mapping trajectories
to the space of the trajectory averages of a function basis on the
state space. Second, we endow the ergodic quotient with a metric
structure that successfully captures how similar the invariant sets
are in the state space. Finally, we parametrize the ergodic quotient
using intrinsic diffusion modes on it. By segmenting the ergodic
quotient based on the diffusion modes, we extract coherent features
in the state space of the dynamical system. The algorithm is validated
by analyzing the Arnold-Beltrami-Childress flow, which was the test-bed
for alternative approaches: the Ulam's approximation of the transfer
operator and the computation of Lagrangian Coherent Structures. Furthermore,
we explain how the method extends the Poincar\'e map analysis for
periodic flows. As a demonstration, we apply the method to a periodically-driven
three-dimensional Hill's vortex flow, discovering unknown coherent
structures in its state space.},
DOI = {10.1016/j.physd.2012.04.006}
}
@Article{BuKe03,
Title = {Estimating good discrete partitions from observed data: {Symbolic} false nearest neighbors},
Author = {{Kennel}, M.~B. and {Buhl}, M.},
Journal = {Phys. Rev. Lett.},
Year = {2003},
Note = {\arXiv{nlin/0304054}},
Pages = {084102},
Volume = {91}
}
@Article{BuMoMe12,
Title = {{Applied Koopmanism}},
Author = {Budi{\v s}i{\'c}, M. and Mohr, R. M. and Mezi{\'c}, I.},
Journal = {Chaos},
Year = {2012},
Note = {\arXiv{1206.3164}},
Pages = {047510},
Volume = {22},
DOI = {10.1063/1.4772195},
Numpages = {33}
}
@Article{Bunimovich85,
Title = {Decay of correlations in dynamical systems with chaotic behavior},
Author = {Bunimovich, L. A.},
Journal = {Sov. Phys. JETP},
Year = {1985},
Pages = {842--852},
Volume = {62},
URL = {http://jetp.ac.ru/cgi-bin/dn/e_062_04_0842.pdf}
}
@Article{BunSin80,
author = {Bunimovich, L. A. and Sinai, Ya. G.},
title = {{Markov} partitions for dispersed billiards},
journal = {Commun. Math. Phys.},
year = {1980},
volume = {78},
pages = {247--280},
note = {Erratum, ibid. {\bf 107}, 357 (1986)},
doi = {10.1007/BF01942372}
}
@Unpublished{BuReSh11,
Title = {Dimension reduction near periodic orbits of hybrid systems},
Author = {{Burden}, S. and {Revzen}, S. and {Shankar Sastry}, S.},
Note = {\arXiv{1109.1780}},
Year = {2011}
}
@Article{Busse04,
Title = {Visualizing the dynamics of the onset of turbulence},
Author = {F. H. Busse},
Journal = {Science},
Year = {2004},
Pages = {1574--1575},
Volume = {305}
}
@Book{ByFu92,
Title = {Mathematics of Classical and Quantum Physics},
Author = {F. W. Byron and R. W. Fuller},
Publisher = {Dover},
Year = {1992},
Address = {New York}
}
@Article{BZMmaw00,
author = {L. Brusch and M. G. Zimmermann and M. van Hecke and M. B{\"{a}}r and A. Torcini},
title = {Modulated amplitude waves and the transition from phase to defect chaos},
journal = {Phys. Rev. Lett.},
year = {2000},
volume = {85},
pages = {86}
}
@Article{C02,
Title = {Subcritical transition in channel flows},
Author = {S. J. Chapman},
Journal = {J. Fluid Mech.},
Year = {2002},
Pages = {35--97},
Volume = {451}
}
@Misc{CaCaKi63,
Title = {{The Ring of Fire}},
Author = {Cash, J. and Carter Cash, J. and Kilgore, M.},
Year = {1963},
Address = {New York},
Publisher = {Columbia Records}
}
@Article{CaCiDeBr12,
Title = {Model of a two-dimensional extended chaotic system: {Evidence} of diffusing dissipative solitons},
Author = {Cartes, C. and Cisternas, J. and Descalzi, O. and Brand, H. R.},
Journal = {Phys. Rev. Lett.},
Year = {2012},
Pages = {178303},
Volume = {109},
DOI = {10.1103/PhysRevLett.109.178303}
}
@Article{CaHa01a,
Title = {Pruning theory and {Thurston}'s classification of surface homeomorphisms},
Author = {de Carvalho, A. and Hall, T.},
Journal = {J. Eur. Math. Soc.},
Year = {2001},
Pages = {287--333},
Volume = {3}
}
@Article{CaHa01b,
Title = {The forcing relation for horseshoe braid types},
Author = {de Carvalho, A. and Hall, T.},
Journal = {Experimental Math.},
Year = {2002},
Pages = {271--288},
Volume = {11}
}
@Article{CaHa02,
Title = {How to prune a horseshoe},
Author = {de Carvalho, A. and Hall, T.},
Journal = {Nonlinearity},
Year = {2002},
Pages = {R19--R68},
Volume = {15}
}
@Article{CaHa03,
Title = {Conjugacies between horseshoe braids},
Author = {de Carvalho, A. and Hall, T.},
Journal = {Nonlinearity},
Year = {2003},
Pages = {1329--1338},
Volume = {16}
}
@Article{CaHa04a,
Title = {Braid forcing and star-shaped train tracks},
Author = {de Carvalho, A. and Hall, T.},
Journal = {Topology},
Year = {2004},
Pages = {247--287},
Volume = {43}
}
@Article{CaHa04b,
Title = {Unimodal generalized pseudo-{Anosov} maps},
Author = {de Carvalho, A. and Hall, T.},
Journal = {Geometry and Topology},
Year = {2004},
Pages = {1127--1188},
Volume = {8}
}
@Article{CaKno97,
author = {T. K. Callahan and E. Knobloch},
title = {Symmetry-breaking bifurcations on cubic lattices},
journal = {Nonlinearity},
year = {1997},
volume = {10},
pages = {1179--1216},
doi = {10.1088/0951-7715/10/5/009}
}
@Article{CaKno99,
Title = {Pattern formation in three-dimensional reaction-diffusion systems},
Author = {T. K. Callahan and E. Knobloch},
Journal = {Physica D},
Year = {1999},
Pages = {339--362},
Volume = {132},
DOI = {10.1016/S0167-2789(99)00041-X}
}
@Article{camporeale06,
Title = {New approach for the study of linear {Vlasov} stability of inhomogeneous systems},
Author = {E. Camporeale and G. L. Delzanno and G. Lapenta and W. Daughton},
Journal = {Phys. Plasmas},
Year = {2006},
Pages = {092110},
Volume = {13}
}
@Article{Cantwell92,
Title = {Exact solution of a restricted {Euler} equation for the velocity gradient tensor},
Author = {B. J. Cantwell},
Journal = {Phys. Fluids A},
Year = {1992},
Pages = {782--793},
Volume = {4},
DOI = {10.1063/1.858295}
}
@Book{Canuto88,
Title = {Spectral Methods in Fluid Dynamics},
Author = {C. Canuto and M. Y. Hussaini and A. Quateroni and T. A. Zhang},
Publisher = {Springer},
Year = {1988},
Address = {New York}
}
@Article{CaPe84,
Author = {A. Carnegie and I. C. Percival},
Journal = {J. Phys. A},
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Volume = {17}
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@Article{car88,
Title = {{P}ainlev\'{e} expansions for nonintegrable evolution equations},
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Journal = {Physica D},
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Volume = {39}
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@Article{carl97int,
Title = {Cycle expansions for intermittent diffusion},
Author = {C. P. Dettmann and P. Cvitanovi\'{c}},
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Pages = {6687},
Volume = {56}
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@Article{CarLit45,
Title = {On nonlinear differential equations of the second order},
Author = {M. L. Cartwright and J. E. Littlewood},
Journal = {J. London Math. Soc.},
Year = {1945},
Pages = {180--189},
Volume = {20}
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@Incollection{Carr12,
Author = {K. M. Carroll},
Booktitle = {{ChaosBook.org/projects}},
Publisher = {Georgia Inst. of Technology},
Year = {2012},
Chapter = {A review of return maps for {R\"ossler} and the complex {Lorenz}},
URL = {http://ChaosBook.org/projects/index.shtml#Carroll}
}
@Inproceedings{carsim,
author = {C. Simo},
title = {On the Analytical and Numerical Approximation of Invariant Manifolds},
booktitle = {Les M\'{e}thodes Modernes de la M\'{e}canique C\'{e}leste (Goutelas '89)},
year = {1989},
editor = {D. Benest and C. Froeschl\'{e}},
pages = {285--329},
address = {Gif-sur-Yvette, France},
publisher = {Editions Frontieres}
}
@Book{Cartan01,
Title = {Riemannian geometry in an orthogonal frame: {From} lectures delivered by \'Elie Cartan at the Sorbonne in 1926-1927},
Author = {Cartan, \'E.},
Publisher = {World Scientific},
Year = {2001},
Address = {River Edge, NJ},
ISBN = {9810247478}
}
@Book{CartanMF,
Title = {La m\'ethode du rep\`ere mobile, la th\'eorie des groupes continus, et les espaces g\'en\'eralis\'es},
Author = {Cartan, \'E.},
Publisher = {Hermann},
Year = {1935},
Address = {Paris},
Series = {{Expos\'es} de {G\'eom\'etrie}},
Volume = {5}
}
@Article{Carvalho99,
Title = {Pruning fronts and the formation of horseshoes},
Author = {de Carvalho, A.},
Journal = {Ergod. Theor. Dynam. Syst.},
Year = {1999},
Pages = {851--894},
Volume = {19},
DOI = {10.1017/S0143385799133972}
}
@Article{Cassanas05a,
Title = {Reduced {G}utzwiller formula with symmetry: {C}ase of a finite group},
Author = {Roch Cassanas},
Journal = {J. Math. Phys.},
Year = {2006},
Note = {\arXiv{math-ph/0506063}},
Pages = {042102},
Volume = {47},
Abstract = {assuming that periodic orbits are nondegenerate in SigmaE/G, we get
a reduced Gutzwiller trace formula which makes periodic orbits of
the reduced space}
}
@Article{Cassanas05b,
Title = {Reduced {G}utzwiller formula with symmetry: {C}ase of a Lie group},
Author = {R. Cassanas},
Note = {\arXiv{math-ph/0509014}},
Abstract = {We consider a classical {H}amiltonian $H$ on $\mathbb{R}^{2d}$, invariant
by a Lie group of symmetry $G$}
}
@Article{castel01,
Title = {Experimental study of chaotic advection regime in a twisted duct flow},
Author = {Castelain, C. and Mokrani, A. and Le Guer, Y. and Peerhossaini, H.},
Journal = {European J. Mechanics B},
Year = {2001},
Pages = {205--232},
Volume = {20},
DOI = {10.1016/S0997-7546(00)01116-X}
}
@Article{Cattell2000,
Title = {Fast algorithms to generate necklaces, unlabeled necklaces, and irreducible polynomials over {GF(2)}},
Author = {K. Cattell and F. Ruskey and J. Sawada and M. Serra and C. R. Miers},
Journal = {J. Algorithms},
Year = {2000},
Pages = {267--282},
Volume = {37}
}
@Article{CB92,
Title = {Three-dimensional convection in a horizontal layer subjected to constant shear},
Author = {R. M. Clever and F. H. Busse},
Journal = {J. Fluid Mech.},
Year = {1992},
Pages = {511--527},
Volume = {234}
}
@Article{CB97,
author = {Clever, R. M. and Busse, F. H.},
title = {Tertiary and quaternary solutions for plane {Couette} flow},
journal = {J.\ Fluid Mech.},
year = {1997},
volume = {344},
pages = {137--153},
abstract = {The manifold of Nagata steady solutions is explored in the parameter
space of the problem and their instabilities are investigated. These
instabilities usually lead to time-periodic solutions whose properties
do not differ much from those of the steady solutions except that
the amplitude varies in time. In some cases travelling wave solutions
which are asymmetric with respect to the midplane of the layer are
found as quaternary states of flow. Similarities with longitudinal
vortices recently observed in experiments are discussed.}
}
@Incollection{CBcontinuous,
Author = {P. Cvitanovi\'{c}},
Booktitle = {{Chaos: Classical and Quantum}},
Publisher = {Niels Bohr Inst.},
Year = {2015},
Address = {Copenhagen},
Chapter = {{Relativity} for cyclists},
URL = {http://ChaosBook.org/paper.shtml#continuous}
}
@Incollection{CBconverg,
Author = {R. Artuso and H. H. Rugh and P. Cvitanovi{\'c}},
Booktitle = {{Chaos: Classical and Quantum}},
Publisher = {Niels Bohr Inst.},
Year = {2015},
Address = {Copenhagen},
Chapter = {{Why} does it work?},
URL = {http://ChaosBook.org/paper.shtml#converg}
}
@Incollection{CBdiffusion,
Author = {R. Artuso and P. Cvitanovi\'{c}},
Booktitle = {{Chaos: Classical and Quantum}},
Publisher = {Niels Bohr Inst.},
Year = {2015},
Address = {Copenhagen},
Chapter = {{Deterministic} diffusion},
URL = {http://ChaosBook.org/paper.shtml#diffusion}
}
@Incollection{CBook:appendApplic,
Author = {P. Cvitanovi\'c and G. Vattay},
Booktitle = {{Chaos: Classical and Quantum}},
Publisher = {Niels Bohr Inst.},
Year = {2015},
Address = {Copenhagen},
Chapter = {{Transport} of vector fields},
URL = {http://ChaosBook.org/paper.shtml#appendApplic}
}
@Incollection{CBtrace,
Author = {P. Cvitanovi{\'c}},
Booktitle = {{Chaos: Classical and Quantum}},
Publisher = {Niels Bohr Inst.},
Year = {2015},
Address = {Copenhagen},
Chapter = {{Trace} formulas},
URL = {http://ChaosBook.org/paper.shtml#trace}
}
@Article{CC92,
Title = {Periodic orbit quantization of the anisotropic {Kepler} problem},
Author = {F. Christiansen and P. Cvitanovi\'c},
Journal = {Chaos},
Year = {1992},
Pages = {61},
Volume = {2}
}
@Article{CCR,
Title = {The spectrum of the period-doubling operator in terms of cycles},
Author = {F. Christiansen and P. Cvitanovi\'c and H. H. Rugh},
Journal = {J. Phys A},
Year = {1990},
Pages = {L713},
Volume = {23}
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@Article{CCW05,
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Journal = {Proc. Amer. Math. Soc.},
Year = {2005},
Pages = {3551--3560},
Volume = {133}
}
@Article{CdV73,
Title = {Spectre du {Laplacien} et longeurs des g\`eod\`esiques p\`eriodiques, II.},
Author = {Colin de Verdi\`ere, Y.},
Journal = {Compositio. Math.},
Year = {1973},
Pages = {159--184},
Volume = {27}
}
@Article{CE89,
Title = {Periodic orbit quantization of chaotic systems},
Author = {Cvitanovi\'c, P. and Eckhardt, B.},
Journal = {Phys. Rev. Lett.},
Year = {1989},
Pages = {823--826},
Volume = {63}
}
@Article{CE91,
author = {P. Cvitanovi\'{c} and B. Eckhardt},
title = {Periodic orbit expansions for classical smooth flows},
journal = {J. Phys. A},
year = {1991},
volume = {24},
pages = {L237--L241},
doi = {10.1088/0305-4470/24/5/005}
}
@Article{CEEks93,
Title = {Analyticity for the {Kuramoto-Sivashinsky} equation},
Author = {P. Collet and J. -P. Eckmann and H. Epstein and J. Stubbe},
Journal = {Physica D},
Year = {1993},
Pages = {321--326},
Volume = {67},
Abstract = {Proved the analyticity of the solution of the KSe on a periodic interval
near the real axis and the width of the region is estimated. Numerical
calculations show that this width should not depend on the size of
the interval.}
}
@Article{CEEksgl93,
Title = {A global attracting set for the {Kuramoto-Sivashinsky} equation},
Author = {P. Collet and J. -P. Eckmann and H. Epstein and J. Stubbe},
Journal = {Commun. Math. Phys.},
Year = {1993},
Pages = {203--214},
Volume = {152},
Abstract = {New bounds are given for the L^2 norm of solutions of the KSe (not
necessarily antisymmetric).}
}
@Book{cellwolf,
Title = {Cellular Automata and Complexity},
Author = {S. Wolfram},
Publisher = {Wesley},
Year = {1994},
Address = {Redwood City}
}
@Article{CenGin13,
author = {M. Cencini and F. Ginelli},
title = {{Lyapunov} analysis: from dynamical systems theory to applications},
journal = {J. Phys. A},
year = {2013},
volume = {46},
pages = {250301},
doi = {10.1088/1751-8113/46/25/250301}
}
@Article{CenVul13,
Title = {Finite size {Lyapunov} exponent: review on applications},
Author = {M. Cencini and A. Vulpiani},
Journal = {J. Phys. A},
Year = {2013},
Pages = {254019},
Volume = {46},
DOI = {10.1088/1751-8113/46/25/254019}
}
@Incollection{CERRS,
Title = {Pinball scattering},
Author = {B. Eckhardt and G. Russberg and P. Cvitanovi\'c and P. E. Rosenqvist and P. Scherer},
Booktitle = {Quantum Chaos: Between Order and Disorder},
Publisher = {Cambridge Univ. Press},
Year = {1995},
Address = {Cambridge},
Editor = {G. Casati and B. Chirikov},
Pages = {483}
}
@Article{CFGT15,
author = {Carmona, V. and Fern{\'a}ndez-S{\'a}nchez, F. and Garc{\'i}a-Medina, E. and Teruel, A. E.},
title = {Noose structure and bifurcations of periodic orbits in reversible
three-dimensional piecewise linear differential systems},
journal = {J. Nonlin. Sci},
year = {2015},
volume = {25},
pages = {1209--1224},
doi = {10.1007/s00332-015-9251-z}
}
@Article{cgl,
Title = {The world of complex {Ginzburg-Landau} equation},
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Year = {2002},
Pages = {99--143},
Volume = {74}
}
@Inproceedings{CGMS02,
author = {A. Chenciner and J. Gerver and R. Montgomery and C. Sim\'o},
title = {Simple choreographic motions of {$N$}-bodies: {A} preliminary study},
booktitle = {Geometry, Mechanics and Dynamics},
year = {2002},
editor = {P. Newton and P. Holmes and A. Weinstein},
pages = {287--308},
address = {New York},
publisher = {Springer}
}
@Article{cgole92,
Title = {A new proof of the {Aubry-Mather}'s theorem},
Author = {C. Gol\'{e}},
Journal = {Math. Z.},
Year = {1992},
Pages = {441--448},
Volume = {210},
Abstract = {A auxiliary infinite gradient-like flow is constructed for the area-preserving
map such that orbits of the original system are rest points of the
new system. In this way, existence of orbits of any rotation number
is proved.}
}
@Article{CGS92,
author = {Cvitanovi\'c, P. and Gaspard, P. and Schreiber, T.},
title = {Investigation of the {Lorentz} gas in terms of periodic orbits},
journal = {Chaos},
year = {1992},
volume = {2},
pages = {85--90},
doi = {10.1063/1.165902}
}
@Article{CGV,
Title = {On the mode-locking universality for critical circle maps},
Author = {P. Cvitanovi\'c and G. H. Gunaratne and M. J. Vinson},
Journal = {Nonlinearity},
Year = {1990},
Pages = {873},
Volume = {3},
Abstract = {The conjectured universality of the Hausdorff dimension of the fractal
set formed by the set of the irrational winding parameter values
for critical circle maps is shown to follow from the universal scalings
for quadratic irrational winding numbers.}
}
@Article{Cha74,
author = {Chazarain, J.},
title = {Formule de {Poisson} pour les vari\'et\'es riemanniennes},
journal = {Invent. Math.},
year = {1974},
volume = {24},
pages = {65--82},
doi = {10.1007/BF01418788}
}
@Article{ChaDeVore79,
Title = {Multiple flow equilibria in the atmosphere and blocking},
Author = {J. G. Charney and J. G. DeVore},
Journal = {J. Meteorology},
Year = {1979},
Pages = {1205--1216},
Volume = {36},
DOI = {10.1175/1520-0469(1979)036<1205:MFEITA>2.0.CO;2}
}
@Article{ChaKer12,
Title = {Invariant recurrent solutions embedded in a turbulent two-dimensional {Kolmogorov} flow},
Author = {Chandler, G. J. and Kerswell, R. R.},
Journal = {J. Fluid M.},
Year = {2013},
Note = {\arXiv{1207.4682}},
Pages = {554--595},
Volume = {722},
DOI = {10.1017/jfm.2013.122}
}
@Article{ChaMann91,
Title = {Evidence of collective behaviour in cellular automata},
Author = {H. Chat\'{e} and P. Manneville},
Journal = {Europhys. Lett.},
Year = {1991},
Pages = {409},
Volume = {14},
Abstract = {A nontrivial collective behaviour occurring in cellular automata for
rather high space dimensions is presented and investigated numerically.
Evidence is given that this dynamical state is neither transient
nor due to the finite size of the lattice. Possible theoretical approaches
are discussed for this phenomenon showing many aspects of low-dimensional
dynamical systems and yet exhibited by many-body nonequilibrium models.},
DOI = {10.1209/0295-5075/14/5/004}
}
@Article{chand03tf,
Title = {Time-frequency analysis of chaotic systems},
Author = {C. Chandre and S. Wiggins and T. Uzer},
Journal = {Physica D},
Year = {2003},
Pages = {171},
Volume = {181},
Abstract = {Use wavelets to extract local frequencies and use the major frequencies
to identify the characteristics of phase space motion.}
}
@Article{chang94,
Title = {Wave evolution on a falling film},
Author = {H. -C. Chang},
Journal = {Ann. Rev. Fluid Mech.},
Year = {1994},
Pages = {103--136},
Volume = {26}
}
@Book{chat_mixing99,
Title = {Mixing - chaos and turbulence},
Author = {H. Chat\'{e} and E. Villermaux and J.-M. Chomaz},
Publisher = {Kluwer},
Year = {1999},
Address = {Dordrecht}
}
@Article{chate94,
Title = {Spatiotemporal intermittency regimes of the one-dimensional complex {Ginzburg-Landau} equation},
Author = {H. Chat\'{e}},
Journal = {Nonlinearity},
Year = {1994},
Pages = {185--204},
Volume = {7},
Abstract = {In the Benjamin-Feir stable region, spatiotemporal intermittency regimes
are identified which consists of patches of linearly stale stable
plane waves separated by localized objects with a well defined dynamics.
In the transition reginon, asymptotic states with an irregular, frozen
spatial structure are shown to occur. The bistable region beyond
the Benjamin-Feir stable region is of the same spatiotemporal intermittency
type with phase turbulence as the ``laminar'' state.}
}
@Article{ChChYu14,
author = {Chen, Y.-C. and Chen, S.-S. and Yuan, J.-M.},
title = {Topological horseshoes in travelling waves of discretized nonlinear wave equations},
journal = {J. Math. Phys.},
year = {2014},
volume = {55},
pages = {042701},
doi = {10.1063/1.4870618}
}
@Article{chemillier04,
Title = {Periodic musical sequences and {Lyndon} words},
Author = {M. Chemillier},
Journal = {Soft Comput.},
Pages = {611--616},
Volume = {8}
}
@Article{CheMon00,
author = {A. Chenciner and R. Montgomery},
title = {A remarkable solution of the 3\mbox{-}{b}ody problem in the case of equal masses},
journal = {Ann. Math.},
year = {2000},
volume = {152},
pages = {881--901}
}
@Article{Chen87,
Title = {Orbit extension methods for finding unstable orbits},
Author = {Q. Chen and Meiss, J. D., and I. C. Percival},
Journal = {Physica D},
Year = {1987},
Pages = {143--154},
Volume = {29}
}
@Article{Chen90b,
Title = {Resonances and transport in the sawtooth map},
Author = {Q. Chen, I. Dana, and Meiss, J. D. and N. W. Murray and I. C. Percival},
Journal = {Physica D},
Year = {1990},
Pages = {217--240},
Volume = {46}
}
@Article{Chenc05,
author = {A. Chenciner},
title = {A note by {P}oincar\'e},
journal = {Regul. Chaotic Dyn.},
year = {2005},
volume = {10},
pages = {119--128}
}
@Unpublished{ChencinerLink,
Title = {Three body problem},
Author = {A. Chenciner},
Note = {\HREF{http://scholarpedia.org/article/Three_body_problem} {Scholarpedia.org}},
Year = {2007}
}
@Article{ChePuShr99,
Title = {Lagrangian tetrad dynamics and the phenomenology of turbulence},
Author = {Chertkov, M. and Pumir, A. and Shraiman, B. I.},
Journal = {Phys. Fluids},
Year = {1999},
Note = {\arXiv{chao-dyn/9905027}},
Pages = {2394--2410},
Volume = { 11}
}
@Article{Cherry28,
author = {Cherry, T. M.},
title = {On periodic solutions of {Hamiltonian} systems of differential equations},
journal = {Proc. Roy. Soc. Lond. Ser A},
year = {1928},
volume = {227},
pages = {137--221}
}
@Article{CheSu90,
Title = {Existence of invariant tori in three-dimensional measure-preserving mappings},
Author = {Cheng, C. Q. and Sun, Y. S.},
Journal = {Celestial Mech. Dynam. Astronom.},
Year = {1990},
Pages = {275--92},
Volume = {47}
}
@Article{chfield,
Title = {{Chaotic Field Theory}: {A} sketch},
Author = {P. Cvitanovi\'{c}},
Journal = {Physica A},
Year = {2000},
Note = {\arXiv{nlin.CD/0001034}},
Pages = {61--80},
Volume = {288}
}
@Article{CHHM98,
Title = {The {Maxwell-Vlasov} equations in {Euler-Poincar\'e} form},
Author = {Cendra, H. and Holm, D. D. and Hoyle, M. J. W. and Marsden, J. E.},
Journal = {J. of Math. Phys.},
Year = {1998},
Pages = {3138--3157},
Volume = {39}
}
@Book{Chicone2006,
Title = {Ordinary Differential Equations with Applications},
Author = {C. Chicone},
Publisher = {Springer},
Year = {2006},
Address = {New York}
}
@Article{Chillingworth2000,
Title = {Generic multiparameter bifurcation from a manifold},
Author = {D. Chillingworth},
Journal = {Dyn. Stab. Syst.},
Year = {2000},
Pages = {101--137},
Volume = {15}
}
@Article{Chirikov79,
Title = {A universal instability of many-dimensional oscillator system},
Author = {B. V. Chirikov},
Journal = {Phys. Rep.},
Year = {1979},
Pages = {265},
Volume = {263--379}
}
@Article{ChKuRo83,
Title = {On averaging, reduction, and symmetry in {Hamiltonian} systems},
Author = {Churchill, R. C. and Kummer, M. and Rod, D. L.},
Journal = {J. Diff. Eqn.},
Year = {1983},
Pages = {359--414},
Volume = {49}
}
@Article{ChoGuck99,
Title = {Computing periodic orbits with high accuracy},
Author = {W. G. Choe and J. Guckenheimer},
Journal = {Computer Methods Appl. Mech. Engineering},
Year = {1999},
Pages = {331--341},
Volume = {170},
Abstract = {This paper introduces a new family of algorithms for computing periodic
orbits of vector fields. These global methods achieve high accuracy
with relatively coarse discretizations of periodic orbits through
the use of automatic differentiation. High degree Taylor series expansions
of trajectories are computed at mesh points. On a fixed mesh, we
construct closed curves that converge smoothly to periodic orbits
as the degree of the Taylor series expansions increase. The algorithms
have been implemented in Matlab together with the use of the automatic
differentiation code ADOL-C. Numerical tests of our codes are compared
with AUTO calculations using the Hodgkin-Huxley equations as a test
problem.},
DOI = {10.1016/S0045-7825(98)00201-1}
}
@Article{Choss01,
Title = {The bifurcation of heteroclinic cycles in systems of hydrodynamic type},
Author = {Chossat, P.},
Journal = {Dyn. Contin. Discr. Impul. Syst., Ser. A.},
Year = {2001},
Pages = {575--590},
Volume = {8}
}
@Article{Choss02,
Title = {The reduction of equivariant dynamics to the orbit space for compact group actions},
Author = {Chossat, P.},
Journal = {Acta Appl. Math.},
Year = {2002},
Pages = {71--94},
Volume = {70},
DOI = {10.1023/A:101397001420}
}
@Article{Choss93,
Title = {Forced reflectional symmetry breaking of an {O(2)}-symmetric homoclinic tangle},
Author = {Chossat, P.},
Journal = {Nonlinearity},
Year = {1993},
Pages = {723--731},
Volume = {6}
}
@Book{Chossat94,
Title = {The {Taylor-Couette} Problem},
Author = {P. Chossat and G. Iooss},
Publisher = {Springer},
Year = {1994},
Address = {New York}
}
@Book{ChossLaut00,
Title = {Methods in Equivariant Bifurcations and Dynamical Systems},
Author = {P. Chossat and R. Lauterbach},
Publisher = {World Scientific},
Year = {2000},
Address = {Singapore},
ISBN = {978-981-02-3828-5}
}
@Article{ChPeCa90,
author = {{Chong}, M.~S. and {Perry}, A.~E. and {Cantwell}, B.~J.},
title = {A general classification of three-dimensional flow fields},
journal = {Phys. Fluids},
year = {1990},
volume = {2},
pages = {765--777}
}
@Article{ChPo96,
Title = {Symbolic encoding in symplectic maps},
Author = {F. Christiansen and A. Politi},
Journal = {Nonlinearity},
Year = {1996},
Pages = {1623--1640},
Volume = {9}
}
@Article{ChPo97,
Title = {Guidelines for the constuction of a generating partition in the standard map},
Author = {F. Christiansen and A. Politi},
Journal = {Physica D},
Year = {1997},
Pages = {32--41},
Volume = {109}
}
@Article{chr95gen,
Title = {Generating partition for the standard map},
Author = {F. Christiansen and A. Politi},
Journal = {Phys. Rev. E},
Year = {1995},
Pages = {3811--3814},
Volume = {51}
}
@Unpublished{Christiansen96,
Title = {Hopf's last hope: spatiotemporal chaos in terms of unstable recurrent patterns},
Author = {F. Christiansen and P. Cvitanovi\'{c} and V. Putkaradze},
Note = {\arXiv{chao-dyn/9606016}},
Year = {1996}
}
@Article{Christiansen97,
author = {F. Christiansen and P. Cvitanovi{\'c} and V. Putkaradze},
title = {Hopf's last hope: {Spatiotemporal} chaos in terms of unstable recurrent patterns},
journal = {Nonlinearity},
year = {1997},
volume = {10},
pages = {55--70},
addendum = {\arXiv{chao-dyn/9606016}},
doi = {10.1088/0951-7715/10/1/004}
}
@Article{ChrLueOtt11,
author = {{Christov}, I. C. and {Lueptow}, R. M. and {Ottino}, J. M.},
title = {{Stretching and folding versus cutting and shuffling: {An} illustrated perspective on mixing and deformations of continua}},
journal = {Amer. J. Physics},
year = {2011},
volume = {79},
pages = {359--367},
note = {\arXiv{1010.2256}},
doi = {10.1119/1.3533213}
}
@Article{ChRu97,
Title = {Computing {Lyapunov} spectra with continuous {Gram-Schmidt} orthonormalization},
Author = {Christiansen, F. and Rugh, H. H.},
Journal = {Nonlinearity},
Year = {1997},
Pages = {1063--1072},
Volume = {10},
Abstract = {We present a continuous method for computing the Lyapunov spectrum
associated with a dynamical system specified by a set of differential
equations. We do this by introducing a stability parameter and augmenting
the dynamical system with an orthonormal k-dimensional frame and
a Lyapunov vector such that the frame is continuously Gram-Schmidt
orthonormalized and at most linear growth of the dynamical variables
is involved. We prove that the method is strongly stable when where
is the kth Lyapunov exponent in descending order and we show through
examples how the method is implemented. }
}
@Misc{CiDiPu14,
Title = {Existence of standing waves for the complex {Ginzburg-Landau} equation},
Author = {{Cipolatti}, R. and {Dickstein}, F. and {Puel}, J. P},
Note = {\arXiv{1404.6461}},
Year = {2014}
}
@Article{cima_algebraic_1997,
Title = {Algebraic properties of the {Lyapunov} and period constants},
Author = {A. Cima},
Journal = {Rocky Mountain J. of Math.},
Year = {1997},
Pages = {471--502},
Volume = {27},
ISSN = {0035-7596}
}
@Article{cima_dynamics_2006,
Title = {Dynamics of some rational discrete dynamical systems via invariants},
Author = {A. Cima},
Journal = {Int. J. Bifur. Chaos},
Year = {2006},
Pages = {631--646},
Volume = {16}
}
@Article{cima_number_2008,
Title = {On the number of critical periods for planar polynomial systems},
Author = {A. Cima},
Journal = {Nonlinear Anal.},
Year = {2008},
Pages = {1889--1903},
Volume = {69}
}
@Article{cima_periodic_2007,
Title = {Periodic orbits in complex {Abel} equations},
Author = {A. Cima and A. Gasull and F. Manosas},
Journal = {J. Diff. Eqn.},
Year = {2007},
Pages = {314--328},
Volume = {232}
}
@Article{cima_relation_1998,
Title = {On the relation between index and multiplicity},
Author = {A. Cima},
Journal = {J. London Mathematical Society},
Year = {1998},
Pages = {757--768},
Volume = {57},
ISSN = {0024-6107}
}
@Article{cima_studying_2008,
Title = {Studying discrete dynamical systems through differential equations},
Author = {A. Cima and A. Gasull and V. Manosa},
Journal = {J. Diff. Eqn.},
Year = {2008},
Pages = {630--648},
Volume = {244}
}
@Article{CLMM96,
Title = {Non-trivial collective behavior in extensively-chaotic dynamical systems: an update},
Author = {Chat{\'e}, H. and Lemaitre, A. and Marcq, P. and Manneville, P.},
Journal = {Physica A},
Year = {1996},
Pages = {447--457},
Volume = {224}
}
@Article{CM87tran,
Title = {Transition to Turbulence via spatiotemporal Intermittency},
Author = {H. Chat\'{e} and Manneville, P.},
Journal = {Phys. Rev. Lett.},
Year = {1987},
Pages = {112},
Volume = {58}
}
@Article{CMRSY10,
Title = {Amplitude-phase synchronization at the onset of permanent spatiotemporal chaos},
Author = {Chian, A. C. and Miranda, R. A. and Rempel, E. L. and Saiki, Y. and Yamada, M.},
Journal = {Phys. Rev. Lett.},
Year = {2010},
Pages = {254102},
Volume = {104},
Abstract = {Amplitude and phase synchronization due to multiscale interactions
in chaotic saddles at the onset of permanent spatiotemporal chaos
is analyzed using the Fourier-Lyapunov representation. By computing
the power-phase spectral entropy and the time-averaged power-phase
spectra, we show that the laminar (bursty) states in the on-off spatiotemporal
intermittency correspond, respectively, to the nonattracting coherent
structures with higher (lower) degrees of amplitude-phase synchronization
across spatial scales.}
}
@Misc{CNS,
Note = {Center for Nonlinear Science,\weblink{www.cns.gatech.edu}}
}
@Article{CoDoe92,
Title = {Energy dissipation in shear driven turbulence},
Author = {P. Constantin and C. R. Doering},
Journal = {Phys. Rev. Lett},
Year = {1992},
Pages = {1648--1651},
Volume = {69},
DOI = {10.1103/PhysRevLett.69.1648}
}
@Article{CoDoe95,
Title = {Variational bounds in dissipative systems},
Author = {P. Constantin and C. R. Doering},
Journal = {Physica D},
Year = {1995},
Pages = {221--228},
Volume = {82},
DOI = {10.1016/0167-2789(94)00237-K}
}
@Article{cohen72,
Title = {Non-linear saturation of the dissipative trapped-ion mode by mode coupling},
Author = {B. I. Cohen and J. A. Krommes and W. M. Tang and M. N. Rosenbluth},
Journal = {Nucl. Fusion},
Year = {1972},
Pages = {971--991},
Volume = {16}
}
@Article{CoKrTeRo76,
Title = {Non-linear saturation of the dissipative trapped ion mode by mode coupling},
Author = {B. I. Cohen and J. A. Krommers and W. M. Tang and M. N. Rosenbluth},
Journal = {Nuclear Fusion},
Year = {1976},
Pages = {971--992},
Volume = {16}
}
@Article{CoLe84,
Title = {Ergodic properties of the {Lozi} mappings},
Author = {P. Collet and Y. Levy},
Journal = {Commun. Math. Phys.},
Year = {1984},
Pages = {461},
Volume = {93}
}
@Book{CoLiSh96,
Title = {Ideals, Varieties and Algorithms},
Author = {D. A. Cox and J. B. Little and D. 0'Shea},
Publisher = {Springer},
Year = {1996},
Address = {New York}
}
@Article{collet80,
Title = {Universal Properties of Maps on an interval},
Author = {P. Collet and J.-P. Eckmann and O. E. Lanford},
Journal = {Commun. Math. Phys.},
Year = {1980},
Pages = {211},
Volume = {76}
}
@Article{Coma06,
Title = {Forced symmetry breaking from {SO(3)} to {SO(2)} for rotating waves on a sphere},
Author = {A. N. Comanici},
Journal = {Nonlinearity},
Year = {2006},
Pages = {999--1019},
Volume = {19},
DOI = {10.1088/0951-7715/19/5/001}
}
@Article{CoMu89,
Title = {Painlev\'e analysis and {Backlund} transformations in the {Kuramoto-Sivashinsky} equation},
Author = {R. Conte and M. Musette},
Journal = {J. Phys. A.},
Year = {1989},
Pages = {169--177},
Volume = {22}
}
@Article{condisfam1,
Title = {Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators},
Author = {J.-A. Sepulchre and R. Mackay},
Journal = {Nonlinearity},
Year = {1997},
Pages = {679--713},
Volume = {10}
}
@Book{constantin_integral_1989,
title = {Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations},
publisher = {Springer},
year = {1989},
author = {P. Constantin and C. Foias and B. Nicolaenko and R. Temam},
address = {New York},
doi = {10.1007/978-1-4612-3506-4},
isbn = {9780387967295}
}
@Article{Constantin97,
Title = {The elusive singularity},
Author = {P. Constantin},
Journal = {Proc. Natl. Acad. Sci. USA},
Year = {1997},
Pages = {12761},
Volume = {94}
}
@Book{contebook,
Title = {The {Painlev\'e} Property: One Century Later},
Author = {R. Conte},
Publisher = {Springer},
Year = {1999},
Address = {New York}
}
@Book{Cornwell97,
Title = {Group Theory in Physics: An introduction},
Author = {Cornwell, J. F.},
Publisher = {Academic},
Year = {1997},
Address = {New York}
}
@Article{CosGre13,
author = {A. B. Costa and J. R. Green},
title = {Extending the length and time scales of {Gram-Schmidt Lyapunov} vector computations},
journal = {J. Comp. Phys.},
year = {2013},
volume = {246},
pages = {113--122},
doi = {10.1016/j.jcp.2013.03.051},
abstract = {Abstract Lyapunov vectors have found growing interest recently due
to their ability to characterize systems out of thermodynamic equilibrium.
The computation of orthogonal Gram-Schmidt vectors requires multiplication
and \{QR\} decomposition of large matrices, which grow as N 2 (with
the particle count). This expense has limited such calculations to
relatively small systems and short time scales. Here, we detail two
implementations of an algorithm for computing Gram-Schmidt vectors.
The first is a distributed-memory message-passing method using Scalapack.
The second uses the newly-released \{MAGMA\} library for GPUs. We
compare the performance of both codes for Lennard-Jones fluids from
N = 100 to 1300 between Intel Nahalem/Infiniband \{DDR\} and \{NVIDIA\}
\{C2050\} architectures. To our best knowledge, these are the largest
systems for which the Gram-Schmidt Lyapunov vectors have been computed,
and the first time their calculation has been GPU-accelerated. We
conclude that Lyapunov vector calculations can be significantly extended
in length and time by leveraging the power of GPU-accelerated linear
algebra.}
}
@Book{Cotton08,
Title = {Chemical applications of group theory},
Author = {Cotton, F. A.},
Publisher = {Wiley},
Year = {2008},
Address = {New York}
}
@Book{CoVanLo88,
Title = {Handbook for Matrix Computations},
Author = {Coleman, T. F. and Van Loan, C.},
Publisher = {SIAM},
Year = {1988},
Address = {Philadelphia}
}
@Article{cox02jcomp,
author = {Cox, S. M. and Matthews, P. C.},
title = {Exponential time differencing for stiff systems},
journal = {J. Comput. Phys.},
year = {2002},
volume = {176},
pages = {430--455},
doi = {10.1006/jcph.2002.6995}
}
@Article{cpmani,
Title = {On the computation of invariant manifolds of fixed points},
Author = {A. J. Homburg and H. M. Osinga and G. Vegter},
Journal = {Z. Angew. Math. Phys.},
Year = {1995},
Pages = {171},
Volume = {46}
}
@Inproceedings{CR93,
author = {Cvitanovi\'c, P. and Rosenqvist, P. E.},
title = {A new determinant for quantum chaos},
booktitle = {From Classical to Quantum Chaos},
year = {1993},
editor = {G. F. Dell'Antonio, S. Fantoni and V. R. Manfredi},
volume = {41},
pages = {57--64},
address = {Bologna},
organization = {Soc. Italiana di Fisica Conf. Proceed.},
publisher = {Ed. Compositori}
}
@Article{crawford89,
Title = {Application of the method of spectral deformation to the {Vlasov-Poisson} system},
Author = {J. D. Crawford and P. D. Hislop},
Journal = {Ann. Phys.},
Year = {1989},
Pages = {265--317},
Volume = {189}
}
@Article{crawford91,
author = {J. D. Crawford},
title = {Introduction to bifurcation theory},
journal = { Rev. Modern Phys.},
year = {1991},
volume = {63},
pages = {991--1037}
}
@Article{crawford97,
author = {J. D. Crawford},
title = {Amplitude equations for electrostatic waves: {Universal} singular behavior in the limit of weak instability},
journal = {Phys. Plasmas},
year = {1995},
volume = {2},
pages = {97--128}
}
@Article{Creagh91,
Title = {Semiclassical trace formulas in the presence of continuous symmetries},
Author = {S. C. Creagh and R. G. Littlejohn},
Journal = {Phys. Rev. A},
Year = {1991},
Pages = {836--850},
Volume = {44},
Abstract = {A semiclassical expression for the symmetry-projected Green?s function
is obtained; it involves a sum over classical periodic orbits on
a symmetry-reduced phase space, weighted by characters of the symmetry
group. These periodic orbits correspond to trajectories on the full
phase space which are not necessarily periodic, but whose end points
are related by symmetry. Examples: the stadium billiard, a particle
in a periodic potential, the Sinai billiard, the quartic oscillator,
and the rotational spectrum of SF6.},
Numpages = {14}
}
@Article{Creagh92,
Title = {Semiclassical trace formulas for systems with non-Abelian symmetry},
Author = {S. C. Creagh and R. G. Littlejohn},
Journal = {J. Phys. A},
Year = {1992},
Pages = {1643--1669},
Volume = {25},
Abstract = {generalizations of the trace formula valid in the presence of a non-Abelian
continuous symmetry. The usual trace formula must be modified in
such cases because periodic orbits occur in continuous families,
whereas the usual trace formula requires that the periodic orbits
be isolated at a given energy. These calculations extend the results
of a previous paper, in which they considered Abelian continuous
symmetries. The most important application of the results is to systems
with full three-dimensional rotational symmetry, and they give this
case special consideration.}
}
@Article{Creagh93,
Title = {Semiclassical mechanics of symmetry reduction},
Author = {S. C. Creagh},
Journal = {J. Phys. A},
Year = {1993},
Pages = {95--118},
Volume = {26},
Abstract = {The author discusses semiclassical approximations that are adapted
to given symmetry classes in quantum mechanics. Arbitrary abelian
symmetries and also rotational symmetry are treated. Semiclassical
approximations are derived for the projected propagator and energy
dependent Green's function associated with a given irreducible representation
of the symmetry group. From these they derive trace formulae, analogous
to the usual trace formula, that determine the energy levels in a
given symmetry class in terms of classical orbits.}
}
@Article{Creagh94,
Title = {Quantum zeta function for perturbed cat maps},
Author = {S. C. Creagh},
Journal = {Chaos},
Year = {1995},
Pages = {477--493},
Volume = {5}
}
@Article{CriKnDeEsp12,
author = {Cristadoro, G. and Knight, G. and Degli Esposti, M.},
title = {Follow the fugitive: an application of the method of images to open systems},
journal = {J. Phys. A},
year = {2013},
volume = {46},
pages = {272001},
note = {\arXiv{1212.0673}},
doi = {10.1088/1751-8113/46/27/272001},
abstract = {Borrowing and extending the method of images we introduce a
theoretical framework that greatly simplifies analytical and numerical
investigations of the escape rate in open systems. As an example, we
explicitly derive the exact size- and position-dependent escape rate in
a Markov case for holes of finite-size . Moreover, a general relation
between the transfer operators of the closed and corresponding open
systems, together with the generating function of the probability of
return to the hole is derived. This relation is then used to compute
the small hole asymptotic behavior, in terms of readily calculable
quantities. As an example we derive logarithmic corrections in the
second order term. Being valid for Markov systems, our framework can
find application in many areas of the physical sciences such as
information theory, network theory, quantum Weyl law and, via Ulam{\textquoteright}s
method, can be used as an approximation method in general dynamical
systems.}
}
@Article{Cristad06,
Title = {Fractal diffusion coefficient from dynamical zeta functions},
Author = {G. Cristadoro},
Journal = {J. Phys. A},
Year = {2006},
Pages = {L151},
Volume = {39},
Abstract = {Dynamical zeta functions provide a powerful method to analyse low-dimensional
dynamical systems when the underlying symbolic dynamics is under
control. On the other hand, even simple one-dimensional maps can
show an intricate structure of the grammar rules that may lead to
a non-smooth dependence of global observables on parameters changes.
A paradigmatic example is the fractal diffusion coefficient arising
in a simple piecewise linear one-dimensional map of the real line.
Using the Baladi-Ruelle generalization of the Milnor-Thurnston kneading
determinant, we provide the exact dynamical zeta function for such
a map and compute the diffusion coefficient from its smallest zero.},
DOI = {10.1088/0305-4470/39/10/L01}
}
@Article{CRMRF02,
Title = {High-dimensional interior crisis in the {Kuramoto-Sivashinsky} equation},
Author = {A. {C.-L.} Chian and E. L. Rempel and E. E. Macau and R. R. Rosa and F. Christiansen},
Journal = {Phys. Rev. E},
Year = {2002},
Pages = {035203},
Volume = {65},
Abstract = {An investigation of interior crisis of high dimensions in an extended
spatiotemporal system exemplified by the {Kuramoto-Sivashinsky} equation
is reported. It is shown that unstable periodic orbits and their
associated invariant manifolds in the Poincar\'e hyperplane can effectively
characterize the global bifurcation dynamics of high-dimensional
systems.}
}
@Phdthesis{Crockett10,
Title = {Orbit Space Reduction for Symmetric Dynamical Systems with an Application to Laser Dynamics},
Author = {Crockett, V. J.},
School = {Univ. of Exeter},
Year = {2010},
Address = {Exeter, UK},
URL = {http://hdl.handle.net/10036/3310}
}
@Article{CroDav06,
author = {J. J. Crofts and R. L. Davidchack},
title = {Efficient detection of periodic orbits in chaotic systems by stabilizing transformations},
journal = {SIAM J. Sci. Comput.},
year = {2006},
volume = {28},
pages = {1275--1288}
}
@PhdThesis{Crofts07thesis,
author = {J. J. Crofts},
title = {Efficient method for detection of periodic orbits in chaotic maps and flows},
school = {Department of Mathematics, Univ of Leicester},
year = {2007},
address = {Leicester, UK},
note = {\arXiv{nlin.CD/0706.1940}}
}
@Article{cross93,
author = {Cross, M. C. and Hohenberg, P. C.},
title = {Pattern formation outside of equilibrium},
journal = {Rev. Mod. Phys.},
year = {1993},
volume = {65},
pages = {851--1112},
doi = {10.1103/RevModPhys.65.851}
}
@Article{cross94sp,
Title = {Spatiotemporal Chaos},
Author = {Cross, M. C. and Hohenberg, P. C.},
Journal = {Science},
Year = {1994},
Pages = {1569--1570},
Volume = {263},
Abstract = {The phenomena of spatiotermporal chaos are introduced based on both
rotating and regular {Rayleigh-B\'{e}nard} convection. The enormous number
of available degrees of freedom is a vivid feature. A set of interesting
questions have been raised, including how to characterize a spatiotemporal
chaotic state. The importance of the study of small subsystems is
emphasized.}
}
@Article{CSRBHK07,
Title = {Chaos in driven Alfv\'en systems: {Unstable} periodic orbits and chaotic saddles},
Author = {Chian, A. C.-L. and Santana, W. M. and Rempel, E. L. and Borotto, F. A. and Hada, T. and Kamide, Y.},
Journal = {Nonlin. Proc. Geophys.},
Year = {2007},
Pages = {17--29},
Volume = {14},
Abstract = { The chaotic dynamics of Alfv\'en waves in space plasmas governed
by the derivative nonlinear Schr\"odinger equation, in the low-dimensional
limit described by stationary spatial solutions, is studied. A bifurcation
diagram is constructed, by varying the driver amplitude, to identify
a number of nonlinear dynamical processes including saddle-node bifurcation,
boundary crisis, and interior crisis. The roles played by unstable
periodic orbits and chaotic saddles in these transitions are analyzed,
and the conversion from a chaotic saddle to a chaotic attractor in
these dynamical processes is demonstrated. In particular, the phenomenon
of gap-filling in the chaotic transition from weak chaos to strong
chaos via an interior crisis is investigated. A coupling unstable
periodic orbit created by an explosion, within the gaps of the chaotic
saddles embedded in a chaotic attractor following an interior crisis,
is found numerically. The gap-filling unstable periodic orbits are
responsible for coupling the banded chaotic saddle (BCS) to the surrounding
chaotic saddle (SCS), leading to crisis-induced intermittency. The
physical relevance of chaos for Alfv\'en intermittent turbulence
observed in the solar wind is discussed. },
DOI = {10.5194/npg-14-17-2007}
}
@Article{Curry78,
Title = {A generalized {Lorenz} system},
Author = {Curry, J. H.},
Journal = {Commun. Math. Phys.},
Year = {1978},
Pages = {193--204},
Volume = {60}
}
@Book{CushBat97,
Title = {Global Aspects of Classical Integrable Systems},
Author = {R. H. Cushman and L. M. Bates},
Publisher = {Birkh{\"{a}}user},
Year = {1997},
Address = {Boston}
}
@Article{CuSoAk02,
Title = {Experimental evidence for soliton explosions},
Author = {Cundiff, S. T. and Soto-Crespo, J. M. and Akhmediev, N.},
Journal = {Phys. Rev. Lett.},
Year = {2002},
Pages = {073903},
Volume = {88},
DOI = {10.1103/PhysRevLett.88.073903},
Numpages = {4}
}
@Article{Cutkosky84,
Title = {The {Gribov} horizon},
Author = {Cutkosky, R. E.},
Journal = {Phys. Rev. D},
Year = {1984},
Pages = {447--454},
Volume = {30},
DOI = {10.1103/PhysRevD.30.447}
}
@Article{CV93,
author = {P. Cvitanovi\'c and G. Vattay },
title = {Entire {Fredholm} determinants for evaluation of semiclassical and thermodynamical spectra},
journal = {Phys. Rev. Lett.},
year = {1993},
volume = {71},
pages = {4138--4141},
note = {\arXiv{chao-dyn/9307012}}
}
@Unpublished{CvGr12,
author = {Cvitanovi\'c, P. and Grigoriev, R. O.},
title = {Slicing a heart to keep it ticking: {Dreams Of Grand Schemes}},
note = {In preparation},
year = {2012}
}
@Unpublished{Cvi07,
Title = {Continuous symmetry reduced trace formulas},
Author = {P. Cvitanovi\'{c}},
Note = {Unpublished},
Year = {2007},
URL = {http://ChaosBook.org/~predrag/papers/Cvi07.pdf}
}
@Article{Cvi92chaos,
Title = {Periodic orbit theory in classical and quantum mechanics},
Author = {Cvitanovi\'c, P.},
Journal = {Chaos},
Year = {1992},
Pages = {1--4},
Volume = {2},
DOI = {10.1063/1.165921}
}
@Article{CviGib10,
Title = {Geometry of turbulence in wall-bounded shear flows: {Periodic} orbits},
Author = {Cvitanovi{\'c}, P. and Gibson, J. F.},
Journal = {Phys. Scr. T},
Year = {2010},
Pages = {014007},
Volume = {142}
}
@InProceedings{CviLip12,
Title = {Knowing when to stop: {How} noise frees us from determinism},
Author = {P. Cvitanovi\'c and D. Lippolis},
Booktitle = {Let's Face Chaos through Nonlinear Dynamics},
Year = {2012},
Address = {Melville, NY},
Editor = {M. Robnik and V. G. Romanovski},
Note = {\arXiv{1206.5506}},
Pages = {82--126},
Publisher = {American Institute of Physics},
DOI = {10.1063/1.4745574}
}
@Article{CviPik93,
author = {Cvitanovic, P. and Pikovsky, A.},
title = {Cycle expansion for power spectrum},
journal = {Proc. SPIE},
year = {1993},
volume = {2038},
pages = {290--298},
doi = {10.1117/12.162683},
abstract = {A cycle expansion method is applied to the calculation of a
power spectrum of chaotic one- dimensional maps. It is shown that the
broad-band part of the spectrum can be represented as a diffusion
constant of some auxiliary process, and this constant is then represented
in terms of periodic orbits. Accuracy of the method is also considered.}
}
@Article{CvitaEckardt,
Title = {Symmetry decomposition of chaotic dynamics},
Author = {Cvitanovi\'c, P. and Eckhardt, B.},
Journal = {Nonlinearity},
Year = {1993},
Note = {\arXiv{chao-dyn/9303016}},
Pages = {277--311},
Volume = {6},
DOI = {10.1088/0951-7715/6/2/008}
}
@Inproceedings{CvitLanCrete02,
author = {P. Cvitanovi{\'c} and Y. Lan},
title = {Turbulent fields and their recurrences},
booktitle = {Proceedings of 10\textsuperscript{th} International Workshop on Multiparticle Production: Correlations and Fluctuations in QCD},
year = {2003},
editor = {N. Antoniou},
pages = {313--325},
address = {Singapore},
publisher = {World Scientific},
note = {\arXiv{nlin.CD/0308006}}
}
@Incollection{CVW96,
Title = {Quantum fluids and classical determinants},
Author = {Cvitanovi{\'c}, P. and Vattay, G. and Wirzba, A.},
Booktitle = {Classical, Semiclassical and Quantum Dynamics in Atoms},
Publisher = {Springer},
Year = {1997},
Address = {New York},
Editor = {H. Friedrich and B. Eckhardt},
Note = {\arXiv{chao-dyn/9608012}},
Pages = {29--62}
}
@InProceedings{CvWiAv12,
Title = {Revealing the state space of turbulent pipe flow by symmetry reduction},
Author = {Cvitanovi{\'c}, P. and Willis, A. P. and Avila, M.},
Booktitle = {Proceed. ICTAM 2012 Intern. Congr. Theor. and Appl. Mech.},
Year = {2012},
Editor = {Jianxiang Wang}
}
@Article{DaGrSaYo94,
Title = {Obstructions to shadowing when a {Lyapunov} exponent is near zero},
Author = {S. P. Dawson and C. Grebogi and T. Sauer and J. A. Yorke},
Journal = {Phys. Rev. Lett.},
Year = {1994},
Pages = {1927},
Volume = {73}
}
@Article{Dahlqv94,
author = {P. Dahlqvist},
title = {Determination of resonance spectra for bound chaotic systems},
journal = {J. Phys. A},
year = {1994},
volume = {27},
pages = {763--785},
doi = {10.1088/0305-4470/27/3/020},
abstract = {We consider the computation of the eigenvalues of the evolution operator-the
resonance spectrum-by means of the zeros of a zeta function. In particular
we address the problems of applying this formalism to bound chaotic
systems, caused by e.g. intermittency and non-completeness of the
symbolic dynamics. For bound intermittent systems we derive an approximation
of the zeta function. With the aid of this zeta function it is argued
that bound systems with long time tails have branch cuts in the zeta
function and traces (of the evolution operator) approaching unity
as a power law. We also show that the dominant time scale can be
much longer than the period of the shortest periodic orbit, as is,
for example, the case for the hyperbola billiard. Isolated zeros
of the zeta function for the hyperbola billiard are evaluated by
means of a cycle expansion. Crucial for the success of this approach
is the identification of a sequence of periodic orbit responsible
for a logarithmic branch cut in the zeta function. Semiclassical
implications are briefly discussed.}
}
@Article{Dahlqvist95,
Title = {Approximate zeta functions for the {Sinai} billiard and related systems},
Author = {P. Dahlqvist},
Journal = {Nonlinearity},
Year = {1995},
Pages = {11},
Volume = {8},
Abstract = {We discuss zeta functions, and traces of the associated weighted evolution
operators for intermittent Hamiltonian systems in general and for
the Sinai billiard in particular. The intermittency of this billiard
is utilized so that the zeta functions may be approximately expressed
in terms of the probability distribution of laminar lengths. In particular
we study a one-parameter family of weights. Depending on the parameter
the trace can be dominated by branch cuts in the zeta function or
by isolated zeros. In the former case the time dependence of the
trace is dominated by a power law and in the latter case by an exponential.
A phase transition occurs when the leading zero collides with a branch
cut. The family considered is relevant for the calculation of resonance
spectra, semiclassical spectra and topological entropy.},
DOI = {10.1088/0951-7715/8/1/002}
}
@Article{DAlesPol90,
Title = {Hierarchical approach to complexity with applications to dynamical systems},
Author = {G. D'Alessandro and A. Politi},
Journal = {Phys. Rev. Lett.},
Year = {1990},
Pages = {1609},
Volume = {64}
}
@Article{DaMacKaSa1991,
author = {M.J. Davis and R.S. MacKay and A. Sannami},
title = {{Markov} shifts in the {H\'enon} family},
journal = {Physica D},
year = {1991},
volume = {52},
pages = {171--178},
doi = {10.1016/0167-2789(91)90119-T},
abstract = {We give evidence for parameter regions in which the set of bounded
orbits of the H\'enon map is a hyperbolic set on which the map is
equivalent to a subshift of finite type. }
}
@Article{DaMuPe88,
Title = {Resonances and diffusion in periodic {Hamiltonian} maps},
Author = {I. Dana and N. V. Murray and I. C. Percival},
Journal = {Phys. Rev. Lett.},
Year = {1988},
Pages = {233},
Volume = {62}
}
@Misc{Dana13,
Title = {Classical and quantum transport in one-dimensional periodically kicked systems},
Author = {Dana, I.},
Note = {\arXiv{1310.7854}, submitted to Canadian J. Chem.},
Year = {2013}
}
@Article{Dang86,
Title = {Steady-state mode interactions in the presence of {O(2)}-symmetry},
Author = {Dangelmayr, G.},
Journal = {Dyn. Sys.},
Year = {1986},
Pages = {159--185},
Volume = {1},
DOI = {10.1080/02681118608806011}
}
@Book{DasBuch,
title = {Chaos: {Classical and Quantum}},
publisher = {Niels Bohr Inst.},
year = {2016},
author = {P. Cvitanovi\'{c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
address = {Copenhagen},
url = {http://ChaosBook.org}
}
@Incollection{DasBuchMirror,
Author = {P. Cvitanovi{\'c}},
Booktitle = {{Chaos: Classical and Quantum}},
Publisher = {Niels Bohr Inst.},
Year = {2015},
Address = {Copenhagen},
Chapter = {{World} in a mirror},
URL = {http://ChaosBook.org/paper.shtml#discrete}
}
@Unpublished{Davidchack_priv,
author = {R. L. Davidchack},
title = {{Kuramoto-Sivashinsky $O(2)$} quotienting},
note = {Unpublished},
year = {2007}
}
@Article{davies2009,
Title = {Laser absorption by overdense plasmas in the relativistic regime},
Author = {Davies, J. R.},
Journal = {Plasma Phys. Control. Fusion},
Year = {2009},
Pages = {014006},
Volume = {51},
Abstract = {Laser absorption is reviewed in the context of fast ignition inertial
fusion. This leads us to consider laser absorption by overdense plasmas
for values of intensity times wavelength squared (I?2) greater than
1010 W (1018 W cm?2 ?m2), which corresponds to the onset of relativistic
electron motion in the laser fields. A collection of published absorption
values obtained in laser?solid experiments and in numerical modelling
of laser?plasma interactions is presented.},
DOI = {10.1088/0741-3335/51/1/014006}
}
@Article{DaVuDel00,
Title = {Spatial heteroclinic bifurcations of time periodic solutions to the {Ginzburg-Landau} equation},
Author = {H. Dang-Vu and C. Delcarte},
Year = {2000},
Pages = {459--472},
Volume = {57}
}
@Article{dawson_collections_1997,
Title = {Collections of heteroclinic cycles in the {Kuramoto-Sivashinsky} equation},
Author = {S. P. Dawson and A. M. Mancho},
Journal = {Physica D},
Year = {1997},
Pages = {231--256},
Volume = {100},
Abstract = {We study the {Kuramoto-Sivashinky} equation with periodic boundary
conditions in the case of low-dimensional behavior. We analyze the
bifurcations that occur in a six-dimensional {(6D)} approximation
of its inertial manifold. We mainly focus on the attracting and structurally
stable heteroclinic connections that arise for these parameter values.
We reanalyze the ones that were previously described via a {4D} reduction
to the center-unstable manifold {(Ambruster} et al., 1988, 1989).
We also find a parameter region for which a manifold of structurally
stable heteroclinic cycles exist. The existence of such a manifold
is responsible for an intermittent behavior which has some features
of unpredictability.}
}
@Misc{DCTSCD14,
author = {Ding, X. and Chat\'e, H. and Cvitanovi\'c, P. and Takeuchi, K. A. and Siminos, E.},
title = {Estimating the dimension of the inertial manifold from unstable periodic orbits},
year = {2016},
addendum = {In preparation}
}
@Article{DDF00,
author = {J.W. Demmel and L. Dieci and M.J. Friedman},
title = {Computing connecting orbits via an improved algorithm for continuing invariant spaces},
journal = {SIAM J. Sci. Comput.},
year = {2000},
volume = {22},
pages = {81--94}
}
@InProceedings{DDS90,
Title = {Automated pattern eduction from turbulent flow diagnostics},
Author = {D. D. Stretch},
Booktitle = {Annual Research Briefs},
Year = {1990},
Pages = {145--157},
Publisher = {Center for Turbulence Research, Stanford University}
}
@Article{DeArKo04,
Title = {Self-propulsion of {N}-hinged ``{Animats}'' at low {Reynolds} number},
Author = {De Ara\'ujo, G. A. and Koiller, J.},
Journal = {Qualitative Theory of Dynamical Systems},
Year = {2004},
Pages = {139--167},
Volume = {4}
}
@Unpublished{Dehaye05,
Title = {Averages over classical compact {L}ie groups and {W}eyl characters},
Author = {P.-O. Dehaye},
Year = {2005}
}
@Article{Deissler1994,
author = {R. J. Deissler and H. R. Brand},
title = {Periodic, quasiperiodic, and chaotic localized solutions of the quintic complex {Ginzburg-Landau} equation},
journal = {Phys. Rev. Lett.},
year = {1994},
volume = {72},
pages = {478--481}
}
@Article{DellAnZwan91,
author = {Dell'Antonio, G. and Zwanziger, D.},
title = {Every gauge orbit passes inside the {Gribov} horizon},
journal = {Commun. Math. Phys.},
year = {1991},
volume = {138},
pages = {291--299},
doi = {10.1007/BF02099494}
}
@Unpublished{deLMeAvHo12,
Title = {Experimental observation of the edge state in pipe flow},
Author = {A. de Lozar and F. Mellibovsky and M. Avila and B. Hof},
Note = {In preparation},
Year = {2012}
}
@Article{Dembo1982,
author = {R. S. Dembo and S. C. Eisenstat and T. Steihaug},
title = {Inexact {Newton} methods},
journal = {SIAM J. Numer. Anal.},
year = {1982},
volume = {19},
pages = {400--408}
}
@Unpublished{DemChaos,
author = {E. Demidov},
title = {Chaotic maps},
note = {www.ibiblio.org/e-notes},
year = {2009}
}
@Book{Demidovich1967,
Title = {Lectures on Stability Theory},
Author = {B. P. Demidovich},
Publisher = {Nauka},
Year = {1967},
Address = {Moscow},
Note = {in Russian}
}
@Article{DeNi79,
author = {Devaney, R. and Nitecki, Z.},
title = {Shift automorphisms in the {H\'enon} mapping},
journal = {Commun. Math. Phys.},
year = {1979},
volume = {67},
pages = {137--146},
doi = {10.1007/BF01221362}
}
@Book{Dennis96,
Title = {Numerical Methods for Unconstrained Optimization and Nonlinear Equations},
Author = {Dennis, J. E. and Schnabel, R. B.},
Publisher = {SIAM},
Year = {1996},
Address = {Philadelphia}
}
@Article{DePoHoo96,
author = {Dellago, C. and Posch, H. A. and Hoover, W. G.},
title = {{Lyapunov} instability in a system of hard disks in equilibrium and nonequilibrium steady states},
journal = {Phys. Rev. E},
year = {1996},
volume = {53},
pages = {1485}
}
@Article{DePSWP12,
Title = {Bifurcation control of a parametric pendulum},
Author = {De Paula, A. S. and Savi, M. A. and Wiercigroch, M.N and Pavlovskaia, E.},
Journal = {Int. J. Bifur. Chaos},
Year = {2012},
Pages = {1250111},
Volume = {22},
DOI = {10.1142/S0218127412501118}
}
@Article{DesBra13,
Title = {Quasi-one-dimensional solutions and their interaction with two-dimensional dissipative solitons},
Author = {Descalzi, O. and Brand, H. R.},
Journal = {Phys. Rev. E},
Year = {2013},
Pages = {022915},
Volume = {87},
DOI = {10.1103/PhysRevE.87.022915}
}
@Article{Descalzi10,
Title = {Transition from modulated to exploding dissipative solitons: {Hysteresis}, dynamics, and analytic aspects},
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Journal = {Phys. Rev. E},
Year = {2010},
Pages = {026203},
Volume = {82}
}
@Article{detect,
author = {Champneys, A. R. and Kuznetsov, Y. A.},
title = {Numerical detection and continuation of codimension-two homoclinic bifurcations},
journal = {Int. J. Bifur. Chaos},
year = {1994},
volume = {4},
pages = {785--822}
}
@Article{Dettm14,
author = {Dettmann, C. P.},
title = {Diffusion in the {Lorentz} gas},
journal = {Commun. Theor. Phys},
year = {2014},
volume = {62},
pages = {521--540},
note = {\arXiv{1402.7010}},
doi = {10.1088/0253-6102/62/4/10},
abstract = {The Lorentz gas, a point particle making mirror-like
reflections from an extended collection of scatterers, has been a useful
model of deterministic diffusion and related statistical properties for
over a century. This survey summarises recent results, including periodic
and aperiodic models, finite and infinite horizon, external fields,
smooth or polygonal obstacles, and in the Boltzmann - Grad limit. New
results are given for several moving particles and for obstacles with
flat points. Finally, a variety of applications are presented.}
}
@Book{devnmap,
Title = {An Introduction to Chaotic Dynamical systems},
Author = {R. L. Devaney},
Publisher = {Wesley},
Year = {1989},
Address = {Redwood City}
}
@Phdthesis{DeWitte13,
Title = {Computational analysis of bifurcations of periodic orbits},
Author = {De Witte, V.},
School = {Ghent University},
Year = {2013}
}
@Article{DFGHMS86,
Title = {Chaotic streamlines in the {ABC} flows},
Author = {Dombre, T. and Frisch, U. and Greene, J. M. and H\'enon, M. and Mehr, A. and Soward, A. M.},
Journal = {J. Fluid Mech.},
Year = {1986},
Pages = {353--391},
Volume = {167},
DOI = {10.1017/S0022112086002859}
}
@Article{DhaLai99,
Title = {Unstable periodic orbits and the natural measure of nonhyperbolic chaotic saddles},
Author = {Dhamala, M. and Lai, Y.-C.},
Journal = {Phys. Rev. E},
Year = {1999},
Pages = {6176--6179},
Volume = {60},
DOI = {10.1103/PhysRevE.60.6176}
}
@Article{DHBEks96,
Title = {Local models of spatio-temporally complex fields},
Author = {H. Dankowicz and P. Holmes and G. Berkooz and J. Elezgaray},
Journal = {Physica D},
Year = {1996},
Pages = {387--407},
Volume = {90}
}
@Article{DhGoKu03,
Title = {{MATCONT: A MATLAB} package for numerical bifurcation analysis of {ODEs}},
Author = {Dhooge, A. and Govaerts, W. and Kuznetsov, Yu. A.},
Journal = {ACM Trans. Math. Softw.},
Year = {2003},
Pages = {141--164},
Volume = {29},
DOI = {10.1145/779359.779362}
}
@Article{DhoPatZhi15,
Title = {The action of the special orthogonal group on planar vectors: integrity bases via a generalization of the symbolic interpretation of Molien functions},
Author = {G. Dhont and F. Patra and B. I. Zhilinski\'is},
Journal = {J. Phys. A},
Year = {2015},
Pages = {035201},
Volume = {48},
DOI = {10.1088/1751-8113/48/3/035201}
}
@Article{DhoZhi13,
Title = {The action of the orthogonal group on planar vectors: invariants, covariants and syzygies},
Author = {G. Dhont and B. I. Zhilinski\'i},
Journal = {J. Phys. A},
Year = {2013},
Pages = {455202},
Volume = {46},
DOI = {10.1088/1751-8113/46/45/455202}
}
@Book{Diacu96,
Title = {Celestial Encounters: The Origins of Chaos and Stability},
Author = {Diacu, F. and Holmes, P.},
Publisher = {Princeton Univ. Press},
Year = {1996},
Address = {Princeton, NJ}
}
@Article{DiEl08,
Title = {{SVD} algorithms to approximate spectra of dynamical systems},
Author = {Dieci, L. and Elia, C.},
Journal = {Math. Comput. Simul.},
Year = {2008},
Pages = {1235--1254},
Volume = {79}
}
@Article{diepenbeek_continuation_????,
Title = {Continuation and bifurcation of periodic orbits in symmetric {H}amiltonian systems.},
Author = {L. U. C. Diepenbeek and F. Dumortier and H. Broer and J. P. Gossez and J. Mawhin and Vanderbauwhede, A. and S. V. Lunel}
}
@Article{diepenbeek_persistence_????,
Title = {Persistence of {Hamiltonian} relative periodic orbits},
Author = {L. U. C. Diepenbeek and F. Dumortier and H. Broer and J. P. Gossez and J. Mawhin and Vanderbauwhede, A. and S. V. Lunel}
}
@Article{diepenbeek_relative_????,
Title = {Relative periodic orbits in symmetric Lagrangian systems},
Author = {L. U. C. Diepenbeek and F. Dumortier and H. Broer and J. P. Gossez and J. Mawhin and Vanderbauwhede, A. and S. V. Lunel}
}
@Misc{DingCvit14,
author = {Ding, X. and Cvitanovi{\'c}, P.},
title = {Periodic eigendecomposition and its application in {Kuramoto-Sivashinsky} system},
year = {2016},
note = {\arXiv{1406.4885}; SIAM J. Appl. Dyn. Syst., to appear}
}
@Article{DiRuVl97,
Title = {On the compuation of {Lyapunov} exponents for continuous dynamical systems},
Author = {Dieci, L. and Russell, R. D. and Van Vleck, E. S.},
Journal = {SIAM J. Numer. Anal.},
Year = {1997},
Pages = {402--423},
Volume = {34}
}
@Book{ditt01cq,
Title = {Classical and Quantum Dynamics: From Classical Paths to Path Integrals},
Author = {W. Dittrich and M. Reuter},
Publisher = {Springer},
Year = {2001},
Address = {New York}
}
@Article{DiVl02,
author = {Dieci, L. and Van Vleck, E. S.},
title = {{Lyapunov} Spectral Intervals: Theory and Computation},
journal = {SIAM J. Numer. Anal.},
year = {2002},
volume = {40},
pages = {516--542}
}
@Article{DJRV07,
author = {Dieci, L. and Jolly, M. S. and Rosa, R. and Van Vleck, E. S.},
title = {Error in approximation of {Lyapunov} exponents on inertial manifolds: {The Kuramoto-Sivashinsky} equation},
journal = {Discrete Contin. Dynam. Systems},
year = {2007},
volume = {9},
pages = {555--580}
}
@Article{DLBconv92,
Title = {Spatio-temporal intermittency in a 1{D} convective pattern: theoretical model and experiments},
Author = {F. Daviaud and J. Lega and P. Berg\'{e} and P. Coullet and M. Dubois},
Journal = {Physica D},
Year = {1992},
Pages = {287--308},
Volume = {55},
Abstract = {Describe the occurrence of the spatio-temporal intermittency in a
1-d convective system that shows time-independent patterns based
on the amplitude equation approach}
}
@Article{dlbd00,
Title = {Estimating generating partitions of chaotic systems by unstable periodic orbits},
Author = {R. L. Davidchack and Y-C Lai and E. M. Bollt and M. Dhamala},
Journal = {Phys. Rev. E},
Year = {2000},
Pages = {1353},
Volume = {61}
}
@Article{DM97,
author = {C. P. Dettmann and G. P. Morriss},
title = {Stability ordering of cycle expansions},
journal = {Phys. Rev. Lett.},
year = {1997},
volume = {78},
pages = {4201--4204},
doi = {10.1103/PhysRevLett.78.4201}
}
@Article{doeb94nls,
Title = {Properties of nonlinear Schr{\"{o}}dinger equations associated with diffeomorphism group representations},
Author = {H.-D. Doebner and G. A. Goldin},
Journal = {J. Phys. A},
Year = {1994},
Pages = {1771--1780},
Volume = {27}
}
@Article{doedel_computation_2003,
author = {Doedel, E. J. and R. C. Paffenroth and H. B. Keller and D. J. Dichmann and Galan, J. and Vanderbauwhede, A.},
title = {Computation of Periodic Solutions of Conservative Systems with Application to the 3\mbox{-}{B}ody Problem},
journal = {Int. J. Bifur. Chaos},
year = {2003},
volume = {13},
pages = {1353--1381},
abstract = {We show how to compute families of periodic solutions of conservative
systems with two-point boundary value problem continuation software.
The computations include detection of bifurcations and corresponding
branch switching. A simple example is used to illustrate the main
idea. Thereafter we compute families of periodic solutions of the
circular restricted 3-body problem. We also continue the figure-8
orbit recently discovered by Chenciner and Montgomery, and numerically
computed by Simu'o, as the mass of one of the bodies is allowed to
vary. In particular, we show how the invariances (phase-shift, scaling
law, and x, y, z translations and rotations) can be dealt with. Our
numerical results show, among other things, that there exists a continuous
path of periodic solutions from the figure-8 orbit to a periodic
solution of the restricted 3-body problem.}
}
@Article{doedel_elemental_2007,
author = {Doedel, E. J. and V. A. Romanov and R. C. Paffenroth and H. B. Keller and D. J. Dichmann and Galan, J. and Vanderbauwhede, A.},
title = {Elemental periodic orbits associated with the libration points in the circular restricted 3\mbox{-}{b}ody problem},
journal = {Int. J. Bifur. Chaos},
year = {2007},
volume = {17},
pages = {2625--2678}
}
@Article{doelman89,
Title = {Slow time-periodic solutions of the {Ginzburg-Landau} equation},
Author = {A. Doelman},
Journal = {Physica D},
Year = {1989},
Pages = {156--172},
Volume = {40}
}
@Article{doelman91n,
Title = {Finite-dimensional models of the {Ginzburg-Landau} equation},
Author = {A. Doelman},
Journal = {Nonlinearity},
Year = {1991},
Pages = {231--250},
Volume = {4}
}
@Article{doercgl87,
Title = {Exact {Lyapunov} dimension of the universal attractor for the complex {Ginzburg-Landau} equation},
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Journal = {Phys. Rev. Lett.},
Year = {1987},
Number = {26},
Pages = {2911--2914},
Volume = {59}
}
@Article{doercgl88,
Title = {Low-dimensional behaviour in the complex {Ginzburg-Landau} equation},
Author = {C. R. Doering and J. D. Gibbon and D. D. Holm and B. Nikolaenko},
Journal = {Nonlinearity},
Year = {1988},
Pages = {279--309},
Volume = {1},
Abstract = {The stability of plane waves of the CGLe is analyzed in a periodic
domain. The mass and energy integrals are estimated to give an upper
bound of the magnitude of the field variable. After that, cone condition
is established for the CGLe which indicates the existence of the
inertial manifold and proves the existence of finite dimensionality
of the attractor in this infinite-dimensional system. The Lyapunov
dimension is estimated. Note the defition of Lyapunov exponents which
is a little different from usual.}
}
@Article{DoLa14,
author = {C. Dong and Y. Lan},
title = {Organization of spatially periodic solutions of the steady {Kuramoto-Sivashinsky} equation},
journal = {Commun. Nonlinear Sci. Numer. Simul.},
year = {2014},
volume = {19},
pages = {2140--2153},
doi = {10.1016/j.cnsns.2013.09.040}
}
@Article{DoLa14a,
author = {C. Dong and Y. Lan},
title = {A variational approach to connecting orbits in nonlinear dynamical systems},
journal = {Phys. Lett. A},
year = {2014},
volume = {378},
pages = {705--712},
doi = {10.1016/j.physleta.2014.01.001},
}
@Article{Dolgopyat04,
author = {Dolgopyat, D.},
title = {On differentiability of {SRB} states for partially hyperbolic systems},
journal = {Invent. Math.},
year = {2004},
volume = {155},
pages = {389--449},
doi = {10.1007/s00222-003-0324-5}
}
@Article{Donaldson09c,
Title = {Vector symmetry reduction},
Author = {Donaldson, A. F.},
Journal = {Electronic Notes in Theoretical Computer Science},
Year = {2009},
Note = {{Proceedings of the Eighth International Workshop on Automated Verification of Critical Systems (AVoCS 2008)}},
Pages = {3--18},
Volume = {250},
DOI = {10.1016/j.entcs.2009.08.014}
}
@Incollection{Donnay88,
author = {Donnay, V. J.},
title = {Geodesic flow on the two-sphere, Part {II}: {Ergodicity}},
booktitle = {Dynamical Systems},
publisher = {Springer},
year = {1988},
editor = {Alexander, J. C.},
volume = {1342},
series = {Lect. Notes Math.},
pages = {112--153},
address = {New York},
doi = {10.1007/BFb0082827}
}
@Article{Donnay89,
Title = {Geodesic flow on the two-sphere, Part {I}: {Positive} measure entropy},
Author = {Donnay, V.},
Journal = {Ergod. Th. \& Dynam. Sys},
Year = {1989},
Pages = {531--553},
Volume = {8}
}
@Misc{dovidio08,
author = {d'Ovidio, F. and J. Isern-Fontanet and C. Lopez and E. Hernandez-Garcia and E. Garcia-Ladona},
title = {Comparison between {Eulerian} diagnostics and finite-size {Lyapunov} exponents computed from altimetry in the {Algerian} basin},
year = {2009},
doi = {10.1016/j.dsr.2008.07.014},
journal = {Deep Sea Research Part I: Oceanographic Research Papers },
pages = {15--31},
url = {http://arXiv.org/abs/0807.3848},
volume = {56}
}
@Article{DR_prl,
Title = {Existence of stable orbits in the {$x^{2}y^{2}$} potential},
Author = {P. Dahlqvist and G. Russberg},
Journal = {Phys. Rev. Lett.},
Year = {1990},
Pages = {2837--2838},
Volume = {65},
DOI = {10.1103/PhysRevLett.65.2837}
}
@Book{DR81,
Title = {Hydrodynamic Stability},
Author = {P. Drazin and W. H. Reid},
Publisher = {Cambridge Univ. Press},
Year = {1981},
Address = {Cambridge}
}
@Article{DR91a,
author = {P. Dahlqvist and G. Russberg },
title = {Periodic orbit quantization of bound chaotic systems},
journal = {J. Phys. A},
year = {1991},
volume = {24},
pages = {4763--4778},
doi = {10.1088/0305-4470/24/20/012}
}
@Article{Dragt05,
Title = {The symplectic group and classical mechanics},
Author = {Dragt, A. J.},
Journal = {Ann. New York Acad. Sci.},
Year = {2005},
Pages = {291--307},
Volume = {1045}
}
@Unpublished{Dragt11,
Title = {Lie Methods for Nonlinear Dynamics with Applications to Accelerator Physics},
Author = {A. J. Dragt},
Year = {2011},
Institution = {Physics Department, U. of Maryland},
URL = {http://www.physics.umd.edu/dsat/dsatliemethods.html}
}
@Techreport{Dragt11bad,
Title = {Lie methods for nonlinear dynamics with applications to accelerator physics},
Author = {Dragt, A. J.},
Institution = {{Physics Department, U. of Maryland}},
Year = {2011},
URL = {http://www.physics.umd.edu/dsat/dsatliemethods.html}
}
@Unpublished{DraHab08,
Title = {How {Wigner} functions transform under symplectic maps},
Author = {Dragt, A. J. and Habib, S.},
Note = {\arXiv{quant-ph/9806056}},
Year = {1998}
}
@Book{Dresselhaus07,
Title = {Group theory: application to the physics of condensed matter},
Author = {Dresselhaus, M. S. and Dresselhaus, G. and Jorio, A.},
Publisher = {Springer},
Year = {2007},
Address = {New York}
}
@Book{Driebe99,
Title = {Fully Chaotic Maps and Broken Time Symmetry},
Author = {Driebe, D.J.},
Publisher = {Springer},
Year = {1999},
Address = {New York},
ISBN = {0792355644}
}
@Article{dromey12,
Title = {Coherent synchrotron emission from electron nanobunches formed in relativistic laser-plasma interactions},
Author = {Dromey, B. and Rykovanov, S. and Yeung, M. and H\"orlein, R. and Jung, D. and Gautier, D. C. and Dzelzainis, T. and Kiefer, D. and Palaniyppan, S. and Shah, R. and Schreiber, J. and Ruhl, H. and Fernandez, J. C. and Lewis, C. L. S. and Zepf, M. and Hegelich, B. M.},
Journal = {Nat. Phys.},
Year = {2012},
Number = {11},
Pages = {804--808},
Volume = {8},
Abstract = {Extreme ultraviolet ({XUV)} and X-ray harmonic spectra produced by
intense laser?solid interactions have, so far, been consistent with
Doppler upshifted reflection from collective relativistic plasma
oscillations?the relativistically oscillating mirror mechanism. Recent
theoretical work, however, has identified a new interaction regime
in which dense electron nanobunches are formed at the plasma?vacuum
boundary resulting in coherent {XUV} radiation by coherent synchrotron
emission ({CSE).} Our experiments enable the isolation of {CSE} from
competing processes, demonstrating that electron nanobunch formation
does indeed occur. We observe spectra with the characteristic spectral
signature of {CSE?a} slow decay of intensity, I, with high-harmonic
order, n, as I(n) n?1.62 before a rapid efficiency rollover. Particle-in-cell
code simulations reveal how dense nanobunches of electrons are periodically
formed and accelerated during normal-incidence interactions with
ultrathin foils and result in {CSE} in the transmitted direction.
This observation of {CSE} presents a route to high-energy {XUV} pulses
and offers a new window on understanding ultrafast energy coupling
during intense laser?solid density interactions.},
DOI = {10.1038/nphys2439},
Language = {en}
}
@Article{DRSB12,
Title = {High-precision continuation of periodic orbits},
Author = {Dena, {\'A}. and Rodr{\'\i}guez, M. and Serrano, S. and Barrio, R.},
Journal = {Abst. Appl. Analysis},
Year = {2012},
Pages = {716024},
Volume = {2012},
DOI = {10.1155/2012/716024}
}
@Article{DruCip00,
Title = {Application of {Lie} algebras to visual servoing},
Author = {T. Drummond and R. Cipolla},
Journal = {Intern. J. Computer Vision},
Year = {2000},
Pages = {21--41},
Volume = {37}
}
@Article{DSM00,
Title = {Self-rotation number using the turning angle},
Author = {H. R. Dullin and D. Sterling and J. D. Meiss},
Journal = {Physica D},
Year = {2000},
Pages = {25--46},
Volume = {145}
}
@Article{DSWatkins,
author = {Watkins, D. S.},
title = {Francis's algorithm},
journal = {Amer. Math. Monthly},
year = {2011},
volume = {118},
pages = {387--403}
}
@Article{du_growth06,
Title = {Growth rates for fast kinematic dynamo instabilities of chaotic fluid flows},
Author = {Y. Du and E. Ott},
Journal = {J. Fluid Mech.},
Year = {2006},
Pages = {265--288},
Volume = {257},
DOI = {10.1017/S0022112093003076}
}
@Article{du87,
Title = {Effect of closed classical orbits on quantum spectra: {Ionization} of atoms in a magnetic field},
Author = {M. L. Du and J. B. Delos},
Journal = {Phys. Rev. Lett.},
Year = {1987},
Pages = {1731},
Volume = {58},
Abstract = {A quantitative theory has been developed which shows that each classical
closed orbit that begins and ends near the nucleus contributes an
oscillatory term to the average oscillator strength.}
}
@Article{duguet07,
author = {Duguet, Y. and Willis, A. P. and Kerswell, R. R.},
title = {Transition in pipe flow: the saddle structure on the boundary of turbulence},
journal = {J. Fluid Mech.},
year = {2008},
volume = {613},
pages = {255--274},
note = {\arXiv{0711.2175}}
}
@Article{duguet08,
Title = {Relative periodic orbits in transitional pipe flow},
Author = {Duguet, Y. and Pringle, C. C. T. and Kerswell, R. R.},
Journal = {Phys.\ Fluids},
Year = {2008},
Note = {\arXiv{0807.2580}},
Pages = {114102},
Volume = {20}
}
@Article{DuIbKo06,
author = {Dumortier, F. and Ibanez, S. and Kokubu, H.},
title = {Cocoon bifurcation in three-dimensional reversible vector fields},
journal = {Nonlinearity},
year = {2006},
volume = {19},
pages = {305--328},
doi = {10.1088/0951-7715/19/2/004}
}
@Article{DuiGui75,
author = {Duistermaat, J. J. and Guillemin, V. W.},
title = {The spectrum of positive elliptic operators and periodic bicharacteristics},
journal = {Invent. Math.},
year = {1975},
volume = {29},
pages = {39--79},
doi = {10.1007/BF01405172}
}
@Book{DuiKol00,
Title = {Lie Groups},
Author = {Duistermaat, J. J. and Kolk, J. A. C.},
Publisher = {Springer},
Year = {2000},
Address = {New York}
}
@Misc{DuisterDSsymm,
Title = {Dynamical Systems with Symmetry},
Author = {J. J. Duistermaat}
}
@Misc{dullin-2008,
Title = {Quadratic volume-preserving maps: {Invariant} circles and bifurcations},
Author = {H. R. Dullin and J. D. Meiss},
Note = {\arXiv{0807.0678}},
Year = {2008}
}
@Article{Dullin99,
Title = {Generic twistless bifurcations},
Author = {H. R. Dullin and J. D. Meiss and D. Sterling},
Journal = {Nonlinearity},
Year = {1999},
Pages = {203--224},
Volume = {13}
}
@Article{DuLoMe12,
author = {H. R. Dullin and H. E. Lomel{\'{i}} and J. D. Meiss},
title = {Symmetry reduction by lifting for maps},
journal = {Nonlinearity},
year = {2012},
volume = {25},
pages = {1709},
note = {\arXiv{1111.3887}},
doi = {10.1088/0951-7715/25/6/1709},
abstract = {We study diffeomorphisms that have one-parameter families of
continuous symmetries. For general maps, in contrast to the symplectic
case, the existence of a symmetry no longer implies the existence of an
invariant. Conversely, a map with an invariant need not have a symmetry.
We show that when a symmetry flow has a global Poincar{\'{e}} section there are
coordinates in which the map takes a reduced, skew-product form, and
hence allows for reduction of dimensionality. We show that the reduction
of a volume-preserving map again is volume preserving. Finally we sharpen
the Noether theorem for symplectic maps. A number of illustrative
examples are discussed and the method is compared with traditional
reduction techniques.}
}
@Article{Duval88,
Title = {Generation d'une section des classes de conjugaison et arbre des mots de {L}yndon de longueur born\'e},
Author = {J.-P. Duval},
Journal = {Theoret. Comput. Sci.},
Year = {1988},
Pages = {255--283},
Volume = {60}
}
@Article{DV00,
Title = {Random {Fibonacci} sequences and the number 1.13198824...},
Author = {Viswanath, D.},
Journal = {Math. Comput.},
Year = {2000},
Pages = {1131--1155},
Volume = {69}
}
@Article{DV01,
Title = {Global errors of numerical {ODE} solvers and {Lyapunov}'s theory of stability},
Author = {D. Viswanath},
Journal = {IMA J. Numer. Anal.},
Year = {2001},
Pages = {387--406},
Volume = {21}
}
@Article{DV02,
Title = {The {Lindstedt-Poincar\'e} technique as an algorithm for finding periodic orbits},
Author = {D. Viswanath},
Journal = {SIAM Review},
Year = {2002},
Pages = {478--496},
Volume = {43}
}
@Article{DV03,
Title = {Symbolic dynamics and periodic orbits of the {Lorenz} attractor},
Author = {D. Viswanath},
Journal = {Nonlinearity},
Year = {2003},
Pages = {1035--1056},
Volume = {16}
}
@Article{DV04,
Title = {The fractal property of the {Lorenz} attractor},
Author = {D. Viswanath},
Journal = {Physica D},
Year = {2004},
Pages = {115--128},
Volume = {190}
}
@Article{DV2000,
Title = {Phase space analysis of a dynamical model for the subcritical transition to turbulence in plane {Couette} flow},
Author = {O. Dauchot and N. Vioujard},
Journal = {European Physical J. B},
Year = {2000},
Pages = {377--381},
Volume = {14}
}
@Unpublished{DVLindstedt,
Title = {An Extension of the {Lindstedt-Poincar\'e} Algorithm for Computing Periodic Orbits},
Author = {D. Viswanath}
}
@Article{DWRGK13,
author = {De Witte, V. and Rossa, F. and Govaerts, W. and Kuznetsov, Yu. A.},
title = {Numerical periodic normalization for codim 2~{b}ifurcations of limit cycles: {Computational} formulas, numerical implementation, and examples},
journal = {SIAM J. Appl. Dyn. Syst.},
year = {2013},
volume = {12},
pages = {722--788},
doi = {10.1137/120874904}
}
@Article{EaPoSa10,
Title = {Group foliation of equations in geophysical fluid dynamics},
Author = {J. Early and J. Pohjanpelto and R. Samelson},
Journal = {Discrete Contin. Dynam. Systems},
Year = {2010},
Pages = {1571--1586},
Volume = {27}
}
@Article{EAS87,
Title = {Chaos: a mixed metaphor for turbulence},
Author = {E. A. Spiegel},
Journal = {Proc. Roy. Soc.},
Year = {1987},
Pages = {87},
Volume = {A413}
}
@Article{Eben02,
Title = {Growth and structure of stochastic sequences},
Author = {E. Ben-Naim and P. L. Krapivsky},
Journal = {J. Phys. A: Math. Gen.},
Year = {2002},
Pages = {L557--L563},
Volume = {35},
Abstract = {A class of stochastic integer sequences are introducd with every element
being a sum of two previous elements. The interplay between randomness
and memory leads to a wide variety of behaviors rangin from stretched
exponential to log-normal to algebraic growth. The set of all possible
sequence values has an intricate structure.}
}
@Article{eck81,
Title = {Roads to turbulence in dissipative dynamical systems},
Author = {J.-P. Eckmann},
Journal = {Rev. Mod. Phys.},
Year = {1981},
Pages = {643},
Volume = {53}
}
@Article{eckcgl,
author = {B. Janiaud and A. Pumir and D. Bensimon and V. Croquette and H. Richter and L. Kramer},
title = {The {E}ckhaus instability for travelling waves},
journal = {Physica D},
year = {1992},
volume = {55},
pages = {269--286}
}
@Article{EckGat00,
Title = {Hydrodynamic {Lyapunov modes} in translation-invariant systems},
Author = {Eckmann, J.-P. and Gat, O.},
Journal = {J. Stat. Phys.},
Year = {2000},
Pages = {775--798},
Volume = {98}
}
@Article{Eckhardt87,
Title = {Fractal properties of scattering singularities},
Author = {B. Eckhardt},
Journal = {J. Phys. A},
Year = {1987},
Pages = {5971},
Volume = {20},
Abstract = {The authors discuss properties of the set of scattering singularities
in regions of irregular scattering. The authors show how a symbolic
organisation of the set can be used to determine the fractal dimension
and the scaling function. These yields information on the distribution
of Lyapunov exponents of bounded orbits. The specific model studied
is the motion of a particle in a plane, elastically reflected by
three circular discs centred on the corners of an equilateral triangle.},
DOI = {10.1088/0305-4470/20/17/030}
}
@Article{eckhardt94,
Title = {Periodic orbit analysis of the {Lorenz} flow},
Author = {B. Eckhardt and G. Ott},
Journal = {Z. Phys. B},
Year = {1994},
Pages = {259--266},
Volume = {93}
}
@Article{EckHoPo89,
author = {B. Eckhardt, G. Hose and E. Pollak},
title = {Quantum mechanics of a classically chaotic system: Observations on scars, periodic orbits, and vibrational adiabaticity},
journal = {Phys. Rev. A},
year = {1989},
volume = {39},
pages = {3776--3793},
doi = {10.1103/PhysRevA.39.3776}
}
@Article{EckKamp86,
Title = {Liapunov exponents from time series},
Author = {Eckmann, J.-P. and Kamphorst, S. O. and Ruelle, D. and Ciliberto, S.},
Journal = {Phys. Rev. A},
Year = {1986},
Pages = {4971--4979},
Volume = {34},
DOI = {10.1103/PhysRevA.34.4971}
}
@Article{ecklp88,
Title = {Liapunov spectra for infinite chains of nonlinear oscillators},
Author = {J.-P. Eckmann and C. E. Wayne},
Journal = {J. Stat. Phys.},
Year = {1988},
Pages = {853--878},
Volume = {50}
}
@Book{eckman80map,
Title = {Iterated Maps on the Interval as Dynamical Systems},
Author = {J.-P. Eckmann and P. Collet},
Publisher = {Birkh{\"{a}}user},
Year = {1980},
Address = {Boston}
}
@Article{EckmannRuelle1985,
author = {J.-P. Eckmann and D. Ruelle},
title = {Ergodic theory of chaos and strange attractors},
journal = {Rev. Mod. Phys.},
year = {1985},
volume = {57},
pages = {617--656},
doi = {10.1103/RevModPhys.57.617}
}
@Article{EckWi90,
Title = {Symbolic description of periodic orbits for the quadratic {Zeeman} effect},
Author = {B. Eckhardt and D. Wintgen},
Journal = {J. Phys. B},
Year = {1990},
Pages = {355},
Volume = {23}
}
@Article{EDASRU14,
author = {Engl, T. and Dujardin, J. and Arg{\"u}elles, A. and
Schlagheck, P. and Richter, K. and Urbina, J. D.},
title = {Coherent backscattering in {Fock} space: {A} signature
of quantum many-body interference in interacting bosonic systems},
journal = {Phys. Rev. Lett.},
year = {2014},
volume = {112},
pages = {140403},
doi = {10.1103/PhysRevLett.112.140403}
}
@Misc{EDUR15,
author = {Engl, T. and Urbina, J. D. and Richter, K.},
title = {The semiclassical propagator in {Fock} space: dynamical echo and many-body interference},
year = {2015},
addendum = {\arXiv{1511.07234}}
}
@Book{Efst05,
Title = {Metamorphoses of {Hamiltonian} systems with symmetries},
Author = {K. Efstathiou},
Publisher = {Springer},
Year = {2005},
Address = {New York}
}
@Article{egolf00,
Title = {Mechanisms of extensive spatiotemporal chaos in {Rayleigh-B\'{e}nard} convection},
Author = {D. A. Egolf and I. V. Melnikov and W. Pesch and R. E. Ecke},
Journal = {Nature},
Year = {2000},
Pages = {733},
Volume = {404},
Abstract = {Numerical study on Boussinesq equation(pseudospectral method with
time splitting scheme). Demonstrate the extensivity of Lyapunov dimension
of the extended system and showed that the generating mechanism is
breaking or connecting rolls mediated by defects at the domain boundary
and exsisting for only a short time.}
}
@Article{egolf94,
Title = {Relation between fractal dimension and spatial correlation length for extensive chaos},
Author = {D. A. Egolf and H. S. Greenside},
Journal = {Nature},
Year = {1994},
Pages = {129},
Volume = {369},
Abstract = {Numerical study on CGLe. Show that even though the fractal dimension
increases with system size the simple scaling relation is not correct,
i.e. the correlation length can not be used to directly calculate
the fractal dimension.}
}
@Article{egolf98,
Title = {Importance of local pattern properties in spiral defect chaos},
Author = {D. A. Egolf and I. V. Melnikov and E. Bodenschatz},
Journal = {Phys. Rev. Lett.},
Year = {1998},
Pages = {3228},
Volume = {80}
}
@Article{EGtran95,
Title = {Characterization of the transition from defect to phase turbulence},
Author = {D. A. Egolf and H. S. Greenside},
Journal = {Phys. Rev. Lett.},
Year = {1995},
Pages = {1751},
Volume = {74},
Abtract = {For the CGLe on a large periodic interval, the transition from defect to phase turbulence is continuous. The conclusion is based on the calculation of the defect density, Lyapunov dimension density and correlation lengths.}
}
@Inbook{eigendensity,
Author = {R. Artuso and H. H. Rugh and P. Cvitanovi{\'c}},
Chapter = {{Why} does it work?},
Editor = {P. Cvitanovi{\'c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
Publisher = {Niels Bohr Inst.},
Year = {2015},
Address = {Copenhagen},
Booktitle = {{Chaos: Classical and Quantum}},
URL = {http://ChaosBook.org/paper.shtml#converg}
}
@Article{Eisenstat1994,
author = {S. C. Eisenstat and H. F. Walker },
title = {Globally convergent inexact {Newton} methods},
journal = {SIAM J. Optim. },
year = {1994},
volume = {4},
pages = {293--442}
}
@Article{EKR87,
Title = {Recurrence plots of dynamical systems},
Author = {Eckmann, J.-P. and Kamphorst, S. O. and Ruelle, D. },
Journal = {Europhys. Lett.},
Year = {1987},
Pages = {973},
Volume = {4},
DOI = {10.1209/0295-5075/4/9/004}
}
@Unpublished{EllGaHoRa09,
Title = {{Lagrange-Poincar\'e} field equations},
Author = {Ellis, D. C. P. and Gay-Balmaz, F. and Holm, D. D. and Ratiu, T. S.},
Note = {Submitted to J. Geometry and Physics, \arXiv{0910.0874}},
Year = {2009},
Abstract = {The Lagrange-Poincare equations of classical mechanics are cast into
a field theoretic context together with their associated constrained
variational principle. An integrability / reconstruction condition
is established that relates solutions of the original problem with
those of the reduced problem. The Kelvin-Noether theorem is formulated
in this context. Applications to the isoperimetric problem, the Skyrme
model for meson interaction, metamorphosis image dynamics, and molecular
strands illustrate various aspects of the theory.}
}
@Book{ELLIOTT,
Title = {Symmetry in Physics},
Author = {Elliott, J. P. and Dawber, P. G.},
Publisher = {MacMillan},
Year = {1979},
Address = {Surrey},
Volume = {2}
}
@Book{ellip,
Title = {Elliptic Functions and Applications},
Author = {D. F. Lawden},
Publisher = {Springer},
Year = {1989},
Address = {New York}
}
@Article{ElWu96,
author = {Elgin, J. and Wu, X.},
title = {Stability of cellular states of the {Kuramoto-Sivashinsky} equation},
journal = {SIAM J. Appl. Math.},
year = {1996},
volume = {56},
pages = {1621--1638},
doi = {10.1137/S0036139994263689}
}
@Article{ElXiDeMa95,
author = {Elder, K. R. and Xi, H. and Deans, M. and Gunton, J. D.},
title = {Spatiotemporal chaos in the damped {Kuramoto-Sivashinsky} equation},
journal = {AIP Conf. Proc.},
year = {1995},
volume = {342},
pages = {702-708},
doi = {10.1063/1.48763}
}
@Article{Elyutin04,
author = {Elyutin, P. V.},
title = {{Lyapunov} exponent for a gas of soft scatterers},
journal = {Phys. Lett. A },
year = {2004},
volume = {331},
pages = {153--156},
doi = {10.1016/j.physleta.2004.08.043}
}
@Article{EM99,
Title = {Transition to turbulence in a shear flow},
Author = {B. Eckhardt and A. Mersmann},
Journal = {Phys. Rev. E},
Year = {1999},
Pages = {509--517},
Volume = {60}
}
@Article{EmmRom89,
Title = {Orbifolds as configuration spaces of systems with gauge symmetries},
Author = {Emmrich, C. and Romer, H.},
Journal = {Commun. Math. Phys.},
Year = {1990},
Pages = {69},
Volume = {129},
DOI = {10.1007/BF02096779}
}
@Article{Enrique03,
Title = {On the dynamics of dominated splitting},
Author = {Pujals, E. R. and Sambarino, M.},
Journal = {Ann. Math.},
Year = {2009},
Pages = {675--740},
Volume = {169}
}
@Misc{EnUrRi15,
author = {Engl, T. and Urbina, J. D. and Richter, K.},
title = {Boson sampling as canonical transformation: {A} semiclassical approach in {Fock} space},
year = {2015},
addendum = {\arXiv{1502.07483}}
}
@Article{EPUR14,
author = {Engl, T. and Pl{\"o}ss, P. and Urbina, J. D. and Richter, K.},
title = {The semiclassical propagator in fermionic {Fock} space},
journal = {Theor. Chem. Acc.},
year = {2014},
volume = {133},
pages = {1563},
doi = {10.1007/s00214-014-1563-9}
}
@Article{ErshPot98,
author = {S. V. Ershov and A. B. Potapov},
title = {On the concept of stationary {Lyapunov} basis},
journal = {Physica D},
year = {1998},
volume = {118},
pages = {167--198}
}
@Article{ERTW91,
Title = {Semiclassical cycle expansion for the helium atom},
Author = {Ezra, G. S. and Richter, K. and Tanner, G. and Wintgen, D.},
Journal = {J. Phys. B},
Year = {1991},
Pages = {L413--L420},
Volume = {24}
}
@Article{esarey09,
Title = {Physics of laser-driven plasma-based electron accelerators},
Author = {Esarey, E. and Schroeder, C. B. and Leemans, W. P.},
Journal = {Rev. Mod. Phys.},
Year = {2009},
Pages = {1229--1285},
Volume = {81},
Abstract = {Laser-driven plasma-based accelerators, which are capable of supporting
fields in excess of 100 {GV}, are reviewed. This includes the laser
wakefield accelerator, the plasma beat wave accelerator, the self-modulated
laser wakefield accelerator, plasma waves driven by multiple laser
pulses, and highly nonlinear regimes. The properties of linear and
nonlinear plasma waves are discussed, as well as electron acceleration
in plasma waves. Methods for injecting and trapping plasma electrons
in plasma waves are also discussed. Limits to the electron energy
gain are summarized, including laser pulse diffraction, electron
dephasing, laser pulse energy depletion, and beam loading limitations.
The basic physics of laser pulse evolution in underdense plasmas
is also reviewed. This includes the propagation, self-focusing, and
guiding of laser pulses in uniform plasmas and with preformed density
channels. Instabilities relevant to intense short-pulse laser-plasma
interactions, such as Raman, self-modulation, and hose instabilities,
are discussed. Experiments demonstrating key physics, such as the
production of high-quality electron bunches at energies of 0.1 {GeV},
are summarized.},
DOI = {10.1103/RevModPhys.81.1229}
}
@Inproceedings{etc12,
author = {Cvitanovi{\'c}, P. and Gibson, J. F.},
title = {Geometry of state space in plane {C}ouette flow},
booktitle = {Advances in Turbulence XII},
year = {2009},
editor = {B. Eckhardt},
series = {Proc. 12\textsuperscript{th} EUROMECH Eur. Turb. Conf., Marburg},
pages = {75--78},
address = {Berlin},
publisher = {Springer}
}
@Article{expholm88,
Title = {Exponentially small splittings of separatrices with applications to {KAM} theory and degenerate bifurcations},
Author = {P. Holmes and J. Marsden and J. Scheurle},
Journal = {Contemp. Math.},
Year = {1988},
Pages = {213--244},
Volume = {81}
}
@Article{EyHaLe04,
Title = {Ruelle's linear response formula, ensemble adjoint schemes and {L\'evy} flights},
Author = {Eyink, G. L. and Haine, T. W. N. and Lea, D. J.},
Journal = {Nonlinearity},
Year = {2004},
Pages = {1867},
Volume = {17},
Abstract = {A traditional subject in statistical physics is the linear response
of a molecular dynamical system to changes in an external forcing
agency, e.g. the Ohmic response of an electrical conductor to an
applied electric field. For molecular systems the linear response
matrices, such as the electrical conductivity, can be represented
by Green-Kubo formulae as improper time-integrals of 2-time correlation
functions in the system. Recently, Ruelle has extended the Green-Kubo
formalism to describe the statistical, steady-state response of a
'sufficiently chaotic' nonlinear dynamical system to changes in its
parameters. This formalism potentially has a number of important
applications. For instance, in studies of global warming one wants
to calculate the response of climate-mean temperature to a change
in the atmospheric concentration of greenhouse gases. In general,
a climate sensitivity is defined as the linear response of a long-time
average to changes in external forces. We show that Ruelle's linear
response formula can be computed by an ensemble adjoint technique
and that this algorithm is equivalent to a more standard ensemble
adjoint method proposed by Lea, Allen and Haine to calculate climate
sensitivities. In a numerical implementation for the 3-variable,
chaotic Lorenz model it is shown that the two methods perform very
similarly. However, because of a power-law tail in the histogram
of adjoint gradients their sum over ensemble members becomes a L?vy
flight , and the central limit theorem breaks down. The law of large
numbers still holds and the ensemble-average converges to the desired
sensitivity, but only very slowly, as the number of samples is increased.
We discuss the implications of this example more generally for ensemble
adjoint techniques and for the important practical issue of calculating
climate sensitivities.},
DOI = {10.1088/0951-7715/17/5/016}
}
@Techreport{EZRide,
Title = {{EZRide}: {A} code for simulating spiral waves in a comoving frame of reference},
Author = {D. Barkley and V. N. Biktashev and A. J. Foulkes},
Institution = {U. Liverpool},
Year = {2010},
URL = {http://www.maths.liv.ac.uk/~vadim/software/EZRide/}
}
@Unpublished{Faddeev09,
Title = {Faddeev-Popov ghosts},
Author = {L. D. Faddeev},
Note = {\HREF{http://scholarpedia.org/article/Faddeev-Popov_ghosts} {Scholarpedia.org}},
Year = {2009},
DOI = {10.4249/scholarpedia.7389}
}
@Article{FaHa12,
Title = {Computing {Lagrangian} coherent structures from their variational theory},
Author = {M. Farazmand and G. Haller},
Journal = {Chaos},
Year = {2012},
Pages = {013128},
Volume = {22},
DOI = {10.1063/1.3690153},
Numpages = {12}
}
@Article{Fang94,
Title = {Dynamics for a two-dimensional antisymmetric map},
Author = {H.P. Fang},
Journal = {J. Phys. A},
Year = {1994},
Pages = {5187--5200},
Volume = {27}
}
@Article{FaOttYo83,
author = {J. D. Farmer and E. Ott and J. A. Yorke},
title = {The dimension of chaotic attractors},
journal = {Physica D},
year = {1983},
volume = {7},
pages = {153--180}
}
@Article{farmer87,
Title = {Predicting chaotic time series},
Author = {J. D. Farmer and J. J. Sidorowich},
Journal = {Phys. Rev. Lett.},
Year = {1987},
Pages = {845},
Volume = {59}
}
@Article{FaRoSj08,
Title = {Semi-classical approach for {Anosov} diffeomorphisms and {Ruelle} resonances},
Author = {Faure, F. and Roy, N. and Sj{\"o}strand, J.},
Journal = {Open Math. J.},
Year = {2008},
Note = {\arXiv{0802.1780}},
Pages = {35--81},
Volume = {1}
}
@Article{faure04,
Title = {A laser-plasma accelerator producing monoenergetic electron beams},
Author = {Faure, J. and Glinec, Y. and Pukhov, A. and Kiselev, S. and Gordienko, S. and Lefebvre, E. and Rousseau, J.-P. and Burgy, F. and Malka, V.},
Journal = {Nature},
Year = {2004},
Number = {7008},
Pages = {541--544},
Volume = {431},
Abstract = {Particle accelerators are used in a wide variety of fields, ranging
from medicine and biology to high-energy physics. The accelerating
fields in conventional accelerators are limited to a few tens of
{MeV} m-1, owing to material breakdown at the walls of the structure.
Thus, the production of energetic particle beams currently requires
large-scale accelerators and expensive infrastructures. Laser?plasma
accelerators have been proposed as a next generation of compact accelerators
because of the huge electric fields they can sustain ({\textgreater}100
{GeV} m-1). However, it has been difficult to use them efficiently
for applications because they have produced poor-quality particle
beams with large energy spreads, owing to a randomization of electrons
in phase space. Here we demonstrate that this randomization can be
suppressed and that the quality of the electron beams can be dramatically
enhanced. Within a length of 3 mm, the laser drives a plasma bubble
that traps and accelerates plasma electrons. The resulting electron
beam is extremely collimated and quasi-monoenergetic, with a high
charge of 0.5 {nC} at 170 {MeV.}},
DOI = {10.1038/nature02963}
}
@Article{Faure07,
author = {F. Faure},
title = {Prequantum chaos: {Resonances} of the prequantum cat map},
journal = {Phys. Rev. Lett.},
year = {1987},
volume = {1},
pages = {255--285},
doi = {10.3934/jmd.2007.1.255},
abstract = {Prequantum dynamics was introduced in the 70s by
Kostant, Souriau and Kirillov as an intermediate between
classical and quantum dynamics. In common with the classical
dynamics, prequantum dynamics transports functions on phase
space, but adds some phases which are important in quantum
interference effects. In the case of hyperbolic dynamical
systems, it is believed that the study of the prequantum
dynamics will give a better understanding of the quantum
interference effects for large time, and of their statistical
properties. We consider a linear hyperbolic map which generates
a chaotic dynamical system on
the torus. The dynamics is lifted to a prequantum fiber bundle.
This gives a unitary prequantum (partially hyperbolic) map. We
calculate its resonances and show that they are related to the
quantum eigenvalues. A remarkable consequence is that quantum
dynamics emerges from long-term behavior of prequantum
dynamics. We present trace formulas, and discuss perspectives
of this approach in the nonlinear case.}
}
@Article{Faure11,
Title = {Semiclassical origin of the spectral gap for transfer operators of a partially expanding map},
Author = {Faure, F.},
Journal = {Nonlinearity},
Year = {2011},
Note = {\arXiv{1206.0282}},
Pages = {1473},
Volume = {24}
}
@Unpublished{FauTsu12,
Title = {Prequantum transfer operator for symplectic {Anosov} diffeomorphism},
Author = {Faure, F. and Tsujii, M.},
Note = {\arXiv{1206.0282}},
Month = jun,
Year = {2012}
}
@Unpublished{FauTsu13,
Title = {Band structure of the {Ruelle} spectrum of contact {Anosov} flows},
Author = {Faure, F. and Tsujii, M.},
Note = {\arXiv{1301.5525}},
Year = {2013}
}
@Article{FE03,
Title = {Traveling waves in Pipe Flow},
Author = {H. Faisst and B. Eckhardt},
Journal = {Phys. Rev. Lett.},
Year = {2003},
Pages = {224502},
Volume = {91}
}
@Article{Feffer83,
Title = {The uncertainty principle},
Author = {C. L. Fefferman},
Journal = {Bull. Amer. Math. Soc.},
Year = {1983},
Pages = {129--206},
Volume = {9}
}
@Article{Fefferman2000,
Title = {Existence and smoothness of the {Navier-Stokes} equation},
Author = {Fefferman, C. L.},
Journal = {The Millennium Prize Problems},
Year = {2000},
Pages = {57--67}
}
@Article{feigen78,
Title = {Quantitative universality for a class of nonlinear transformations},
Author = {M. J. Feigenbaum},
Journal = {J. Stat. Phys.},
Year = {1978},
Pages = {25},
Volume = {19}
}
@Article{Feiste1994,
Title = {18 {GHZ} all-optical frequency locking and clock recovery using a self-pulsating two-section laser},
Author = {U. Feiste and D. J. As and A. Erhardt},
Journal = {IEEE Photon. Technol. Lett.},
Year = {1994},
Pages = {106--108},
Volume = {6}
}
@Article{FelsOlver98,
Title = {Moving coframes: {I. A} practical algorithm},
Author = {M. Fels and P. J. Olver},
Journal = {Acta Appl. Math.},
Year = {1998},
Pages = {161--213},
Volume = {51}
}
@Article{FelsOlver99,
Title = {Moving coframes: {II. Regularization} and theoretical foundations},
Author = {M. Fels and P. J. Olver},
Journal = {Acta Appl. Math.},
Year = {1999},
Pages = {127--208},
Volume = {55}
}
@Article{Feniche71,
author = {Fenichel, N.},
title = {Persistence and smoothness of invariant manifolds for flows},
journal = {Indiana Univ. Math. J.},
year = {1971},
volume = {21},
pages = {193--226}
}
@Misc{ffmSliceGitHub,
Title = {{FFM Slice}},
Author = {Budanur, N. B.},
Note = {\HREF{https://github.com/burakbudanur/ffmSlice} {GitHub.com/\-burakbudanur/\-ffmSlice}},
Year = {2014},
Institution = {Georgia Inst.\ of Technology},
URL = {https://github.com/burakbudanur/ffmSlice}
}
@Book{Field07,
Title = {Dynamics and Symmetry},
Author = {Field, M. J.},
Publisher = {Imperial College Press},
Year = {2007},
Address = {London}
}
@Article{Field70,
author = {M. J. Field},
title = {Equivariant dynamical systems},
journal = {Bull. Amer. Math. Soc.},
year = {1970},
volume = {76},
pages = {1314--1318}
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Year = {{1996}},
Pages = {{R8}},
Volume = {{120}},
Abstract = {{This work comprises a general study of symmetry breaking for compact
Lie groups in the context of equivariant bifurcation theory. We start
by extending the theory developed by Field and Richardson {[}37]
for absolutely irreducible representations of finite groups to general
irreducible representations of compact Lie groups. In particular,
we allow for branches of relative equilibria and phenomena such as
the Hopf bifurcation. We also present a general theory of determinacy
for irreducible Lie group actions along the lines previously described
in Field {[}29]. In the main result of this work, we show that branching
patterns for generic equivariant bifurcation problems defined on
irreducible representations persist under perturbations by sufficiently
high order non-equivariant terms. We give applications of this result
to normal form computations yielding, for example, equivariant Hopf
bifurcations and show how normal form computations of branching and
stabilities are valid when we take account of the non-normalized
tail.}}
}
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doi = {10.1016/0022-0396(83)90011-6},
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attention is paid to the idea that if the Jacobian determinant of a map
is greater than one and a ball is mapped into itself, then generically,
the attractor will have positive two-dimensional measure, and most of
this paper is devoted to presenting cases with such Jacobians for which
the attractors are proved to have non-empty interior. }
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Title = {Matrix quantization of turbulence},
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DOI = {10.1063/1.3517079},
Numpages = {13}
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vertical. Tracers in this flow follow chaotic trajectories composed
of correlated L\'evy flights with varying velocities. Locations of
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of tracers is divided into transient and long-term regimes, each
with different growth exponents.}
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Year = {1988},
Pages = {197},
Volume = {67}
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@Article{FOP15,
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Journal = {Phys. Rev. E},
Year = {2010},
Note = {\arXiv{1001.4454}},
Pages = {046702},
Volume = {81},
Abstract = {We describe an approach to numerical simulation of spiral waves dynamics
of large spatial extent, using small computational grids.}
}
@Unpublished{FouBik10a,
Title = {Riding a spiral wave: {Numerical} simulation of spiral waves in a co-moving frame of reference},
Author = {A. J. Foulkes and V. N. Biktashev},
Note = {\arXiv{1001.4454}},
Year = {2010}
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journal = {Physica D},
year = {1982},
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pages = {139--163}
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note = {\arXiv{nlin/0408056}},
year = {2004}
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pages = {2074--2084},
note = {\arXiv{1101.3037}},
doi = {10.1016/j.cnsns.2011.07.007}
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year = {1993},
volume = {3},
pages = {619--636},
note = {\arXiv{chao-dyn/9307014}},
doi = {10.1063/1.165992}
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@Phdthesis{FreddyThesis,
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School = {Copenhagen Univ.},
Year = {1992},
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Journal = {Physica D},
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Note = {\arXiv{1204.0871}},
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Volume = {247},
DOI = {10.1016/j.physd.2012.12.005}
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Pages = {103103},
Volume = {20},
Abstract = {In this paper, we investigate the saturation mechanisms of backward stimulated Raman scattering ({BSRS)} induced by nonlinear kinetic effects. In particular, we stress the importance of accounting for both the nonlinear frequency shift of the electron plasma wave and the growth of sidebands, in order to understand what stops the coherent growth of Raman scattering. Using a Bernstein-Greene-Kruskal approach, we provide an estimate for the maximum amplitude reached by a {BSRS-driven} plasma wave after the phase of monotonic growth. This estimate is in very good agreement with the results from kinetic simulations of stimulated Raman scattering using both a Vlasov and a Particle in Cell code. Our analysis, which may be generalized to a multidimensional geometry, should provide a means to estimate the limits of backward Raman amplification or the effectiveness of strategies that aim at strongly reducing Raman reflectivity in a fusion plasma.},
DOI = {10.1063/1.4823714}
}
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Author = {Froehlich, S.},
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Year = {2010},
Chapter = {Reducing continuous symmetries with linear slice},
URL = {http://ChaosBook.org/projects/index.shtml#Froehlich}
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organization = {IEEE}
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Year = {2009},
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Abstract = {The general public has been made aware of the research field of "chaos"
by the book of that title by James Gleick. This paper will focus
on the achievements of Sonya Kovalevskaya, Mary Cartwright, and Mary
Tsingou, whose pioneer works were not mentioned in Gleick's book.}
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year = {1992},
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@Inbook{Gaspard92a,
Author = {Gaspard, P. },
Chapter = {From dynamical chaos to diffusion},
Editor = {Gy{\"o}rgyi, G. and Kondor, I. and Sasv{\'a}ri, L. and T{\'e}l, T.},
Pages = {322--334},
Publisher = {World Scientific},
Year = {1992},
Address = {Singapore},
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title = {Semiclassical quantization of the scattering from a classically chaotic repellor},
journal = {J. Chem. Phys.},
year = {1989},
volume = {90},
pages = {2242--2254},
doi = {10.1063/1.456018}
}
@Article{GaspRice89c,
author = {P. Gaspard and S. A. Rice},
title = {Exact quantization of the scattering from a classically chaotic repellor},
journal = {J. Chem. Phys.},
year = {1989},
volume = {90},
pages = {2255--2262},
doi = {10.1063/1.456019}
}
@Article{gasull_stability_2008,
Title = {On the stability of periodic orbits for differential systems in $\mathbb{R}^n$},
Author = {A. Gasull and H. Giacomini and M. Grau},
Journal = {Discrete Contin. Dynam. Systems},
Year = {2008},
Note = {\arXiv{math.DS/0610151}},
Pages = { 495 -- 509 },
Volume = {8}
}
@Book{gatermannHab,
title = {Computer Algebra Methods for Equivariant Dynamical Systems},
publisher = {Springer},
year = {2000},
author = {Gatermann, K.},
isbn = {9783540671619},
address = {New York}
}
@Article{GatGuy99,
author = {K. Gatermann and F. Guyard},
title = {Gr\"obner bases, invariant theory and equivariant dynamics},
journal = {J. Symbolic Comp.},
year = {1999},
volume = {28},
pages = {275--302},
doi = {10.1006/jsco.1998.0277}
}
@Book{gauge,
Title = {Chaos and Gauge Field Theory},
Author = {T. S. Bir\'{o} and S. G. Matinyan and B. M{\"{u}}ller},
Publisher = {World Scientific},
Year = {1994},
Address = {Singapore}
}
@Article{GaXiLa11,
author = {{Gao}, A. and {Xie}, J. and {Lan}, Y.},
title = {Accelerating cycle expansions by dynamical conjugacy},
journal = {J. Stat. Phys.},
year = {2012},
volume = {146},
pages = {56--66},
note = {\arXiv{1106.1045}},
doi = {10.1007/s10955-011-0369-6}
}
@Unpublished{GBHMRV10,
Title = {Invariant higher-order variational problems},
Author = {{Gay-Balmaz}, F. and {Holm}, D.~D. and {Meier}, D.~M. and {Ratiu}, T.~S. and {Vialard}, {F.-X.}},
Note = {\arXiv{1012.5060}},
Year = {2010}
}
@Article{GDTR08,
Title = {Open-flow mixing: {Experimental} evidence for strange eigenmodes},
Author = {E. Gouillart and O. Dauchot and {J.-L.} Thiffeault and S. Roux},
Journal = {Phys. Fluids},
Year = {2009},
Note = {\arXiv{0807.1723}},
Pages = {023603},
Volume = {21},
Abstract = {We investigate experimentally the mixing dynamics in a channel flow
with a finite stirring region undergoing chaotic advection. We study
the homogenization of dye in two variants of an eggbeater stirring
protocol that differ in the extent of their mixing region. In the
first case, the mixing region is separated from the side walls of
the channel, while in the second it extends to the walls. For the
first case, we observe the onset of a permanent concentration pattern
that repeats over time with decaying intensity. A quantitative analysis
of the concentration field of dye confirms the convergence to a self-similar
pattern, akin to the strange eigenmodes previously observed in closed
flows. We model this phenomenon using an idealized map, where an
analysis of the mixing dynamics explains the convergence to an eigenmode.
In contrast, for the second case the presence of no-slip walls and
separation points on the frontier of the mixing region leads to non-self-similar
mixing dynamics.}
}
@Article{geddes04,
Title = {High-quality electron beams from a laser wakefield accelerator using plasma-channel guiding},
Author = {Geddes, C. G. R. and Toth, Cs and van Tilborg, J. and Esarey, E. and Schroeder, C. B. and Bruhwiler, D. and Nieter, C. and Cary, J. and Leemans, W. P.},
Journal = {Nature},
Year = {2004},
Pages = {538--541},
Volume = {431},
Abstract = {Laser-driven accelerators, in which particles are accelerated by the
electric field of a plasma wave (the wakefield) driven by an intense
laser, have demonstrated accelerating electric fields of hundreds
of {GV} m-1 (refs 1?3). These fields are thousands of times greater
than those achievable in conventional radio-frequency accelerators,
spurring interest in laser accelerators as compact next-generation
sources of energetic electrons and radiation. To date, however, acceleration
distances have been severely limited by the lack of a controllable
method for extending the propagation distance of the focused laser
pulse. The ensuing short acceleration distance results in low-energy
beams with 100 per cent electron energy spread, which limits potential
applications. Here we demonstrate a laser accelerator that produces
electron beams with an energy spread of a few per cent, low emittance
and increased energy (more than 109 electrons above 80 {MeV).} Our
technique involves the use of a preformed plasma density channel
to guide a relativistically intense laser, resulting in a longer
propagation distance. The results open the way for compact and tunable
high-brightness sources of electrons and radiation.},
DOI = {10.1038/nature02900}
}
@Unpublished{GeDeAl12,
Title = {Faster than expected escape for a class of fully chaotic maps},
Author = {Georgiou, O. and Dettmann, C. P. and Altmann, E. G.},
Note = {\arXiv{1207.7000}},
Year = {2012}
}
@Article{Geist01051990,
author = {Geist, K. and Parlitz, U. and Lauterborn, W.},
title = {Comparison of different methods for computing {Lyapunov} exponents},
journal = {Progr. Theor. Phys.},
year = {1990},
volume = {83},
pages = {875--893},
doi = {10.1143/PTP.83.875}
}
@Unpublished{GeOgSk11,
Title = {Efficient integration of the variational equations of multi-dimensional {Hamiltonian} systems: {Application} to the {Fermi-Pasta-Ulam} lattice},
Author = {Gerlach, E. and Eggl, S. and {Skokos}, C. },
Note = {\arXiv{1104.3127}},
Year = {2011}
}
@Book{Georgi99,
Title = {{Lie} Algebras in Particle Physics},
Author = {H. Georgi},
Publisher = {Perseus Books},
Year = {1999},
Address = {Reading, MA}
}
@Article{GeZe84,
Title = {Models of representations of classical groups and their hidden symmetries},
Author = {Gel'fand, I. M. and Zelevinskii, A. V.},
Journal = {Funct. Anal. Appl.},
Year = {1984},
Pages = {183--198},
Volume = {18}
}
@Unpublished{GGJLF10,
Title = {Connecting curves for dynamical systems},
Author = {R. Gilmore and J.-M. Ginoux and T. Jones and C. Letellier and U. S. Freitas},
Note = {\arXiv{1003.1703}},
Year = {2010}
}
@Article{GGLTL09,
Title = {Detecting unstable periodic orbits of nonlinear mappings by a novel quantum-behaved particle swarm optimization non-{Lyapunov} way},
Author = {F. Gao and H. Gao and Zh. Li and H. Tong and J.-J. Lee},
Journal = {Chaos Solit. Fract.},
Year = {2009},
Pages = {2450--2463},
Volume = {42},
Abstract = {... detecting UPOs of nonlinear map is one of the most challenging
problems of nonlinear science in both numerical computations and
experimental measures. In this paper, a new method is proposed to
detect the UPOs in a non-Lyapunov way. Firstly three special techniques
are added to quantum-behaved particle swarm optimization (QPSO),
a novel mbest particle, contracting the searching space self-adaptively
and boundaries restriction (NCB), then the new method NCB--QPSO is
proposed. It can maintain an effective search mechanism with fine
equilibrium between exploitation and exploration. Secondly, the problems
of detecting the UPOs are converted into a non-negative functions'
minimization through a proper translation in a non-Lyapunov way.
Thirdly the simulations to 6 benchmark optimization problems and
different high order UPOs of 5 classic nonlinear maps are done by
the proposed method. And the results show that NCB--QPSO is a successful
method in detecting the UPOs, and it has the advantages of fast convergence,
high precision and robustness.},
DOI = {10.1016/j.chaos.2009.03.119}
}
@Article{GHCV08,
Title = {Heteroclinic connections in plane {Couette} flow},
Author = {Halcrow, J. and Gibson, J. F. and Cvitanovi{\'c}, P. and Viswanath, D.},
Journal = {J. Fluid Mech.},
Year = {2009},
Note = {\arXiv{0808.1865}},
Pages = {365--376},
Volume = {621}
}
@Article{GHCW07,
Title = {Visualizing the geometry of state-space in plane {Couette} flow},
Author = {J. F. Gibson and J. Halcrow and P. Cvitanovi{\'c}},
Journal = {J. Fluid Mech.},
Year = {2008},
Note = {\arXiv{0705.3957}},
Pages = {107--130},
Volume = {611},
DOI = {10.1017/S002211200800267X}
}
@Article{GiacoOtto05,
Title = {New bounds for the {Kuramoto-Sivashinsky} equation},
Author = {L. Giacomelli and F. Otto},
Journal = {Commun. Pure Appl. Math.},
Year = {2005},
Pages = {297--318},
Volume = {58}
}
@Article{GibMcCLE82,
Title = {The real and complex {Lorenz} equations in rotating fluids and lasers},
Author = {Gibbon, J. D. and McGuinness, M. J.},
Journal = {Physica D},
Year = {1982},
Pages = {108--122},
Volume = {5}
}
@Article{GibMcCLE83,
Title = {The real and complex {Lorenz} equations},
Author = {Gibbon, J. D. and McGuinness, M. J.},
Journal = {Proc. R. Irish. Acad., Ser. A},
Year = {1983},
Pages = {17--22},
Volume = {83}
}
@Book{GiBo,
Title = {Quantum chaos: Between order and disorder},
Author = {G. Casati and B. V. Chirikov},
Publisher = {Cambridge Univ. Press},
Year = {1995},
Address = {Cambridge}
}
@Techreport{GibsonMovies,
Title = {Movies of plane {Couette}},
Author = {Gibson, J. F. and Cvitanovi{\'c}, P.},
Institution = {Georgia Inst. of Technology},
Year = {2015},
URL = {http://ChaosBook.org/tutorials}
}
@Phdthesis{GibsonPhD,
Title = {Dynamical-systems Models of Wall-bounded, Shear-flow Turbulence},
Author = {J. F. Gibson},
School = {Cornell Univ.},
Year = {2002}
}
@Article{GiChLiPo12,
author = {Ginelli, F. and Chat\'{e}, H. and Livi, R. and Politi, A.},
title = {Covariant {Lyapunov} vectors},
journal = {J. Phys. A},
year = {2013},
volume = {46},
pages = {254005},
note = {\arXiv{1212.3961}},
doi = {10.1088/1751-8113/46/25/254005}
}
@Article{GiKaChPoAl11,
Title = {Chaos in the {Hamiltonian} mean-field model},
Author = {Ginelli, F. and Takeuchi, K. A. and Chat\'{e}, H. and Politi, A. and Torcini A.},
Journal = {Phys. Rev. E},
Year = {2011},
Note = {\arXiv{1109.6452}},
Pages = {066211},
Volume = {84}
}
@Unpublished{GiKuLeTa11,
Title = {Simulating rare events in dynamical processes},
Author = {{Giardina}, C. and {Kurchan}, J. and {Lecomte}, V. and {Tailleur}, J.},
Note = {\arXiv{1106.4929}},
Year = {2011}
}
@Article{GilbLef08,
Title = {Heat conductivity from molecular chaos hypothesis in locally confined billiard systems},
Author = {Gilbert, T. and Lefevere, R.},
Journal = {Phys. Rev. Lett.},
Year = {2008},
Note = {\arXiv{0808.1179}},
Pages = {200601},
Volume = {101},
Abstract = {We discuss the transport properties of a class of Hamiltonian
dynamics with local confinement, in which interactions between
neighboring particles occur through hard core elastic
collisions. Such dynamics may be described as high-dimensional
billiards. We consider the case where the collisions are rare
and, for large systems, derive a Boltzmann-like equation for
the evolution of the probability densities. We solve this
equation in the linear regime and compute the heat conductivity
in the approximate stationary state and with the help of the
Green-Kubo formula. We demonstrate the validity of the
molecular chaos hypothesis by comparing our theoretical
predictions to the results of numerical simulations performed
on a new class of models, which are defocusing chaotic
billiards, likened to higher-dimensional stadia.},
DOI = {10.1103/PhysRevLett.101.200601}
}
@Book{Gill82,
Title = {Atmosphere-ocean dynamics},
Author = {A. E. Gill},
Publisher = {Academic},
Year = {1982},
Address = {London}
}
@Book{gilmore08,
Title = {Lie Groups, Physics, and Geometry},
Author = {R. Gilmore},
Publisher = {Cambridge Univ. Press},
Year = {2008},
Address = {Cambridge}
}
@Book{gilmore2003,
Title = {The Topology of Chaos},
Author = {R. Gilmore and M. Lefranc},
Publisher = {Wiley},
Year = {2003},
Address = {New York}
}
@Book{gilmore74,
Title = {Lie Groups, {L}ie Algebras, and some of their Applications},
Author = {R. Gilmore},
Publisher = {Wiley},
Year = {1974},
Address = {New York}
}
@Article{ginelli-2007-99,
author = {Ginelli, F. and Poggi, P. and Turchi, A. and Chat\'e, H. and Livi, R. and Politi, A.},
title = {Characterizing dynamics with covariant {Lyapunov} vectors},
journal = {Phys. Rev. Lett.},
year = {2007},
volume = {99},
pages = {130601},
note = {\arXiv{0706.0510}},
doi = {10.1103/PhysRevLett.99.130601}
}
@Article{GioAdr98,
Title = {Nonuniform stationary measure of the invariant unstable foliation in {H}amiltonian and fluid mixing systems},
Author = {M. Giona and A. Adrover},
Journal = {Phys. Rev. Lett.},
Year = {1998},
Pages = {3864},
Volume = {81},
Abstract = {demonstrate the existence of a nonuniform stationary measure associated
with the space-filling properties of the unstable invariant foliation
in 2D differentiable area-preserving systems and derives a sequence
of analytical approximations for it. In fluid convection systems,
it is just the convected material line.}
}
@Article{GioCer05,
Title = {Connecting the spatial structure of periodic orbits and invariant manifolds in hyperbolic area-preserving systems},
Author = {M. Giona and S. Cerbelli},
Journal = {Phys. Lett. A},
Year = {2005},
Pages = {200--207},
Volume = {347},
Abstract = {This Letter discusses the equivalence between the Bowen measure associated
with the set Per ( n ) of periodic points of period n of hyperbolic
area-preserving maps of a smooth manifold, and the measure associated
with the intersections between stable and unstable manifolds of hyperbolic
points. In typical cases of physical interest (i.e., nonuniformly
hyperbolic systems) these measures are found to be highly nonuniform
(multifractal).},
DOI = {10.1016/j.physleta.2005.08.005}
}
@Unpublished{GiPaPe11,
Title = {Global attractor and asymptotic dynamics in the {Kuramoto} model for coupled noisy phase oscillators},
Author = {{Giacomin}, G. and {Pakdaman}, K. and {Pellegrin}, X.},
Note = {\arXiv{1107.4501}},
Year = {2011}
}
@Article{GiPo91,
Title = {Homoclinic tangencies, generating partitions and curvature of invariant manifolds},
Author = {F. Giovannini and A. Politi},
Journal = {J. Phys. A},
Year = {1991},
Pages = {1837--1848},
Volume = {24},
Abstract = {A method to compute the curvature of the unstable manifold is introduced
and applied to H\'enon map and Duffing attractor, showing that it
allows the authors to locate the homoclinic tangencies and, in turn,
to construct a generating partition.}
}
@Article{GiPo92,
Title = {Generating partitions in {H\'enon}-type maps},
Author = {F. Giovannini and A. Politi},
Journal = {Phys. Lett. A},
Year = {1992},
Pages = {332--336},
Volume = {161}
}
@Article{GKOS15,
Title = {Invariant manifolds and global bifurcations},
Author = {J. Guckenheimer and B. Krauskopf and H. M. Osinga and B. Sandstede},
Journal = {Chaos},
Year = {2015},
Volume = {25},
DOI = {10.1063/1.4915528},
URL = {http://www.dam.brown.edu/people/sandsted/publications/gkos-survey.pdf}
}
@Article{GLEM07,
Title = {Matrix exponential-based closures forthe turbulent subgrid-scale stress tensor},
Author = {{Li}, Y. and {Chevillard}, L. and {Eyink}, G.and {Meneveau}, C.},
Journal = {Phys. Rev. E},
Year = {2009},
Note = {\arXiv{0704.3781}},
Pages = {016305},
Volume = {79}
}
@Book{GL-Gil07b,
Title = {The Symmetry of Chaos},
Author = {R. Gilmore and C. Letellier},
Publisher = {Oxford Univ. Press},
Year = {2007},
Address = {Oxford}
}
@Article{GL-Let01,
Title = {Covering dynamical systems: {T}wo-fold covers},
Author = {C. Letellier and R. Gilmore},
Journal = {Phys. Rev. E},
Year = {2001},
Pages = {016206},
Volume = {63}
}
@Article{GL-Mir93,
Title = {The proto-{Lorenz} system},
Author = {R. Miranda and E. Stone},
Journal = {Phys. Letters A},
Year = {1993},
Pages = {105},
Volume = {178}
}
@Article{GloKR85,
Title = {Continuation-conjugate gradient methods for the least-squares solution of nonlinear boundary value problems},
Author = {R. Glowinski and H. B. Keller and L. Rheinhart},
Journal = {SIAM J. Sci. Statist. Comput.},
Year = {1985},
Pages = {793--832},
Volume = {6}
}
@Article{GM00aut,
Title = {Computing periodic orbits and their bifurcations with automatic differentiation},
Author = {J. Guckenheimer and B. Meloon},
Journal = {SIAM J. Sci. Comput.},
Year = {2000},
Pages = {951--985},
Volume = {22}
}
@Misc{Goldobin12,
Title = {Limit distribution of averages over unstable periodic orbits forming chaotic attractor},
Author = {Goldobin, D. S.},
Note = {\arXiv{1208.1691}},
Year = {2012}
}
@Article{goldstein01,
author = {Goldstein, H. and Poole, C. P. and Safko, J. L.},
title = {Classical Mechanics},
year = {2001},
address = {Reading, MA},
edition = {3\textsuperscript{rd}},
publisher = {Wesley}
}
@Book{goldstein59,
Title = {Classical Mechanics},
Author = {Goldstein, H.},
Publisher = {Wesley},
Year = {1959},
Address = {Reading, MA}
}
@Book{goldstein80,
title = {Classical Mechanics},
publisher = {Wesley},
year = {1980},
author = {Goldstein, H.},
address = {Reading, MA},
edition = {2\textsuperscript{nd}}
}
@Book{golubI,
Title = {Singularities and Groups in Bifurcation Theory, vol. I},
Author = {M. Golubitsky and D. G. Schaeffer},
Publisher = {Springer},
Year = {1984},
Address = {New York}
}
@Book{golubII,
Title = {Singularities and Groups in Bifurcation Theory, vol. II},
Author = {M. Golubitsky and I. Stewart and D. G. Schaeffer},
Publisher = {Springer},
Year = {1988},
Address = {New York}
}
@Book{golubitsky2002sp,
Title = {The Symmetry Perspective},
Author = {Golubitsky, M. and Stewart, I.},
Publisher = {Birkh{\"a}user},
Year = {2002},
Address = {Boston}
}
@Article{golubord,
Title = {Order and disorder in fluid motion},
Author = {J. P. Gollub},
Journal = {Proc. Natl. Acad. Sci. USA},
Year = {1995},
Pages = {6705},
Volume = {92},
Abstract = {A brief review of experiments on film flows, surface waves, and thermal
convection. In 1-d, the transition from cellular states to spatiotemporal
chaos is described. In 2-d, periodic and quasiperiodic patterns including
defect-mediating ones are discussed. Statistical emphasis on the
transport and mixing phenomena in fluids is put and some open problems
are stated.}
}
@Article{golubsym84,
Title = {Symmetries and pattern selection in {Rayleigh-B\'{e}nard} convection},
Author = {M. Golubitsky and J. W. Swift and Knobloch, J.},
Journal = {Physica D},
Year = {1984},
Pages = {249},
Volume = {10}
}
@Article{Good94,
Title = {Stability of the {Kuramoto-Sivashinsky} and related systems},
Author = {J. Goodman},
Journal = {Commun. Pure Appl. Math.},
Year = {1994},
Pages = {293--306},
Volume = {47},
DOI = {10.1002/cpa.3160470304}
}
@Article{GoPa11,
Title = {Numerical simulation of asymptotic states of the damped {Kuramoto-Sivashinsky} equation},
Author = {Gomez, H. and Paris, J.},
Journal = {Phys. Rev. E},
Year = {2011},
Pages = {046702},
Volume = {83},
DOI = {10.1103/PhysRevE.83.046702}
}
@Article{GoSi94,
Title = {The one-dimensional complex {Ginzburg-Landau} equation in the low dissipation limit},
Author = {Goldman, D. and Sirovich, L.},
Journal = {Nonlinearity},
Year = {1994},
Pages = {417--439},
Volume = {7}
}
@Article{GoSuOr87,
Title = {Stability and {Lyapunov} stability of dynamical systems: {A} differential approach and a numerical method},
Author = {Goldhirsch, I. and Sulem, P. L. and Orszag, S. A.},
Journal = {Physica D},
Year = {1987},
Pages = {311--337},
Volume = {27}
}
@Book{Govaerts00,
Title = {Numerical Methods for Bifurcations of Dynamical Equilibria},
Author = {W. J. F. Govaerts},
Publisher = {SIAM},
Year = {2000},
Address = {Philadelphia}
}
@Book{GoVanLo96,
Title = {Matrix Computations},
Author = {Golub, G. H. and Van Loan, C. F.},
Publisher = {J. Hopkins Univ. Press},
Year = {1996},
Address = {Baltimore, MD}
}
@Misc{GoZa10,
Title = {Noise reduces disorder in chaotic dynamics},
Author = {Goldobin, D. S. and Zaks, M. A. },
Note = {\arXiv{1008.0073}},
Year = {2010}
}
@Article{GranatK06,
Title = {Direct eigenvalue reordering in a product of matrices in periodic {Schur} form},
Author = {R. Granat and B. K\r{a}gstr\"{o}m},
Journal = {SIAM J. Matrix Anal. Appl.},
Year = {2006},
Pages = {285--300},
Volume = {28}
}
@Inproceedings{GranatRBA,
author = {Granat, R. and Jonsson, I. and K{\aa}gstr\"{o}m, B.},
title = {Recursive blocked algorithms for solving periodic triangular {Sylvester}-type matrix equations},
booktitle = {Proc. 8\textsuperscript{th} Intern. Conf. Applied Parallel Computing: State of the Art in Scientific Computing},
year = {2007},
series = {PARA'06},
pages = {531--539}
}
@Article{grant67,
author = {F. C. Grant and M. R. Feix},
title = {{Fourier-{Hermit}e} solutions of the {Vlasov} equations in the linearized limit},
journal = {Phys. Fluids},
year = {1967},
volume = {10},
pages = {696--702}
}
@Article{GraPro83,
Title = {Estimation of the {Kolmogorov} entropy from a chaotic signal},
Author = {Grassberger, P. and Procaccia, I.},
Journal = {Phys. Rev. A},
Year = {1983},
Pages = {2591--2593},
Volume = {28}
}
@Article{GraPro83a,
author = {Grassberger, P. and Procaccia, I.},
title = {Measuring the strangeness of strange attractors},
journal = {Physica D},
year = {1983},
volume = {9},
pages = {189--208}
}
@Article{grass89,
Title = {On the symbolic dynamics of {H\'enon} map},
Author = {P. Grassberger and H. Kantz and U. Moenig},
Journal = {J. Phys. A},
Year = {1989},
Pages = {5217--5230},
Volume = {22},
Abstract = {Discuss Biham-Wenzel method in {H\'enon} map and point out its defficiency:
1. It converges to a limit cycle sometimes. 2. Two symbol sequence
converges to the same cycle.}
}
@Article{GrBrMa08,
Title = {Climate response of linear and quadratic functionals using the fluctuation-dissipation theorem},
Author = {A. Gritsun and G. Branstator and A. Majda},
Journal = {J. Atmos. Sci.},
Year = {2008},
Pages = {2824--2841},
Volume = {65},
DOI = {10.1175/2007JAS2496.1}
}
@Article{greb90shad,
Title = {Shadowing of physical trajectories in chaotic dynamics: {Containment} and refinement},
Author = {C. Grebogi and S. M. Hammel and J. A. Yorke and T. Sauer},
Journal = {Phys. Rev. Lett.},
Year = {1990},
Pages = {1527--1530},
Volume = {65}
}
@Article{gree79,
Title = {A method for determining a stochastic transition},
Author = {J. M. Greene},
Journal = {J. Math. Phys.},
Year = {1979},
Pages = {1183--1201},
Volume = {20}
}
@Article{gree98,
Title = {Two-dimensional measure-preserving mappings},
Author = {J. M. Greene},
Journal = {J. Math. Phys.},
Year = {1968},
Pages = {760},
Volume = {9}
}
@Article{GreeKim87,
Title = {The calculation of {Lyapunov} spectra},
Author = {J. M. Greene and J.-S. Kim},
Journal = {Physica D},
Year = {1987},
Pages = {213--225},
Volume = {24}
}
@Book{Greensite11,
Title = {An introduction to the confinement problem},
Author = {Greensite, J.},
Publisher = {Springer},
Year = {2011},
Address = {New York},
Pages = {101--129}
}
@Book{greshosani,
Title = {Incompressible Flow and the Finite Element Method},
Author = {P. M. Gresho and R. L. Sani},
Publisher = {Wiley},
Year = {2000},
Address = {New York}
}
@Article{Gribov77,
Title = {Quantization of nonabelian gauge theories},
Author = {Gribov, V. N.},
Journal = {Nucl. Phys.},
Year = {1978},
Pages = {1--19},
Volume = {B139},
DOI = {10.1016/0550-3213(78)90175-X}
}
@Article{Gritsun08,
Title = {Unstable periodic trajectories of a barotropic model of the atmosphere},
Author = {Gritsun, A. S.},
Journal = {Russian J. Numer. Analysis and Math. Modelling},
Year = {2008},
Pages = {345--367},
Volume = {23},
Abstract = {Unstable periodic trajectories of a chaotic dissipative system belong
to the attractor of the system and are its important characteristics.
Many chaotic systems have an infinite number of periodic solutions
forming the skeleton of the system attractor. This allows one to
approximate the system trajectories and statistical characteristics
by using periodic solutions. The least unstable orbits may generate
local maxima of the system state distribution functions on the attractor.
With respect to atmospheric systems this means that orbits may determine
dynamic circulation regimes and typical variability modes of the
system. In some cases, given a small number of periodic solutions,
one can describe the dynamics on the attractor of the system and
the basic statistics with sufficient precision. Thus, the information
concerning periodic trajectories of a particular dynamic system may
be very important for analysis of its behavior. A search for periodic
trajectories is reduced to the solution of a system of nonlinear
equations with respect to the initial condition of an orbit and its
period. The choice of a numerical solution method and an initial
guess is an important aspect here. In this paper we consider the
problem of the calculation of periodic trajectories for a barotropic
model of the atmosphere. Several methods for determination of periodic
orbits of the model are formulated and implemented, including the
Newton method with step suppression, the Newton method with a second
order tensor correction, the quasi-Newton method with step suppression,
the quasi-Newton method with minimization of the error functional
and approximate Hessian inversion by the LBFG scheme and the GMRES
method. A comparison of the efficiency of these numerical methods
and different choices of initial conditions is performed. Various
factors influencing the rate of convergence of the methods are considered.},
DOI = {10.1515/RJNAMM.2008.021}
}
@Article{Gritsun10,
Title = {Statistical characteristics of barotropic atmospeheric system and its unstable periodic solutions},
Author = {Gritsun, A.},
Journal = {Doklady Earth Sciences},
Year = {2010},
Pages = {1688--1691},
Volume = {435},
Abstract = {This paper is devoted to the problem of approximating an invariant
measure and statistical characteristics of barotropic atmospheric
model with the help of its periodic trajectories. In this procedure
orbits are taken into account according to their weights defined
by the orbit instability characteristics. The method comes from the
dynamical systems theory where in several specific case (for hyperbolic
systems in particular) unstable periodic orbits define the system
invariant measure. In our study we show that the system PDF could
be reconstructed with the error less than 10\% provided that the
optimal orbit weight function is chosen.},
DOI = {10.1134/S1028334X10120287}
}
@Article{Gritsun11,
Title = {Connection of periodic orbits and variability patterns of circulation for the barotropic model of atmospheric dynamics},
Author = {Gritsun, A.},
Journal = {Doklady Earth Sciences},
Year = {2011},
Pages = {636--640},
Volume = {438},
Abstract = {We have investigated the relationship between periodic trajectories
of barotropic atmospheric model and the modes of the model variability.
In particular, we have studied the nature of ``25 day'' mode of variability
(Branstator, 1987; Kushnir 1987). This mode arises as a first complex
empirical orthogonal function (or ``Hilbert EOF'' according to (H.
von Storch, Zwiers)) for a given system and is a dominant rotational
component of the system dynamics. It was shown that the mode structure
coincides with several least unstable periodic orbits of the system.
The phase portrait of the system in the plane of the first complex
EOF has regular shape with maximum of the probability density function
in the vicinity of these weakly unstable periodic orbits.},
DOI = {10.1134/S1028334X11050035}
}
@Article{Gritsun13,
Title = {Statistical characteristics, circulation regimes and unstable periodic orbits of a barotropic atmospheric model},
Author = {Gritsun, A.},
Journal = {Philos. Trans. Royal Soc. A},
Year = {2013},
Volume = {371},
Abstract = {The theory of chaotic dynamical systems gives many tools that can
be used in climate studies. The widely used ones are the Lyapunov
exponents, the Kolmogorov entropy and the attractor dimension characterizing
global quantities of a system. Another potentially useful tool from
dynamical system theory arises from the fact that the local analysis
of a system probability distribution function (PDF) can be accomplished
by using a procedure that involves an expansion in terms of unstable
periodic orbits (UPOs). The system measure (or its statistical characteristics)
is approximated as a weighted sum over the orbits. The weights are
inversely proportional to the orbit instability characteristics so
that the least unstable orbits make larger contributions to the PDF.
Consequently, one can expect some relationship between the least
unstable orbits and the local maxima of the system PDF. As a result,
the most probable system trajectories (or `circulation regimes' in
some sense) may be explained in terms of orbits. For the special
classes of chaotic dynamical systems, there is a strict theory guaranteeing
the accuracy of this approach. However, a typical atmospheric model
may not qualify for these theorems. In our study, we will try to
apply the idea of UPO expansion to the simple atmospheric system
based on the barotropic vorticity equation of the sphere. We will
check how well orbits approximate the system attractor, its statistical
characteristics and PDF. The connection of the most probable states
of the system with the least unstable periodic orbits will also be
analysed.},
DOI = {10.1098/rsta.2012.0336}
}
@Article{GrKa85,
Title = {Generating partitions for the dissipative {H\'enon} map},
Author = {P. Grassberger and H. Kantz},
Journal = {Phys. Lett. A},
Year = {1985},
Pages = {235--238},
Volume = {113},
Abstract = {Discuss Biham-Wenzel method in {H\'enon} map and point out its defficiency:
1. It converges to a limit cycle sometimes. 2. Two symbol sequence
converges to the same cycle.}
}
@Article{GrMaViFe81,
Title = {Universal behaviour in families of area-preserving maps},
Author = {J. M. Greene and R. S. MacKay and F. Vivaldi and M. J. Feigenbaum},
Journal = {Physica D},
Year = {1981},
Pages = {468},
Volume = {3}
}
@Unpublished{GrPo12,
Title = {Implementing {Turing} machines in dynamic field architectures},
Author = {{beim Graben}, P. and Potthast, R.},
Note = {\arXiv{1204.5462}},
Year = {2012}
}
@Article{Guck79,
author = {Guckenheimer, J.},
title = {Sensitive dependence to initial conditions for one dimensional maps},
journal = {Commun. Math. Phys.},
year = {1979},
volume = {70},
pages = {133--160},
doi = {10.1007/BF01982351},
abstract = {This paper studies the iteration of maps of the interval which have
negative Schwarzian derivative and one critical point. The maps in
this class are classified up to topological equivalence. The equivalence
classes of maps which display sensitivity to initial conditions for
large sets of initial conditions are characterized.}
}
@Book{guckb,
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Author = {J. Guckenheimer and P. Holmes},
Publisher = {Springer},
Year = {1983},
Address = {New York}
}
@Book{GuiSte90,
Title = {Symplectic Techniques in Physics},
Author = {V. Guillemin and S. Sternberg},
Publisher = {Cambridge Univ. Press},
Year = {1990},
Address = {Cambridge}
}
@Article{GuiUri87,
Title = {Reduction, the trace formula, and semiclassical asymptotics},
Author = {Guillemin, V. and Uribe, A.},
Journal = {Proc. Natl. Acad. Sci. USA},
Year = {1987},
Pages = {7799--7801},
Volume = {84}
}
@Article{GuiUri89,
author = {Guillemin, V. and Uribe, A.},
title = {Circular symmetry and the trace formula},
journal = {Invent. Math.},
year = {1989},
volume = {96},
pages = {385--423},
doi = {10.1007/BF01393968}
}
@Article{GuiUri90,
Title = {Reduction and the trace formula},
Author = {Guillemin, V. and Uribe, A.},
Journal = {J. Diff. Geom.},
Year = {1990},
Pages = {315},
Volume = {32},
url = {http://projecteuclid.org/euclid.jdg/1214445310}
}
@Book{Gurtin81,
Title = {An Introduction to Continuum Mechanics},
Author = {Gurtin, M. E.},
Publisher = {Academic},
Year = {1981},
Address = {New York}
}
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Title = {The quantization of a classically ergodic system},
Author = {M. C. Gutzwiller},
Journal = {Physica D},
Year = {1982},
Pages = {183},
Volume = {5},
Abstract = {uses Ising model to improve summations, introduces symmetrizations
of the spectra, computes the spectrum}
}
@Book{gutbook,
Title = {Chaos in Classical and Quantum Mechanics},
Author = {M. C. Gutzwiller},
Publisher = {Springer},
Year = {1990},
Address = {New York}
}
@Misc{GutOsi15,
author = {Gutkin, B. and {Osipov}, V.},
title = {Classical foundations of many-particle quantum chaos},
year = {2015},
addendum = {\arXiv{1503.02676}}
}
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Title = {Phase-integral approximation in momentum space and the bound states of an atom},
Author = {M. C. Gutzwiller},
Journal = {J. Math. Phys.},
Year = {1967},
Pages = {1979--2000},
Volume = {8}
}
@Article{gutzwiller69,
Title = {Phase-integral approximation in momentum space and the bound states of an atom. {II}},
Author = {M. C. Gutzwiller},
Journal = {J. Math. Phys.},
Year = {1969},
Pages = {1004--1021},
Volume = {10}
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@Article{gutzwiller70,
Title = {Energy spectrum according to classical mechanics},
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Year = {1970},
Number = {6},
Pages = {1791--1806},
Volume = {11}
}
@Article{gutzwiller71,
Title = {Periodic orbits and classical quantization conditions},
Author = {M. C. Gutzwiller},
Journal = {J. Math. Phys.},
Year = {1971},
Pages = {343--358},
Volume = {12}
}
@Article{Guyard99,
Author = {F. Guyard},
Journal = {J. Symbolic Comp.},
Year = {1999},
Pages = {275--302},
Volume = {28},
DOI = {10.1006/jsco.1998.0277}
}
@Book{HaCh08,
Title = {Nonlinear Dynamical Systems and Control: A {Lyapunov}-Based Approach},
Author = {Haddad, W. M. and Chellaboina, V.},
Publisher = {Princeton Univ. Press},
Year = {2008},
Address = {Princeton, NJ}
}
@Inbook{HaCho10,
Title = {Connection vector fields and optimized coordinates for swimming systems at low and high {Reynolds} numbers},
Author = {Hatton, R. L. and Choset, H.},
Pages = {1--8},
Publisher = {ASME},
Year = {2010},
Address = {New York},
Booktitle = {ASME 2010 Dynamic Systems and Control Conference}
}
@Misc{HaFaHa13,
Title = {Detecting invariant manifolds, attractors, and generalized {KAM} tori in aperiodically forced mechanical systems},
Author = {{Hadjighasem}, A. and {Farazmand}, M. and {Haller}, G.},
Note = {\arXiv{1302.1732}},
Year = {2013}
}
@Article{Hagen1839,
Title = {{\"U}ber die Bewegung des Wassers in engen cylindrischen R{\"o}hren},
Author = {G. Hagen},
Journal = {Ann. Phys.},
Year = {1839},
Pages = {423--442},
Volume = {122},
DOI = {10.1002/andp.18391220304}
}
@Article{HaiMaj10,
author = {M. Hairer and A. J. Majda},
title = {A simple framework to justify linear response theory},
journal = {Nonlinearity},
year = {2010},
volume = {23},
pages = {909},
note = {\arXiv{0909.4313}},
doi = {10.1088/0951-7715/23/4/008}
}
@Article{HakenZeroLyap83,
Title = {At least one {Lyapunov} exponent vanishes if the trajectory of an attractor does not contain a fixed point},
Author = {H. Haken},
Journal = {Phys. Lett. A},
Year = {1983},
Pages = {71--72},
Volume = {94},
Abstract = {We treat a set of coupled ordinary nonlinear differential equations
and show that for each trajectory which belongs to an attractor (or
to its basin) and which does not contain a fixed point, at least
one Lyapunov exponent vanishes.}
}
@Article{HaKiWa95,
Title = {Regeneration mechanisms of near-wall turbulence structures},
Author = {J. Hamilton and J. Kim and F. Waleffe},
Journal = {J. Fluid Mech.},
Year = {1995},
Pages = {317--348},
Volume = {287}
}
@Phdthesis{HalcrowThesis,
Title = {Geometry of turbulence: An exploration of the state-space of plane {Couette} flow},
Author = {Halcrow, J.},
School = {School of Physics, Georgia Inst. of Technology},
Year = {2008},
Address = {Atlanta},
URL = {http://ChaosBook.org/projects/theses.html}
}
@Book{haledb,
Title = {Dynamics and Bifurcations},
Author = {J. Hale and H. Ko\c{c}ak},
Publisher = {Springer},
Year = {1991},
Address = {New York}
}
@Book{haleinf84,
Title = {An Introduction to Infinite Dimensional Systems-Geometric Theory},
Author = {J. K. Hale and L. T. Magalhaes},
Publisher = {Springer},
Year = {1984},
Address = {New York}
}
@Book{haleob,
Title = {Ordinary Differential Equations},
Author = {J. Hale},
Publisher = {Wiley},
Year = {1969},
Address = {New York}
}
@Book{Hall03,
Title = {Lie Groups, {Lie} Algebras, and Representations},
Author = {Hall, B. C},
Publisher = {Springer},
Year = {2003},
Address = {New York}
}
@Article{Hall94,
Title = {The creation of horseshoes},
Author = {T. Hall},
Journal = {Nonlinearity},
Year = {1994},
Pages = {861--924},
Volume = {7}
}
@Article{Haller11,
Title = {A variational theory of hyperbolic {Lagrangian Coherent Structures}},
Author = {Haller, G.},
Journal = {Physica D},
Year = {2011},
Pages = {574--598},
Volume = {240},
DOI = {10.1016/j.physd.2010.11.010}
}
@Article{Haller11err,
Title = {Erratum and addendum to ``A variational theory of hyperbolic Lagrangian coherent structures''},
Author = {Farazmand, M. and Haller, G.},
Journal = {Physica D},
Year = {2011},
Pages = {439--441}
}
@Article{haller2000,
Title = {Lagrangian coherent structures and mixing in two-dimensional turbulence},
Author = {Haller, G. and Yuan, G.},
Journal = {Physica D},
Year = {2000},
Pages = {352--370},
Volume = {147},
Abstract = {We introduce a Lagrangian definition for the boundaries of coherent
structures in two-dimensional turbulence. The boundaries are defined
as material lines that are linearly stable or unstable for longer
times than any of their neighbors. Such material lines are responsible
for stretching and folding in the mixing of passive tracers. We derive
an analytic criterion that can be used to extract coherent structures
with high precision from numerical or experimental data sets. The
criterion provides a rigorous link between the Lagrangian concept
of hyperbolicity, the {Okubo?Weiss} criterion, and vortex boundaries.
We apply the results to simulations of two-dimensional barotropic
turbulence.},
DOI = {10.1016/S0167-2789(00)00142-1}
}
@Article{HaMe98,
Title = {Reduction of three-dimensional, volume-preserving flows with symmetry},
Author = {G. Haller and I. Mezi\'c},
Journal = {Nonlinearity},
Year = {1998},
Pages = {319--339},
Volume = {11}
}
@Book{hamer,
Title = {Group Theory and Its Application to Physical Problems},
Author = {M. Hamermesh},
Publisher = {Dover},
Year = {1962},
Address = {New York}
}
@Article{HamGomez02,
Title = {Reversible polynomial automorphisms of the plane: {The} involutory case},
Author = {A. G\'omez and J. D. Meiss},
Journal = {Phys. Lett. A},
Year = {2003},
Note = {\arXiv{nlin.CD/0209055}},
Pages = {49--58},
Volume = {312}
}
@Article{HamSteDuMei04,
Title = {Symbolic codes for rotational orbits},
Author = {H. R. Dullin and J. D. Meiss and D. G. Sterling},
Journal = {SIAM J. Appl. Dyn. Sys.},
Year = {2005},
Note = {\arXiv{nlin.CD/0408015}},
Pages = {515--562},
Volume = {4}
}
@Article{HamSteDuMei99,
Title = {Homoclinic bifurcations for the {H\'enon} map},
Author = {D. G. Sterling and H. R. Dullin and J. D. Meiss},
Journal = {Physica D},
Year = {1999},
Note = {\arXiv{chao-dyn/9904019}},
Pages = {153--184},
Volume = {134}
}
@Phdthesis{hansen,
Title = {Symbolic Dynamics in Chaotic systems},
Author = {Hansen, K. T.},
School = {Univ. of Oslo},
Year = {1993},
Address = {Oslo, Norway},
URL = {http://ChaosBook.org/projects/KTHansen/thesis}
}
@Article{hansen1d,
Title = {Bifurcation structures in maps of {H\'enon} type},
Author = {P. Cvitanovi\'{c} and Hansen, K. T.},
Journal = {Nonlinearity},
Year = {1998},
Pages = {1233},
Volume = {11}
}
@Article{hansen92,
Title = {Remarks on the symbolic dynamics for the {H\'enon} map},
Author = {Hansen, K. T.},
Journal = {Phys. Lett. A},
Year = {1992},
Pages = {100--104},
Volume = {165},
Abstract = {Discuss the change of the symbol sequence of one particular periodic
orbit in the {H\'enon} map.}
}
@Article{Hansen92-1,
author = {Hansen, K. T.},
title = {Symbolic dynamics. 1.~{Finite} dispersive billiards},
journal = {Nonlinearity},
year = {1993},
volume = {6},
pages = {753 -- 769}
}
@Article{Hansen92-2,
author = {Hansen, K. T.},
title = {Symbolic dynamics. 2.~{Bifurcations} in billiards and smooth potentials},
journal = {Nonlinearity},
year = {1993},
volume = {6},
pages = {771--778}
}
@Article{Hansen92b,
Title = {Pruning of orbits in four-disk and hyperbola billiards},
Author = {Hansen, K. T.},
Journal = {Chaos},
Year = {1992},
Pages = {71--75},
Volume = {2},
Mrnumber = {93a:58053}
}
@Article{Hansen95,
Title = {Alternative method to find orbits in chaotic systems},
Author = {Hansen, K. T.},
Journal = {Phys. Rev. E},
Year = {1995},
Note = {\arXiv{chao-dyn/9507003}},
Pages = {2388--2391},
Volume = {52},
DOI = {10.1103/PhysRevE.52.2388}
}
@Book{hao89,
Title = {Elementary Symbolic Dynamics and Chaos in Dissipative Systems},
Author = {B.-L. Hao},
Publisher = {World Scientific},
Year = {1989},
Address = {Singapore}
}
@Book{hao90b,
Title = {Chaos {II}},
Author = {B.-L. Hao},
Publisher = {World Scientific},
Year = {1990},
Address = {Singapore}
}
@Book{hao98,
Title = {Applied Symbolic Dynamics and Chaos},
Author = {B.-L. Hao and W.-M. Zheng},
Publisher = {World Scientific},
Year = {1998},
Address = {Singapore}
}
@Book{Harrison70,
Title = {Solid State Theory},
Author = {Harrison, W. A.},
Publisher = {Dover},
Year = {1980},
Address = {New York}
}
@Article{Hart79,
author = {Hart, J. E.},
title = {Finite amplitude baroclinic instability},
journal = {Ann. Rev. Fluid Mech.},
year = {1979},
volume = {11},
pages = {147--172},
doi = {10.1146/annurev.fl.11.010179.001051}
}
@Book{Harter93,
Title = {Principles of Symmetry, Dynamics, and Spectroscopy},
Author = {Harter, W. G.},
Publisher = {Wiley},
Year = {1993},
Address = {New York}
}
@Book{hartmanb,
Title = {Ordinary Differential Equations},
Author = {P. Hartman},
Publisher = {Wiley},
Year = {1964},
Address = {New York}
}
@Article{Hasha05,
Title = {A search for baroclinic structures},
Author = {A. E. Hasha},
Journal = {Proceed. 2005 WHOI Summer. Geophysical Fluid Dynamics Program},
Year = {2005}
}
@Article{HaShu04a,
Title = {An algorithm to prune the area-preserving {H\'enon} map},
Author = {Hagiwara, R. and Shudo, A.},
Journal = {J. Phys. A},
Year = {2004},
Number = {44},
Pages = {10521},
Volume = {37},
Abstract = {An explicit algorithm to provide the pruning front for the area-preserving
H\'enon map is presented. The procedure terminates within finitely
many steps when the map has hyperbolic structure. The only information
required to specify the pruning front is a bifurcation diagram of
homoclinic orbits, and it is obtained by tracking orbits from the
anti-integrable limit. The pruned region thus determined is used
to construct the Markov partition of the map, and the topological
entropy is evaluated as an application.},
DOI = {10.1088/0305-4470/37/44/005}
}
@Article{HaShu04b,
Title = {Grammatical complexity for two-dimensional maps},
Author = {Hagiwara, R. and Shudo, A.},
Journal = {J. Phys. A},
Year = {2004},
Pages = {10545},
Volume = {37},
Abstract = {We calculate the grammatical complexity of the symbol sequences generated
from the {H\'enon} map and the Lozi map using the recently developed
methods to construct the pruning front. When the map is hyperbolic,
the language of symbol sequences is regular in the sense of the Chomsky
hierarchy and the corresponding grammatical complexity takes finite
values. It is found that the complexity exhibits a self-similar structure
as a function of the system parameter, and the similarity of the
pruning fronts is discussed as an origin of such self-similarity.
For non-hyperbolic cases, it is observed that the complexity monotonically
increases as we increase the resolution of the pruning front.}
}
@Article{Heinzl96,
author = {Heinzl, T.},
title = {{Hamiltonian} approach to the {Gribov} problem},
journal = {Nucl. Phys. Proc. Suppl.},
year = {1997},
volume = {54A},
pages = {194--197},
note = {\arXiv{hep-th/9609055}}
}
@Article{HeiZha07,
author = {J. Heidel and F. Zhang},
title = {Nonchaotic and chaotic behavior in three-dimensional quadratic systems: {Five}-one conservative cases},
journal = {Int. J. Bifur. Chaos},
year = {2007},
volume = {17},
pages = {2049--2072},
doi = {10.1142/S021812740701821X}
}
@Article{HeLa96,
Title = {A scaling theory for horizontally homogeneous, baroclinically unstable flow on a {Beta} plane},
Author = {Held, I. M. and Larichev, V. D.},
Journal = {J. Atmos. Sci.},
Year = {1996},
Pages = {946--952},
Volume = {53},
DOI = {10.1175/1520-0469(1996)053<0946:ASTFHH>2.0.CO;2}
}
@Article{hell78,
Title = {Periodic solutions of arbitrary period, variational methods},
Author = {R. Helleman and T. Bountis},
Journal = {LNP},
Year = {1978},
Pages = {353},
Volume = {93}
}
@Article{HeMuAlBrHa07,
Title = {Periodic-orbit theory of level correlations},
Author = {Heusler, S. and M\"uller, S. and Altland, A. and Braun, P. and Haake, F.},
Journal = {Phys. Rev. Lett.},
Year = {2007},
Pages = {044103},
Volume = {98}
}
@Article{HenCviLip14,
author = {J. M. Heninger and D. Lippolis and P. Cvitanovi\'c},
title = {Neighborhoods of periodic orbits and the stationary distribution of a noisy chaotic system},
journal = {Phys. Rev. E},
year = {2015},
volume = {92},
pages = {062922},
note = {\arXiv{1507.00462}},
doi = {10.1103/PhysRevE.92.062922}
}
@Misc{HenLipCvi15,
author = {J. M. Heninger and D. Lippolis and P. Cvitanovi\'c},
title = {Perturbation theory for the {Fokker-Planck} operator in chaos},
year = {2015},
note = {to be submitted to Comm. Nonlinear Sci. Numer. Simul.},
url = {http://arxiv.org/abs/1602.03044}
}
@Article{henon,
Title = {A two-dimensional mapping with a strange attractor},
Author = {M. H\'enon},
Journal = {Commun. Math. Phys.},
Year = {1976},
Pages = {69},
Volume = {50}
}
@Book{henonrtb2,
Title = {Generating Families in the Restricted Three-Body Problem {II}. Quantitative Study of Bifurcations},
Author = {M. H\'{e}non},
Publisher = {Springer},
Year = {2001},
Address = {New York}
}
@Article{HePaPi85,
Title = {Stationary external {Rossby} waves in vertical shear},
Author = {Held, I. M. and Panetta, R. L. and Pierrehumbert, R. T.},
Journal = {J. Atmos. Sci.},
Year = {1985},
Pages = {865--883},
Volume = {42},
DOI = {10.1175/1520-0469(1985)042<0865:SERWIV>2.0.CO;2}
}
@Article{HePiPa86,
Title = {Dissipative destabilization of external {Rossby} waves},
Author = {Held, I. M. and Pierrehumbert, R. T. and Panetta, R. L.},
Journal = {J. Atmos. Sci.},
Year = {1986},
Pages = {388--396},
Volume = {43},
DOI = {10.1175/1520-0469(1986)043<0388:DDOERW>2.0.CO;2}
}
@Article{HerGot10,
Title = {The large core limit of spiral waves in excitable media},
Author = {S. Hermann and G. A. Gottwald},
Journal = {SIAM J. Appl. Dyn. Sys.},
Year = {2010},
Note = {\arXiv{1003.5830}},
Pages = {536--567},
Volume = {9}
}
@Article{heron87,
Title = {Dimension of the attractors associated to the {Ginzburg-Landau} partial differential equation},
Author = {J. M. Ghidaglia and B. H\'{e}ron},
Journal = {Physica D},
Year = {1987},
Pages = {282--304},
Volume = {28}
}
@Article{HeRuSch08,
author = {Hertsch, A. and Rudolph, G. and Schmidt, M.},
title = {{Gauge Orbit Types for Theories with Classical Compact Gauge Group O(n), SO(n) or Sp(n)}},
journal = {Ann. Henri Poincare},
year = {2011},
volume = {12},
pages = {351--395},
note = {\arXiv{0812.0228}},
doi = {10.1007/s00023-011-0081-8}
}
@Article{HGC08,
Title = {Equilibrium and traveling-wave solutions of plane {Couette} flow},
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Year = {2009},
Note = {\arXiv{0808.3375}},
Pages = {243--266},
Volume = {638}
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@Article{HHKR50yChaos,
Title = {Fifty years of chaos: {Applied} and theoretical},
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Journal = {Chaos},
Year = {2012},
Pages = {047501},
Volume = {22},
DOI = {10.1063/1.4769035}
}
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Title = {The correlation spectrum for hyperbolic analytic maps},
Author = {H. H. Rugh},
Journal = {Nonlinearity},
Year = {1992},
Pages = {1237},
Volume = {5},
DOI = {10.1088/0951-7715/5/6/003}
}
@Article{hhrugh94,
Title = {On the asymptotic form and the reality of spectra of {Perron-Frobenius} operators},
Author = {H. H. Rugh},
Journal = {Nonlinearity},
Year = {1994},
Pages = {1055},
Volume = {7},
Abstract = {We study the eigenvalue spectrum of the generalized Perron-Frobenius
operator for 1-D maps having two expanding branches. We show that
if one branch 'dominates' the other, the dominating branch determines
the asymptotic form of the spectrum. In particular, we obtain sufficient
conditions for the reality of the spectrum of the usual Perron-Frobenius
operator.},
DOI = {10.1088/0951-7715/7/3/015}
}
@Article{hhsys,
Title = {The Applicability of the third integral of motion: {Some} numerical experiments},
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Volume = {69}
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@Article{Hilbert93,
author = {Hilbert, D.},
title = {\"Uber die vollen Invariantensysteme},
journal = {Math. Ann.},
year = {1893},
volume = {42},
pages = {313--373},
doi = {10.1007/BF01444162}
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@Book{hille,
Title = {Ordinary Differential Equations in the Complex Domain},
Author = {E. Hille},
Publisher = {Dover},
Year = {1997},
Address = {New York}
}
@Incollection{hindmarsh1983,
Title = {{ODEPACK}, A Systematized Collection of {ODE} Solvers},
Author = {Hindmarsh, A. C.},
Booktitle = {Scientific Computing},
Publisher = {North-Holland},
Year = {1983},
Address = {Amsterdam},
Editor = {Stepleman, R. S.},
Pages = {55--64},
Volume = {1}
}
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Title = {Application of spectral deformation to the {Vlasov--Poisson} system. {II. Mathematical} results},
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Year = {1989},
Pages = {2819--2837},
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Journal = {Phys. Rev. A},
Year = {2001},
Pages = {044102},
Volume = {63},
DOI = {10.1103/PhysRevA.63.044102}
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year = {2011},
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pages = {747--776},
doi = {10.1007/s00220-011-1262-5}
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@Article{HMR98,
Title = {The {Euler-Poincar\'e} equations and semidirect products with applications to continuum theories},
Author = {Holm, D. D. and J. E. Marsden and Ratiu, T. S.},
Journal = {Advances in Mathematics},
Year = {1998},
Pages = {1--81},
Volume = {137}
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@Article{HNks86,
author = {J. M. Hyman and B. Nicolaenko},
title = {The {Kuramoto-Sivashinsky} equation: a bridge between {PDE}'s and dynamical systems},
journal = {Physica D},
year = {1986},
volume = {18},
pages = {113--126},
doi = {10.1016/0167-2789(86)90166-1}
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@Article{HNZks86,
Title = {Order and complexity in the {Kuramoto-Sivashinsky} model of weakly turbulent interfaces},
Author = {J. M. Hyman and B. Nicolaenko and S. Zaleski},
Journal = {Physica D},
Year = {1986},
Pages = {265--292},
Volume = {23},
Abstract = {The properties of the KSe is discussed in even larger domains. Still
the dynamical behavior is dominated by the skeleton of low dimensional
vector fields. More complex dynamics happens in large domain where
intermittency is accurately identified only by high-precision numerical
method. The presence of strange fixed points overrun the cellular
fixed points in large scales which generates complex spatial structure
to the solutions. The Large scale statistical behavior of the system
is computed.}
}
@Article{hof2006flt,
Title = {Finite lifetime of turbulence in shear flows},
Author = {Hof, B. and Westerweel, J. and Schneider, T. M. and Eckhardt, B.},
Journal = {Nature},
Year = {2006},
Number = {7107},
Pages = {59--62},
Volume = {443}
}
@Book{HoJo90,
Title = {Matrix Analysis},
Author = {Horn, R. A. and Johnson, C. R.},
Publisher = {Cambridge Univ. Press},
Year = {1990},
Address = {Cambridge}
}
@Article{Holmes05,
Title = {Ninety plus thirty years of nonlinear dynamics: {Less} is more and more is different},
Author = {Holmes, P.},
Journal = {Int. J. Bifur. Chaos},
Year = {2005},
Pages = {2703--2716},
Volume = {15}
}
@Book{Holmes96,
title = {{Turbulence, Coherent Structures, Dynamical Systems and Symmetry}},
publisher = {Cambridge Univ. Press},
year = {1996},
author = {P. Holmes and J. L. Lumley and G. Berkooz},
address = {Cambridge},
isbn = {9781107008250}
}
@Book{Holton79,
title = {An introduction to dynamic meteorology},
publisher = {Academic},
year = { 1979 },
author = { Holton, J. R. },
address = {New York}
}
@Article{homann-2007,
Title = {Impact of the floating-point precision and interpolation scheme on the results of {DNS} of turbulence by pseudo-spectral codes},
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Journal = {Comp. Phys. Commun.},
Year = {2007},
Note = {\arXiv{0705.3144}},
Pages = {560--565},
Volume = {177}
}
@Article{homcont,
author = {Champneys, A. R. and Kuznetsov, Y. A. and Sandstede, B.},
title = {A numerical toolbox for homoclinic bifurcation analysis},
journal = {Int. J. Bifur. Chaos},
year = {1996},
volume = {6},
pages = {867--887}
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@Article{HooHoo12,
author = {Hoover, W. G. and Hoover, C. G.},
title = {Local {Gram--Schmidt} and {covariant {Lyapunov} vectors} and exponents for three harmonic oscillator problems},
journal = {Commun. Nonlinear Sci. Numer. Simul.},
year = {2012},
volume = {17},
pages = {1043--1054},
note = {\arXiv{1106.2367}}
}
@Unpublished{HooHoo13,
Title = {Why the instantaneous values of the ''{covariant}'' {Lyapunov} exponents depend upon the chosen state-space norm},
Author = {Hoover, W. G. and Hoover, C. G.},
Note = {\arXiv{1309.2342}},
Year = {2013}
}
@Article{hoopks85,
Title = {Nonlinear instability at the interface between two viscous fluids},
Author = {A. P. Hooper and R. Grimshaw},
Journal = {Phys. Fluids},
Year = {1985},
Pages = {37--45},
Volume = {28}
}
@Article{HooPoFoDeZh02,
author = {Hoover, W. G. and Posch, H. A. and Forster, C. and Dellago, C. and Zhou, M.},
title = {{Lyapunov} modes of two-dimensional many-body systems; soft disks, hard disks, and rotors},
journal = {J. Stat. Phys.},
year = {2002},
volume = {109},
pages = {765--776},
doi = {10.1023/A:1020474901341}
}
@Article{hopf42,
Title = {Abzweigung einer periodischen L\"osung},
Author = {Hopf, E.},
Journal = {Bereich. S\"achs. Acad. Wiss. Leipzig, Math. Phys. Kl},
Year = {1942},
Pages = {19},
Volume = {94}
}
@Article{hopf48,
Title = {A mathematical example displaying features of turbulence},
Author = {E. Hopf},
Journal = {Commun. Appl. Math.},
Year = {1948},
Pages = {303--322},
Volume = {1}
}
@Article{hopf50,
Title = {The partial differential equation {$u_t + uu_x = u_{xx}$}},
Author = {E. Hopf},
Journal = {Commun. Pure Appl. Math.},
Year = {1950},
Pages = {201--230},
Volume = {3}
}
@Article{hopf52,
Title = {Statistical hydromechanics and functional calculus},
Author = {E. Hopf},
Journal = {J. Rat. Mech. Anal.},
Year = {1952},
Pages = {87--123},
Volume = {1}
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@Article{HoWi05,
Title = {Baroclinic instabilities of the two-layer quasigeostrophic {Alpha} model},
Author = {Holm, D. D. and Wingate, B. A.},
Journal = {J. Phys. Oceanogr.},
Year = {2005},
Pages = {1287--1296},
Volume = {35},
DOI = {10.1175/JPO2741.1}
}
@Book{hoyll06,
Title = {Pattern Formation: An Introduction to Methods},
Author = {R. Hoyle},
Publisher = {Cambridge Univ. Press},
Year = {2006},
Address = {Cambridge}
}
@Article{HSGsize98,
Title = {Size-dependent transition to high-dimensional chaotic dynamics in a two-dimensional excitable media},
Author = {M. C. Strain and H. S. Greenside},
Journal = {Phys. Rev. Lett.},
Year = {1998},
Pages = {2306--2309},
Volume = {80}
}
@Article{HuKoYa11,
Title = {Micro-swimmers with hydrodynamic interactions},
Author = {Huber, G. and Koehler, S. A. and Yang, J.},
Journal = {Math. Computer Modelling},
Year = {2011},
Pages = {1518--1526},
Volume = {53}
}
@Article{Hunter2007,
author = {Hunter, J. D.},
title = {Matplotlib: A {2D} graphics environment},
journal = {Comput. Sci. Eng.},
year = {2007},
volume = {9},
pages = {90--95},
abstract = {Matplotlib is a 2D graphics package used for Python
for application development, interactive scripting, and
publication-quality image generation across user
interfaces and operating systems.}
}
@Book{Husemoller1993,
Title = {Fibre Bundles},
Author = {D. Husemoller},
Publisher = {Springer},
Year = {1993},
Address = {New York}
}
@Book{Huyg1673,
Title = {L'Horloge \`a Pendule},
Author = {C. Huygens},
Publisher = {Swets \& Zeitlinger},
Year = {1673},
Address = {Amsterdam}
}
@Article{HW90,
Title = {Spectral properties of strongly perturbed {Coulomb} systems: {Fluctuation} properties},
Author = {H\"onig, A. and Wintgen, D.},
Journal = {Phys. Rev. A},
Year = {1989},
Pages = {5642--5657},
Volume = {39},
DOI = {10.1103/PhysRevA.39.5642}
}
@Article{IkeSak13,
author = {Ikegami, T. and Sakurai, T.},
title = {Contour integral eigensolver for non-{H}ermitian systems: {A {Rayleigh}-Ritz-Type} approach},
journal = {Taiwanese J. Math.},
year = {2010},
volume = {14},
pages = {825--837},
abstract = {{The Rayleigh-Ritz-type approach of the contour integral (CIRR)
eigensolver is extended to be generally applicable to non-Hermitian
systems. The CIRR method can extract only the eigenvalues in a given
domain, which was previously formulated for non-degenerated Hermitian
systems. In this method, the Ritz space for the domain is constructed by
numerical evaluation of a contour integral. The effect of the numerical
approximation is analyzed from the viewpoint of a filter operator, which
supports the use of moderate approximations. The numerical accuracy of
the original moment-based approach is also assured. A block version of
the CIRR method is proposed with a detailed algorithm, which allows us
to resolve degenerated systems.}}
}
@Article{Ilyash92,
Title = {Global analysis of the phase portrait for the {Kuramoto-Sivashinsky} equation},
Author = {Il\'{}yashenko, Yu. S.},
Journal = {J. Dynam. Diff. Eq.},
Year = {1992},
Pages = {585--615},
Volume = {4}
}
@Book{ince,
Title = {Ordinary Differential Equations},
Author = {E. L. Ince},
Publisher = {Dover},
Year = {1956},
Address = {New York}
}
@Book{infdymnon,
title = {{Infinite-Dimensional Dynamical Systems in Mechanics and Physics}},
publisher = {Springer},
year = {2013},
author = {Temam, R.},
address = {New York},
edition = {2},
doi = {10.1007/978-1-4612-0645-3},
isbn = {9781461206453}
}
@Article{InKoTaYa12,
Title = {Covariant {Lyapunov} analysis of chaotic {Kolmogorov} flows},
Author = {Inubushi, M. and Kobayashi, M. U. and Takehiro, S.-i. and Yamada, M.},
Journal = {Phys. Rev. E},
Year = {2012},
Pages = {016331},
Volume = {85},
DOI = {10.1103/PhysRevE.85.016331}
}
@Article{InKoTaYa12a,
author = {Inubushi, M. and Kobayashi, M. U. and Takehiro, S.-i. and Yamada, M.},
title = {Covariant {Lyapunov} analysis of chaotic {Kolmogorov} flows and time-correlation function},
journal = {Procedia IUTAM},
year = {2012},
volume = {5},
pages = {244--248},
note = {IUTAM Symposium on 50 Years of Chaos: Applied and Theoretical},
doi = {10.1016/j.piutam.2012.06.033},
abstract = {We study a hyperbolic/non-hyperbolic transition of the flows on two-dimensional
torus governed by the incompressible Navier-Stokes equation (Kolmogorov
flows) using the method of covariant Lyapunov analysis developed
by Ginelli et al. (2007). As the Reynolds number is increased, chaotic
Kolmogorov flows become non-hyperbolic at a certain Reynolds number,
where some new physical property is expected to appear in the long-time
statistics of the fluid motion. Here we focus our attention on behaviors
of the time-correlation function of vorticity across the transition
point, and that the long-time asymptotic form of the correlation
function changes at the Reynolds number close to that of the hyperbolic/non-hyperbolic
transition, which suggests that the time-correlation function reflects
the transition to non-hyperbolicity.}
}
@Techreport{Inubushi13,
Title = {Covariant {Lyapunov} analysis of {Navier-Stokes} turbulence},
Author = {Inubushi, M.},
Institution = {Kyoto Univ.},
Year = {2013},
Address = {Kyoto},
Number = {RIMS Preprint 1770},
URL = {http://www.kurims.kyoto-u.ac.jp/preprint/}
}
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Title = {Invariant measurement of strange sets in terms of cycles},
Author = {P. Cvitanovi\'{c}},
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Year = {1988},
Pages = {2729},
Volume = {61}
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@Book{IooAde98,
Title = {Topics in Bifurcation Theory and Applications},
Author = {Iooss, G. and Adelmeyer, M.},
Publisher = {World Scientific},
Year = {1998},
Address = {Singapore},
ISBN = {978-981-02-3728-8}
}
@Book{isham99,
Title = {Modern Differential Geometry for Physicists},
Author = {C. J. Isham},
Publisher = {World Scientific},
Year = {1999},
Address = {Singapore}
}
@Article{Ishii97,
Title = {Towards a kneading theory for {Lozi mappings}. {I}. {A} solution of the pruning front conjecture and the first tangency problem},
Author = {Ishii, Y.},
Journal = {Nonlinearity},
Year = {1997},
Pages = {731--747},
Volume = {10},
Abstract = {We construct a kneading theory a la Milnor - Thurston for Lozi mappings
(piecewise affine homeomorphisms of the plane). As a two-dimensional
analogue of the kneading sequence, the pruning front and the primary
pruned region are introduced, and the admissibility criterion for
symbol sequences known as the pruning front conjecture is proven
under a mild condition on the parameters. Using this result, we show
that topological properties of the dynamics of the Lozi mapping are
determined by its pruning front and primary pruned region only. This
gives us a solution to the first tangency problem for the Lozi family,
moreover the boundary of the set of all horseshoes in the parameter
space is shown to be algebraic.},
Mrnumber = {98h:58117}
}
@Article{Ishii97a,
Title = {Towards a kneading theory for {Lozi mappings}. {II}: {Monotonicity} of the topological entropy and {Hausdorff} dimension of attractors},
Author = {Ishii, Y.},
Journal = {Commun. Math. Phys.},
Year = {1997},
Pages = {375--394},
Volume = {190}
}
@Article{Ishimura02,
author = {N. Ishimura},
title = {Remarks on third-order {ODEs} relevant to the {Kuramoto-Sivashinsky} equation},
journal = {J. Diff. Eqn.},
year = {2002},
volume = {178},
pages = {466--477},
doi = {10.1006/jdeq.2001.4018},
}
@Article{IshSa98,
author = {Ishii, Y. and Sands, D.},
title = {{Monotonicity of the {Lozi} family near the tent-maps}},
journal = {Commun. Math. Phys.},
year = {1998},
volume = {198},
pages = {397--406},
doi = {10.1007/s002200050482}
}
@Article{IT01,
Title = {The dynamics of bursting process in wall turbulence},
Author = {T. Itano and S. Toh},
Journal = {J. Phys. Soc. Japan},
Year = {2001},
Pages = {703--716},
Volume = {70}
}
@Article{itoh82,
Title = {Analysis of the phase unwrapping algorithm},
Author = {K. Itoh},
Journal = {Appl. Phys.},
Year = {1982},
Pages = {2470},
Volume = {21},
DOI = {10.1364/AO.21.002470}
}
@Article{Jac46,
Title = {{\"U}ber ein leichtes Verfahren die in der Theorie der S{\"a}cul{\"a}rst{\"o}rungen vorkommenden Gleichungen numerisch aufzul{\"o}sen},
Author = {Jacobi, C. G. J.},
Journal = {J. Reine Angew. Math. ({Crelle})},
Year = {1846},
Pages = {51--94},
Volume = {30}
}
@Article{Jacobi1841,
Title = {De functionibus alternantibus earumque divisione per productum e differentiis elementorum conflatum},
Author = {Jacobi, C. G. J.},
Journal = {J. Reine Angew. Math. ({Crelle})},
Year = {1841},
Pages = {439--452},
Volume = {22}
}
@Book{Jacobi1969,
Title = {Collected Works},
Author = {Jacobi, C. G. J.},
Publisher = {Amer. Math. Soc.},
Year = {1969},
Address = {Providence R.I.}
}
@Article{jaeger_kantz,
author = {L. Jaeger and H. Kantz},
title = {Homoclinic tangencies and non-normal Jacobians - {Effects} of noise in nonhyperbolic chaotic systems},
journal = {Physica D},
year = {1997},
volume = {105},
pages = {79--96},
doi = {10.1016/S0167-2789(97)00247-9},
abstract = {A general study of the effects of noise interacting with the deterministic
dynamics of nonhyperbolic chaotic 2-D maps and 3-flows is presented.
Noise in these systems can have dramatic effects on the invariant
measure of the system. In regions of homoclinic tangencies perturbations
are transported away from the neighborhood of the attractor, leading
to deformations which can be one to two orders of magnitude larger
than the noise level. A qualitative understanding is attained by
a study of the universal properties of the unstable and stable foliation
of the phase space in the vicinity of homoclinic tangencies. Through
the investigation of nontrivial structures of the tangent space at
homoclinic tangencies related to the non-normal Jacobians we obtain
a quantitative description of these noise-induced deformations of
the attractor. Local expansion rates which are much larger than the
maximal Lyapunov exponent can be used as a measure of the system's
structural instability under perturbations. We exemplify our general
results on the {H\'enon}, Ikeda and Duffing system. A new and effective
algorithm to calculate the homoclinic tangencies in the entire phase
space based on the results of this paper is presented.}
}
@Article{Jaime2012,
author = {Cisternas, J. and Descalzi, O. and Cartes, C.},
title = {The transition to explosive solitons and the destruction of invariant tori},
journal = {Cent. Eur. J. Phys.},
year = {2012},
volume = {10},
pages = {660--668},
doi = {10.2478/s11534-012-0023-1}
}
@Article{Jaime2013,
Title = {Intermittent explosions of dissipative solitons and noise-induced crisis},
Author = {Cisternas, J. and Descalzi, O.},
Journal = {Phys. Rev. E},
Year = {2013},
Pages = {022903},
Volume = {88},
DOI = {10.1103/PhysRevE.88.022903},
Numpages = {8}
}
@Article{Jakobson81,
author = {M. V. Jakobson},
title = {Absolutely continuous invariant measures for one-parameter families of one-dimensional maps},
journal = {Comm. Math. Phys.},
year = {1981},
volume = {81},
pages = {39--88},
doi = {10.1007/BF01941800}
}
@Book{Jammer66,
Title = {The Conceptual Development of Quantum Mechanics},
Author = {M. Jammer},
Publisher = {McGraw-Hill},
Year = {1966},
Address = {New York}
}
@Book{JBG97,
Title = {Poincar\'e and the Three Body Problem},
Author = {J. Barrow-Green},
Publisher = {Amer. Math. Soc.},
Year = {1997},
Address = {Providence R.I.}
}
@Book{Jee11,
Title = {An Introduction to Tensors and Group Theory for Physicists},
Author = {Jeevanjee, N.},
Publisher = {Birkh\"auser},
Year = {2011},
Address = {Boston},
ISBN = {9780817647148}
}
@Book{jeffb,
Title = {Methods of Mathematical Physics},
Author = {H. Jeffreys and B. Jeffreys},
Publisher = {Cambridge Univ. Press},
Year = {1999},
Address = {Cambridge}
}
@Article{jensen84,
Title = {Transition to chaos by interaction of resonances in dissipative systems},
Author = {M. H. Jensen and P. Bak and T. Bohr},
Journal = {Phys. Rev. A},
Year = {1984},
Pages = {1960},
Volume = {30}
}
@Article{JGB86,
Title = {On the origin of streamwise vortices in a turbulent boundary layer},
Author = {P. S. {Jang} and R. L. {Gran} and D. J. {Benney}},
Journal = {J. Fluid Mech.},
Year = {1986},
Pages = {109--123},
Volume = {169}
}
@Book{jhos,
Title = {Oscillations in Nonlinear Systems},
Author = {J. K. Hale},
Publisher = {McGraw-Hill},
Year = {1963},
Address = {New York}
}
@Article{JKSN05,
Title = {Characterization of near-wall turbulence in terms of equilibrium and `bursting' solutions},
Author = {J. Jimenez and G. Kawahara and M. P. Simens and M. Nagata},
Journal = {Phys. Fluids},
Year = {2005},
Pages = {015105},
Volume = {17}
}
@Article{johnston04,
Title = {Accurate, stable and efficient {N}avier-{S}tokes solvers based on explicit treatment of the pressure term},
Author = {H. Johnston and J.-G. Liu},
Journal = {J. Comp. Phys.},
Year = {2004},
Number = {1},
Pages = {221--259},
Volume = {199}
}
@Article{jolly_evaluating_2000,
Title = {Evaluating the dimension of an inertial manifold for the {Kuramoto-Sivashinsky} equation},
Author = {M. Jolly and R. Rosa and R. Temam},
Journal = {Advances in Differential Equations},
Year = {2000},
Pages = {31--66},
Volume = {5}
}
@Article{jolly1990,
author = {M. S. Jolly and I. G. Kevrekidis and E. S. Titi},
title = {Approximate inertial manifolds for the {Kuramoto-Sivashinsky} equation: {Analysis} and computations},
journal = {Physica D},
year = {1990},
volume = {44},
pages = {38--60},
doi = {10.1016/0167-2789(90)90046-R}
}
@Incollection{jones_dynamics_1992,
author = {Bielawski, R. and G\'orniewicz, L. and Plaskacz, S.},
title = {Topological approach to differential inclusions on closed subset of {$R^n$}},
booktitle = {Dynamics Reported},
publisher = {Springer},
year = {1992},
editor = {Jones, C. K. R. T. and Kirchgraber, U. and Walther, H.-O.},
volume = {1},
isbn = {978-3-642-64758-1},
pages = {225--250},
address = {New York},
doi = {10.1007/978-3-642-61243-5_6}
}
@Article{JoPro87,
Title = {Strong spatial resonance and travelling waves in {Benard} convection},
Author = {C. A. Jones and M. R. E. Proctor},
Journal = {Phys. Lett. A},
Year = {1987},
Pages = {224--228},
Volume = {121},
DOI = {10.1016/0375-9601(87)90008-9}
}
@Book{JoSa,
Title = {Classical dynamics - A contemporary approach},
Author = {J. V. Jos\'e and E. J. Salatan},
Publisher = {Cambridge Univ. Press},
Year = {1998},
Address = {Cambridge}
}
@Article{JosSat72,
author = {Joseph, D. D. and Sattinger, D. H.},
title = {Bifurcating time periodic solutions and their stability},
journal = {Arch. Rational Mech. Anal.},
year = {1972},
volume = {45},
pages = {79--109},
doi = {10.1007/BF00253039}
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@Article{JRT01,
Title = {Accurate Computations on Inertial Manifolds},
Author = {M. Jolly and R. Rosa and R. Temam},
Journal = {{SIAM} J. Sci. Comp.},
Year = {2001},
Pages = {2216},
Volume = {22}
}
@Article{JT63,
Title = {Transverse velocity components in fully developed unsteady flows},
Author = {D. D. Joseph and L. N. Tao},
Journal = {J. Appl. Mech.},
Year = {1963},
Pages = {147--148},
Volume = {30}
}
@Article{JuRi90,
Author = {C. Jung and P. Richter},
Journal = {J. Phys. A},
Year = {1990},
Pages = {2847},
Volume = {23}
}
@Article{JuScho87,
Author = {C. Jung and H. J. Scholz},
Journal = {J. Phys. A},
Year = {1987},
Pages = {3607},
Volume = {20}
}
@Article{K31,
author = {B. O. Koopman},
title = {Hamiltonian systems and transformations in {Hilbert} space},
journal = {Proc. Natl. Acad. Sci.},
year = {1931},
volume = {17},
pages = {315--318}
}
@InProceedings{KaCoCa02,
Title = {Are bred vectors the same as {Lyapunov} vectors?},
Author = {{Kalnay}, E. and {Corazza}, M. and {Cai}, M.},
Booktitle = {EGS XXVII General Assembly, Nice, 21-26 April 2002},
Year = {2002}
}
@Article{KaGrPrLaSi02,
Title = {Unexpected robustness against noise of a class of nonhyperbolic chaotic attractors},
Author = {Kantz, H. and Grebogi, C. and Prasad, A. and Lai, Y.-C. and Sinde, E.},
Journal = {Phys. Rev. E},
Year = {2002},
Pages = {026209},
Volume = {65}
}
@Article{KaiTom80,
Title = {Statistical mechanics of deterministic chaos},
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Journal = {Progr. Theor. Phys.},
Year = {1980},
Pages = {1532--1550},
Volume = {64},
DOI = {10.1143/PTP.64.1532}
}
@Article{KaMaPaa96,
Title = {Spatial dynamics of time-periodic solutions for the {Ginzburg-Landau} equation},
Author = {T. Kapitula and S. Maier-Paape},
Journal = {Z. Angew. Math. Phys.},
Year = {1996},
Pages = {265--305},
Volume = {47}
}
@Incollection{KapYor79,
author = {Kaplan, J. L. and Yorke, J. A.},
title = {Numerical solution of a generalized eigenvalue problem for even mappings},
booktitle = {Functional Differential Equations and Approximation of Fixed Points},
publisher = {Springer},
year = {1979},
editor = {Peitgen, H.-O. and Walther, H.-O.},
volume = {730},
series = {Lect. Notes Math.},
pages = {228--237},
address = {New York},
doi = {10.1007/BFb0064320}
}
@Incollection{KapYor79a,
Title = {Chaotic behavior of multidimensional difference equations},
Author = {Kaplan, J. L. and Yorke, J. A.},
Booktitle = {Functional Differential Equations and Approximation of Fixed Points},
Publisher = {Springer},
Year = {1979},
Address = {Berlin},
Editor = {Peitgen, H.-O. and Walther, H.-O.},
Pages = {204--227},
Series = {Lect. Notes Math.},
Volume = {730},
DOI = {10.1007/BFb0064320}
}
@Article{KaRaYa13,
Title = {The problem of spurious {Lyapunov} exponents in time series analysis and its solution by covariant {Lyapunov} vectors},
Author = {H. Kantz and G. Radons and H.-l. Yang},
Journal = {J. Phys. A},
Year = {2013},
Pages = {254009},
Volume = {46}
}
@Article{Karma1994,
author = {A. Karma},
title = {Electrical alternans and spiral wave breakup in cardiac tissue},
journal = {Chaos},
year = {1994},
volume = {4},
pages = {461--472},
doi = {10.1063/1.166024}
}
@Article{KarTel97,
author = {K\'{a}roly, G. and T\'{e}l, T.},
title = {Chaotic tracer scattering and fractal basin boundaries in a blinking vortex-sink system},
journal = {Phys. Rep.},
year = {1997},
volume = {290},
pages = {125--147},
abstract = {unphysical and the openness of the flow comes from having sinks and
unbounded trajectories}
}
@Article{KaSa05,
Title = {Statistics of unstable periodic orbits of a chaotic dynamical system with a large number of degrees of freedom},
Author = {Kawasaki, M. and Sasa, S.},
Journal = {Phys. Rev. E},
Year = {2005},
Note = {\arXiv{9801020}},
Pages = {037202},
Volume = {72}
}
@Article{Kato03,
Title = {Unstable periodic solutions embedded in a shell model turbulence},
Author = {S. Kato and M. Yamada},
Journal = {Phys. Rev. E},
Year = {2003},
Pages = {025302},
Volume = {68},
Abstract = {A UPO is found in the chaotic region of one shell turbulence model
(GOY). The scaling exponents of the structure function are calculated
to verify that the statistics are well approximated by the UPO.}
}
@Book{Katok95,
Title = {Introduction to the Modern Theory of Dynamical Systems},
Author = {A. Katok and B. Hasselblatt},
Publisher = {Cambridge Univ. Press},
Year = {1995},
Address = {Cambridge}
}
@Article{kavousanakis_projective_2007,
Title = {Projective and coarse projective integration for problems with continuous symmetries},
Author = {M. E. Kavousanakis and R. Erban and A. G. Boudouvis and C. W. Gear and I. G. Kevrekidis},
Journal = {J. Comp. Phys.},
Year = {2007},
Pages = {382--407},
Volume = {225},
Abstract = {Temporal integration of equations possessing continuous symmetries
(e.g. systems with translational invariance associated with traveling
solutions and scale invariance associated with self-similar solutions)
in a "co-evolving" frame (i.e. a frame which is co-traveling, co-collapsing
or co-exploding with the evolving solution) leads to improved accuracy
because of the smaller time derivative in the new spatial frame.
The slower time behavior permits the use of projective and coarse
projective integration with longer projective steps in the computation
of the time evolution of partial differential equations and multiscale
systems, respectively. These methods are also demonstrated to be
effective for systems which only approximately or asymptotically
possess continuous symmetries. The ideas of projective integration
in a co-evolving frame are illustrated on the one-dimensional, translationally
invariant Nagumo partial differential equation {(PDE).} A corresponding
kinetic Monte Carlo model, motivated from the Nagumo kinetics, is
used to illustrate the coarse-grained method. A simple, one-dimensional
diffusion problem is used to illustrate the scale invariant case.
The efficiency of projective integration in the co-evolving frame
for both the macroscopic diffusion {PDE} and for a random-walker
particle based model is again demonstrated.}
}
@Article{kaw1970,
Title = {Relativistic nonlinear propagation of laser beams in cold overdense plasmas},
Author = {Kaw, P. and Dawson, J.},
Journal = {Phys. Fluids},
Year = {1970},
Pages = {472--481},
Volume = {13},
Abstract = {The nonlinear propagation of a very intense laser beam in a cold overdense
plasma is analytically investigated; the laser beam is assumed to
be intense enough to cause the directed component of electron velocity
to be comparable to the velocity of light. Special attention has
been given to the case when coupled longitudinal?cum?transverse modes
propagate in the plasma. An interesting result is that the beam can
readily propagate through the overdense plasma, in contrast to what
would be expected on the basis of a linear theory of laser propagation.},
DOI = {10.1063/1.1692942}
}
@Article{KawKida01,
Title = {Periodic motion embedded in plane {Couette} turbulence: {Regeneration} cycle and burst},
Author = {G. Kawahara and S. Kida},
Journal = {J. Fluid Mech.},
Year = {2001},
Pages = {291--300},
Volume = {449},
Abstract = {Two time-periodic solutions are found in a 3-d constrained plane Couette
flow, multishooting method being used. The turbulent state mainly
follows the periodic orbit with strong variations. The gentle one
is related to the bursting behavior of the system. Heteroclinic orbits
between these two periodic orbits are found.}
}
@Article{Kaz83,
Title = {Rydberg atoms in weak magnetic fields},
Author = {Kazantsev, A. P. and Pokrovsky, V. L. and Bergou, J.},
Journal = {Phys. Rev. A},
Year = {1983},
Pages = {3659--3662},
Volume = {28},
DOI = {10.1103/PhysRevA.28.3659}
}
@Article{Kazantsev01,
Title = {Sensitivity of the attractor of the barotropic ocean model to external influences: {Approach} by unstable periodic orbits},
Author = {Kazantsev, E.},
Journal = {Nonlin. Proc. Geophys.},
Year = {2001},
Pages = {281--300},
Volume = {8},
DOI = {10.5194/npg-8-281-2001}
}
@Article{Kazantsev01a,
Title = {Sensitivity of attractor to external influences: {Approach} by unstable periodic orbits},
Author = {Kazantsev, E.},
Journal = {Chaos Solit. Fract.},
Year = {2001},
Pages = {1989--2005},
Volume = {12},
Abstract = {A description of a deterministic chaotic system in terms of unstable
periodic orbits (UPOs) is used to develop a method of an a priori
estimate of the sensitivity of statistical averages of the solution
to small external influences. This method allows us to determine
the forcing perturbation which maximizes the norm of the perturbation
of a statistical moment of the solution on the attractor. The method
was applied to the Lorenz model. The estimates of perturbations of
two statistical moments were compared with directly calculated values.
The comparison shows that some 100 UPOs are sufficient to realize
this approach and get a good accuracy. The linear approach remains
valid up to rather high norms of the forcing perturbation.},
DOI = {10.1016/S0960-0779(00)00154-5}
}
@Article{Kazantsev98,
Title = {Unstable periodic orbits and attractor of the barotropic ocean model},
Author = {Kazantsev, E.},
Journal = {Nonlin. Proc. Geophys.},
Year = {1998},
Pages = {193--208},
Volume = {5},
DOI = {10.5194/npg-5-193-1998}
}
@Unpublished{Kazantsev98a,
Title = {Unstable periodic orbits and attractor of the {Lorenz} model},
Author = {Kazantsev, E.},
Note = {Projet NUMATH, Rapport de recherche n. 3344},
Year = {1998},
Institution = {INRIA Lorraine}
}
@Incollection{Keating97,
Title = {Resummation and the turning points of zeta functions},
Author = {Keating, J.P.},
Booktitle = {Classical, Semiclassical and Quantum Dynamics in Atoms},
Publisher = {Springer},
Year = {1997},
Address = {Berlin},
Editor = {B. Eckhardt and H. Friedrich},
Pages = {83--93},
DOI = {10.1007/BFb0105970}
}
@Article{Keefe85,
Title = {Dynamics of perturbed wavetrain solutions to the {Ginzburg-Landau} equation},
Author = {Keefe, L. R.},
Journal = {Stud. Appl. Math.},
Year = {1985},
Pages = {91--153},
Volume = {73}
}
@Book{kellbv,
Title = {Numerical Methods for Two-Point Boundary-Value Problems},
Author = {H. B. Keller},
Publisher = {Dover},
Year = {1992},
Address = {New York}
}
@InProceedings{Keller77,
Title = {Numerical solution of bifurcation and nonlinear eigenvalue problems},
Author = {H. B. Keller},
Booktitle = {Applications of Bifurcation Theory},
Year = {1977},
Address = {New York},
Editor = {P. H. Rabinowitz},
Pages = {359--384},
Publisher = {Academic}
}
@InProceedings{Keller79,
Title = {Global homotopies and {Newton} methods},
Author = {H. B. Keller},
Booktitle = {Recent Advances in Numerical Analysis},
Year = {1979},
Address = {New York},
Editor = {C. de Boor and G. H. Golub},
Pages = {73--94},
Publisher = {Academic}
}
@Article{KellerEBK,
Title = {Corrected {Bohr-Sommerfeld} quantum conditions for nonseparable systems},
Author = {J. B. Keller},
Journal = {Ann. Phys.},
Year = {1958},
Pages = {180--188},
Volume = {4}
}
@Article{KellerEpsLogEps,
author = {G. Keller and P. J. Howard and Klages, R.},
title = {Continuity properties of transport coefficients in simple maps},
journal = {Nonlinearity},
year = {2008},
volume = {21},
pages = {1719},
doi = {10.1088/0951-7715/21/8/003},
abstract = {We consider families of dynamics that can be described in
terms of Perron--Frobenius operators with exponential mixing properties.
For piecewise C2 expanding interval maps we rigorously prove
continuity properties of the drift and of the diffusion
coefficient D under parameter variation. Our main result is that D
is Lipschitz continuous up to quadratic logarithmic corrections.
For a special class of piecewise linear maps we provide more precise
estimates at specific parameter values. Our analytical findings are
quantified numerically for the latter class of maps by using exact
expansions for the transport coefficients that can be evaluated
numerically. We numerically observe strong local variations of all
continuity properties.}
}
@Article{Kerswell05,
Title = {Recent progress in understanding the transition to turbulence in a pipe},
Author = {R. R. Kerswell},
Journal = {Nonlinearity},
Year = {2005},
Pages = {R17--R44},
Volume = {18},
Abstract = {The problem of understanding the nature of fluid flow through a circular
straight pipe remains one of the oldest problems in fluid mechanics.
So far no explanation has been substantiated to rationalize the transition
process by which the steady unidirectional laminar flow state gives
way to a temporally and spatially disordered three-dimensional (turbulent)
solution as the flow rate increases. Recently, new travelling wave
solutions have been discovered which are saddle points in phase space.
These plausibly represent the lowest level in a hierarchy of spatio-temporal
periodic flow solutions which may be used to construct a cycle expansion
theory of turbulent pipe flows. We summarize this success against
the backdrop of past work and discuss its implications for future
research.}
}
@Misc{Kerswell12,
Title = {Misunderstanding the turbulence in a pipe},
Author = {R. R. Kerswell},
Note = {In preparation},
Year = {2012}
}
@Article{KeTu06,
Title = {Recurrence of travelling waves in transitional pipe flow},
Author = {Kerswell, R. R. and Tutty, O.R.},
Journal = {J.\ Fluid Mech.},
Year = {2007},
Note = {\arXiv{physics/0611009}},
Pages = {69--102},
Volume = {584},
Abstract = { We find that travelling waves with low wall shear stresses (lower
branch solutions) are on a surface which separates initial conditions
which uneventfully relaminarise and those which lead to a turbulent
evolution. Evidence for recurrent travelling wave visits is found
in both 5D and 10D long periodic pipes but only for those travelling
waves with low-to-intermediate wall shear stress and for less than
about 10\%\ of the time in turbulent flow. Dynamical structures such
as periodic orbits need to be isolated and included in any such expansion.}
}
@Article{kev01ks,
Title = {The {Oseberg} transition: visualization of global bifurcations for the {Kuramoto-Sivashinsky} equation},
Author = {M. E. Johnson and M. S. Jolly and I. G. Kevrekidis},
Journal = {Int. J. Bifur. Chaos},
Year = {2001},
Pages = {1--18},
Volume = {11}
}
@Article{Kim87,
Title = {Turbulence statistics in fully developed channel flow at low {Reynolds} number},
Author = {J. Kim and P. Moin and R. Moser},
Journal = {J. Fluid Mech.},
Year = {1987},
Pages = {133--166},
Volume = {177}
}
@Article{Kimball01,
Title = {Chaotic properties of the soft-disk {Lorentz} gas},
Author = {Kimball, J. C.},
Journal = {Phys. Rev. E},
Year = {2001},
Pages = {066216},
Volume = {63},
DOI = {10.1103/PhysRevE.63.066216}
}
@Article{kirby_reconstructing_1992,
Title = {Reconstructing phase space from {PDE} simulations},
Author = {M. Kirby and D. Armbruster},
Journal = {Z.\ Angew.\ Math.\ Phys.},
Year = {1992},
Pages = {999--1022},
Volume = {43},
Abstract = {We propose the {Karhunen-Lo\'eve} {(K-L)} decomposition as a tool
to analyze complex spatio-temporal structures in {PDE} simulations
in terms of concepts from dynamical systems theory. Taking the {Kuramoto-Sivashinsky}
equation as a model problem we discuss the {K-L} decomposition for
4 different values of its bifurcation parameter a. We distinguish
two modes of using the {K-L} decomposition: As an analytic and synthetic
tool respectively. Using the analytic mode we find unstable fixed
points and stable and unstable manifolds in a parameter regime with
structurally stable homoclinic orbits (a=17.75). Choosing the data
for a {K-L} analysis carefully by restricting them to certain burst
events, we can analyze a more complicated intermittent regime at
a=68. We establish that the spatially localized oscillations around
a so called `strange' fixed point which are considered as fore-runners
of spatially concentrated zones of turbulence are in fact created
by a very specific limit cycle (a=83.75) which, for a=87, bifurcates
into a modulated traveling wave. Using the {K-L} decomposition synthetically
by determining an optimal Galerkin system, we present evidence that
the {K-L} decomposition systematically destroys dissipation and leads
to blow up solutions.}
}
@Article{Kirwan88,
Title = {The topology of reduced phase spaces of the motion of vortices on a sphere},
Author = {Kirwan, F.},
Journal = {Physica D},
Year = {1988},
Pages = {99--123},
Volume = {30}
}
@Article{Kiselev98,
author = {Kiselev, A. D.},
title = {Symmetry breaking and bifurcations in complex {Lorenz} model},
journal = {J. Phys. Studies},
year = {1998},
volume = {2},
pages = {30--37},
url = {https://www.researchgate.net/profile/Alexei_Kiselev2/publication/252760787_Symmetry_breaking_and_bifurcations_in_complex_Lorenz_model_(in_English)/links/00b7d52ebbf6c2aa32000000.pdf}
}
@InProceedings{KK05,
Title = {Unstable periodic motion in plane {Couette} system: The skeleton of turbulence},
Author = {G. Kawahara and S. Kida and M. Nagata},
Booktitle = {One Hundred Years of Boundary Layer Research},
Year = {2005},
Address = {Dordrecht},
Publisher = {Kluwer}
}
@Article{KKCSG07,
Title = {Fractal properties of anomalous diffusion in intermittent maps},
Author = {Korabel, N. and Klages, R. and Chechkin, A. V. and Sokolov, I. M. and Gonchar, V. Yu.},
Journal = {Phys. Rev. E},
Year = {2007},
Pages = {036213},
Volume = {75},
DOI = {10.1103/PhysRevE.75.036213}
}
@Article{KKPW89,
author = {Katok, A. and Knieper, G. and Pollicott, M. and Weiss, H.},
title = {Differentiability and analyticity of topological entropy for {Anosov} and geodesic flows},
journal = {Invent. Math.},
year = {1989},
volume = {98},
pages = {581--597},
doi = {10.1007/BF01393838}
}
@Article{KlBo011,
Title = {Convergence analysis of {Davidchack and Lai's} algorithm for finding periodic orbits},
Author = {A. Klebanoff and E. Bollt},
Journal = {Chaos Solit. Fract.},
Year = {2001},
Pages = {1305--1322},
Volume = {12},
Abstract = {We rigorously study a recent algorithm due to Davidchack and Lai (DL)
[Davidchack RL, Lai Y-C. Phys Rev E 1999;60(5):6172-5] for efficiently
locating complete sets of hyperbolic periodic orbits for chaotic
maps. We give theorems concerning sufficient conditions on convergence
and also describing variable sized basins of attraction of initial
seeds, thus pointing out a particularly attractive feature of the
DL-algorithm. We also point out the true role of involutary matrices
which is different from that implied by Schmelcher and Diakonos [Schmelcher
P, Diakonos FK. Phys Rev E 1998;57(3):2739-46] and propagated by
Davidchack and Lai.},
DOI = {10.1016/S0960-0779(00)00099-0}
}
@Inproceedings{Kleiser80,
author = {L. Kleiser and U. Schuman},
title = {Treatment of incompressibility and boundary conditions in 3-{D} numerical spectral simulations of plane channel flows},
booktitle = {Proc. 3\textsuperscript{rd} GAMM Conf. Numerical Methods in Fluid Mechanics},
year = {1980},
editor = {E. Hirschel},
pages = {165--173},
address = {Viewweg, Braunschweig},
organization = {GAMM}
}
@Article{KLH94,
Title = {Bounds for threshold amplitudes in subcritical shear flows},
Author = {G. {Kreiss} and A. {Lundbladh} and D.~S. {Henningson}},
Journal = {J. Fluid Mech.},
Year = {1994},
Pages = {175--198},
Volume = {270}
}
@Article{KlPe86,
Title = {A numerical study of baroclinic instability at large super-criticality},
Author = {Klein, P. and Pedlosky, J.},
Journal = {J. Atmos. Sci.},
Year = {1986},
Pages = {1243--1262},
Volume = {43},
DOI = {10.1175/1520-0469(1986)043<1263:ANSOBI>2.0.CO;2}
}
@Article{KlPe92,
Title = {The role of dissipation mechanisms in the nonlinear dynamics of unstable baroclinic waves},
Author = {Klein, P. and Pedlosky, J.},
Journal = {J. Atmos. Sci.},
Year = {1992},
Pages = {29--48},
Volume = {49},
DOI = {10.1175/1520-0469(1992)049<0029:TRODMI>2.0.CO;2}
}
@Article{KMOSTY10,
Title = {Chaotic behavior in classical {Yang-Mills} dynamics},
Author = {Kunihiro, T. and M\"uller, B. and Ohnishi, A. and Sch\"afer, A. and Takahashi, T. T. and Yamamoto, A.},
Journal = {Phys. Rev. D},
Year = {2010},
Pages = {114015},
Volume = {82},
DOI = {10.1103/PhysRevD.82.114015}
}
@Article{Knauf87,
Title = {Ergodic and topological properties of {Coulombic} periodic potentials},
Author = {Knauf, A.},
Journal = {Comm. Math. Phys.},
Year = {1987},
Pages = {89--112},
Volume = {110}
}
@Article{KniKla11a,
Title = {Capturing correlations in chaotic diffusion by approximation methods},
Author = {Knight, G. and Klages, R.},
Journal = {Phys. Rev. E},
Year = {2011},
Pages = {041135},
Volume = {84},
DOI = {10.1103/PhysRevE.84.041135}
}
@Article{KniKla11b,
author = {Knight, G. and Klages, R.},
title = {Linear and fractal diffusion coefficients in a family of one-dimensional chaotic maps},
journal = {Nonlinearity},
year = {2011},
volume = {24},
pages = {227},
doi = {10.1088/0951-7715/24/1/011},
abstract = {We analyse deterministic diffusion in a simple,
one-dimensional setting consisting of a family of four parameter
dependent, chaotic maps defined over the real line. When iterated under
these maps, a probability density function spreads out and one can
define a diffusion coefficient. We look at how the diffusion
coefficient varies across the family of maps and under parameter
variation. Using a technique by which Taylor--Green--Kubo formulae are
evaluated in terms of generalized Takagi functions, we derive exact,
fully analytical expressions for the diffusion coefficients. Typically,
for simple maps these quantities are fractal functions of control
parameters. However, our family of four maps exhibits both fractal and
linear behaviour. We explain these different structures by looking at
the topology of the Markov partitions and the ergodic properties of the
maps.}
}
@Article{knobloch_general_1996,
Title = {A general reduction method for periodic solutions in conservative and reversible systems},
Author = {Knobloch, J. and Vanderbauwhede, A.},
Journal = {J. Diff. Eqn.},
Year = {1996},
Pages = {71},
Volume = {8}
}
@InProceedings{knobloch_hopf_1994,
Title = {Hopf bifurcation at k-fold resonances in equivariant reversible systems},
Author = {Knobloch, J. and Vanderbauwhede, A.},
Booktitle = {Dynamics, Bifurcation and Symmetry, New Trends and New Tools},
Year = {1994},
Address = {Dordrecht},
Editor = {P. Chossat},
Pages = {167},
Publisher = {Kluwer}
}
@Article{knobloch_hopf_1996,
Title = {Hopf bifurcation at k-fold resonances in conservative systems},
Author = {Knobloch, J. and Vanderbauwhede, A.},
Journal = {Progr. Nonlinear Differential Equations Appl.},
Year = {1996},
Pages = {155--170},
Volume = {19}
}
@Article{KNSks90,
Title = {Back in the saddle again: a computer assisted study of the {Kuramoto-Sivashinsky} equation},
Author = {I. G. Kevrekidis and B. Nicolaenko and J. C. Scovel},
Journal = {SIAM J. Appl. Math.},
Year = {1990},
Number = {3},
Pages = {760--790},
Volume = {50},
Abstract = {The initial bifurcations of the KSe are examined analytically and
numerically. The heteroclinic connections between symmetric solutions
proved to play an important role in the dynamics.}
}
@Article{Knyazev02,
author = {A. V. Knyazev and M. E. Argentati},
title = {Principal angles between subspaces in an {A-based} scalar product: {Algorithms} and perturbation estimates},
journal = {SIAM J. Sci. Comput.},
year = {2002},
volume = {23},
pages = {2008--2040},
doi = {10.1137/S1064827500377332}
}
@Article{KoEhlMo96,
Title = {Problems and progress in microswimming},
Author = {Koiller, J. and Ehlers, K. and Montgomery, R.},
Journal = {J. Nonlin. Sci.},
Year = {1996},
Pages = {507--541},
Volume = {6},
DOI = {10.1007/BF02434055}
}
@Article{Koenig97,
Title = {Linearization of vector fields on the orbit space of the action of a compact {Lie} group},
Author = {M. K{\oe}nig},
Journal = {Math. Proc. Cambridge Philos. Soc.},
Year = {1997},
Pages = {401--424},
Volume = {121},
DOI = {10.1017/S0305004196001314}
}
@Article{KoHoGu06,
author = {P. Holmes and R. J. Full and D. Koditschek and J. Guckenheimer},
title = {The dynamics of legged locomotion: {Models}, analyses, and challenges},
journal = {SIAM Review},
year = {2006},
volume = {48},
pages = {207--304},
doi = {10.1137/S0036144504445133}
}
@Article{KohTak07,
Title = {Finding periodic orbits of higher-dimensional flows by including tangential components of trajectory motion},
Author = {Koh, Y. W. and Takatsuka, K.},
Journal = {Phys. Rev. E},
Year = {2007},
Pages = {066205},
Volume = {76},
DOI = {10.1103/PhysRevE.76.066205}
}
@Book{Koks06,
Title = {Explorations in Mathematical Physics: The Concepts Behind an Elegant Language},
Author = {Koks, D.},
Publisher = {Springer},
Year = {2006},
Address = {New York}
}
@Article{kolm91,
Title = {The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers},
Author = {A. N. Kolmogorov},
Journal = {Proc. R. Soc. Lond. A},
Year = {1991},
Number = {1890},
Pages = {9--13},
Volume = {434},
Abstract = {Study the distribution of velocity difference of neighboring points
based on the locally homogeneous and locally isotropic hypothesized
poperties of turbulence motion. Two more similarity hypothesis suggested
the existence of a universal distribution for different viscosity
and energy dissipation rate. The large distance asymptotic behavior
of the distribution is also deduced.}
}
@Article{kook89,
Title = {Periodic orbits for reversible symplectic mappings},
Author = {H-T. Kook and J. D. Meiss},
Journal = {Physica D},
Year = {1989},
Pages = {65--86},
Volume = {35},
Abstract = {By constructing a 2N-dimensional symplectic map for a Langrangian
system, the orbit structure of the phase space is discussed for a
time-reversible system. Periodic orbits are classfied according to
the rotation number, symmetry, and morse index. Unstable orbits repel
other periodic orbits to form resonances where a orbit can be trapped
for long with the same rotation number. The chain of resonances with
the same commensurability forms channel which provides the communication
bridge between different resonances. The connection of the orbit
structure with the continued fraction representation of real numbers
is discussed.}
}
@Article{kooknewt,
Title = {Application of {Newton}'s method to {Lagrangian} mappings},
Author = {H-T Kook and J. D. Meiss},
Journal = {Physica D},
Year = {1989},
Pages = {317--326},
Volume = {36},
Abstract = {An algorithm of {Newton}'s method is presented to find periodic orbits
for {Lagrangian} mappings. The method is based on block-diagonalization
of the Hessian Matrix of the action function.}
}
@Incollection{koopmania,
Author = {P. Cvitanovi\'{c}},
Booktitle = {{Chaos: Classical and Quantum}},
Publisher = {Niels Bohr Inst.},
Year = {2015},
Address = {Copenhagen},
Chapter = {{Implementing} evolution},
URL = {http://ChaosBook.org/paper.shtml#appendMeasure}
}
@Article{KoRe96,
Title = {Groups and nonlinear dynamical systems. {Dynamics} on the {SU(2)} group},
Author = {Kowalski, K. and Rembielinski, J.},
Journal = {Physica D},
Year = {1996},
Note = {\arXiv{chao-dyn/9801019}},
Pages = {237--251},
Volume = {99}
}
@Article{KoRe98,
Title = {Groups and nonlinear dynamical systems. {Chaotic} dynamics on the {SU(2)xSU(2)} group},
Author = {K. Kowalski and J. Rembielinski},
Journal = {Chaos Solit. Fract.},
Year = {1998},
Note = {\arXiv{chao-dyn/9801020}},
Pages = {437--448},
Volume = {9}
}
@Unpublished{KoSa11,
Title = {Manifold structures of unstable periodic orbits and the appearance of periodic windows in chaotic systems},
Author = {Kobayashi, M. U. and Saiki, Y.},
Note = {submitted},
Year = {2011}
}
@Article{kostelich97,
Title = {Unstable dimension variability: a source of nonhyperbolicity in chaotic systems},
Author = {E. J. Kostelich and I. Kan and C. Grebogi and E. Ott and J. A. Yorke},
Journal = {Physica D},
Year = {1997},
Pages = {81--90},
Volume = {109}
}
@Article{Kowa97,
Title = {Nonlinear dynamical systems and classical orthogonal polynomials},
Author = {Kowalski, K.},
Journal = {J. Math. Phys.},
Year = {1997},
Note = {\arXiv{solv-int/9801018}},
Pages = {2483--2505},
Volume = {38}
}
@Article{Kras04,
Title = {Growing random sequences},
Author = {I. Krasikov and G. J. Rodgers and C. E. Tripp},
Journal = {J. Phys. A: Math. Gen.},
Year = {2004},
Pages = {2365--2370},
Volume = {37},
Abstract = {The random sequence has been generalized both in the form of addition
and in the probability distributions. Critical parameter values are
discovered around which the system has very different behavior.}
}
@Article{krauskopf_survey_2005,
Title = {A survey of methods for computing (un)stable manifolds of vector fields},
Author = {B. Krauskopf and H. M. Osinga and E. J. Doedel and M. E. Henderson and J. Guckenheimer and A. Vladimirsky and M. Dellnitz and O. Junge},
Journal = {Int. J. Bifur. Chaos},
Year = {2005},
Pages = {763--791},
Volume = {15},
Abstract = {The computation of global invariant manifolds has seen renewed interest
in recent years. We survey different approaches for computing a global
stable or unstable manifold of a vector field, where we concentrate
on the case of a two-dimensional manifold. All methods are illustrated
with the same example of the two-dimensional stable manifold of the
origin in the Lorenz system.}
}
@Article{KreEck12,
Title = {Periodic orbits near onset of chaos in plane {Couette} flow},
Author = {T. Kreilos and B. Eckhardt},
Journal = {Chaos},
Year = {2012},
Note = {\arXiv{1205.0347}},
Pages = {047505},
Volume = {22},
DOI = {10.1063/1.4757227}
}
@Unpublished{KreEck13,
Title = {Symmetry related dynamics in parallel shear flows},
Author = {T. Kreilos and B. Eckhardt},
Note = {\arXiv{1309.4590}, submitted to J. Fluid Mech.},
Year = {2013}
}
@Article{Kressner2006,
Title = {A periodic {Krylov-Schur} algorithm for large matrix products},
Author = {D. Kressner},
Journal = {Numer. Math.},
Year = {2006},
Pages = {461--483},
Volume = {103},
DOI = {10.1007/s00211-006-0682-1}
}
@Article{KRSR67,
author = {S. J. Kline and W. C. Reynolds and F. A. Schraub and P. W. Rundstadler},
title = {The structure of turbulent boundary layers},
journal = {J. Fluid Mech.},
year = {1967},
volume = {30},
pages = {741--773}
}
@Article{Krupa90,
Title = {Bifurcations of relative equilibria},
Author = {M. Krupa},
Journal = {{SIAM} J. Math. Anal.},
Year = {1990},
Pages = {1453--1486},
Volume = {21},
DOI = {10.1137/0521081}
}
@Article{KrupaRobHetCyc97,
Title = {Robust Heteroclinic Cycles},
Author = {M. Krupa},
Journal = {J. Nonlin. Sci.},
Year = {1997},
Pages = {129--176},
Volume = {7},
Abstract = {Examines the theoretical and applied research of robust cycles. Formation
of heteroclinic cycles in higher codimension; Stability of robust
cycles; Concept of heteroclinic cycles.}
}
@Article{kruskal62,
Title = {Asymptotic theory of {H}amiltonian and other systems with all solutons nearly periodic},
Author = {M. Kruskal},
Journal = {J. Math. Phys.},
Year = {1962},
Number = {4},
Pages = {806},
Volume = {3},
Abstract = {Asymptotic expansion to all orders. A systematic way to construct
the adiabatic invariants.}
}
@Article{kruskal65,
Title = {Interaction of solitons in a collisionless plasma and the recurrence of initial states},
Author = {N. J. Zabusky and M. D. Kruskal},
Journal = {Phys. Rev. Lett.},
Year = {1965},
Pages = {240},
Volume = {15}
}
@Article{KrWaHaDa12,
author = {J. A. Krakos and Q. Wang and S. R. Hall and D. L. Darmofal},
title = {Sensitivity analysis of limit cycle oscillations},
journal = {J. Comp. Phys.},
year = {2012},
volume = {231},
pages = {3228--3245},
doi = {10.1016/j.jcp.2012.01.001}
}
@Book{KryBog47,
Title = {Introduction to Nonlinear Mechanics},
Author = {Krylov, N. M. and Bogoliubov, N. N.},
Publisher = {Princeton Univ. Press},
Year = {1947},
Address = {Princeton, NJ}
}
@Article{ks05com,
author = {A.-K. Kassam and L. N. Trefethen},
title = {Fourth-order time stepping for stiff {PDE}s},
journal = {SIAM J. Sci. Comput.},
year = {2005},
volume = {26},
pages = {1214--1233},
doi = {10.1137/S1064827502410633}
}
@Article{kschang86,
author = {H.-C. Chang},
title = {Travelling waves on fluid interfaces: {Normal} form analysis of the {Kuramoto-Sivashinsky} equation},
journal = {Phys. Fluids},
year = {1986},
volume = {29},
number = {10},
pages = {3142},
abstract = {The lowest order normal form analysis is give to the KSe and the approximate
analytical wavelength-amplitude and wavespeed-amplitude relation
are given. They agree quite well with the numerical calculations.},
doi = {10.1063/1.865965}
}
@Article{ksgreene88,
author = {J. M. Greene and J.-S. Kim},
title = {The steady states of the {Kuramoto-Sivashinsky} equation},
journal = {Physica D},
year = {1988},
volume = {33},
pages = {99--120},
abtract = {Use Fourier modes to discuss the energy transport from long wavelength modes to short ones. The generation of steady states with periodic boundary condition is studied systematically with a bifurcation analysis. Their stability is investigated and the scaling in the limit of large system size is presented.},
doi = {10.1016/S0167-2789(98)90013-6}
}
@Article{ksgrim91,
Title = {The non-existence of a certain class of travelling wave solutions of the {Kuramoto-Sivashinsky} equation},
Author = {R. Grimshaw and A. P. Hooper},
Journal = {Physica D},
Year = {1991},
Pages = {231--238},
Volume = {50},
Abstract = {If the effect of short-wave stability is small, there are no regular
shocks.}
}
@Article{ksham95,
Title = {Hamiltonian structure and integrability of the stationary {Kuramoto-Sivashinsky} equation},
Author = {S. Bouquet},
Journal = {J. Math. Phys.},
Year = {1995},
Number = {3},
Pages = {1242},
Volume = {36},
Abstract = {The stationary KSe can be transformed into a time-depent 1-d {H}amiltonian
system. There exist a large class of initial conditions under which
the system will explode in a finite time like 120/(x-x_0)^3.}
}
@Article{kshooper88,
author = {A. P. Hooper and R. Grimshaw},
title = {Travelling wave solutions of the {Kuramoto-Sivashinsky} equation},
journal = {Wave Motion},
year = {1988},
volume = {10},
pages = {405--420},
abstract = {Numerically calculated connections of the KSe},
doi = {10.1016/0165-2125(88)90045-5}
}
@Article{kskent92,
Title = {Travelling-waves of the {Kuramoto-Sivashinsky} equation: {Period-multiplying} bifurcations},
Author = {P. Kent and J. Elgin},
Journal = {Nonlinearity},
Year = {1992},
Pages = {899--919},
Volume = {5},
Abstract = {Some properties of the KSe are discussed. A unified argument about
the connections and periodic orbits is given. In particular, the
K-bifurcation of the system is conjectured to arises in the 1:n resonances
of a fixed point.}
}
@Article{kstroy89,
Title = {The existence of steady solutions of the {Kuramoto-Sivashinsky} equation},
Author = {W. C. Troy},
Journal = {J. Diff. Eqn.},
Year = {1989},
Pages = {269--313},
Volume = {82},
Abstract = {For c=1, the existence of two periodic orbits and two heteroclinic
orbits are proved and suggestions for proving the existence of more
complicated orbits are given.}
}
@Article{kstroy92,
Title = {Steady solutions of the {Kuramoto-Sivashinsky} equation for small wave speed},
Author = {J. Jones and W. C. Troy and A. D. MacGillivary},
Journal = {J. Diff. Eqn.},
Year = {1992},
Pages = {28--55},
Volume = {96},
Abstract = {For small c>0, at least one odd periodic solution exists. For any
c>0, there existes no monotone bounded travelling wave solution.
The normal form analysis may not be able to give correct results
to all orders.}
}
@Article{ksyang97,
Title = {On travelling-wave solutions of the {Kuramoto-Sivashinsky} equation},
Author = {T.-S. Yang},
Journal = {Physica D},
Year = {1997},
Pages = {25--42},
Volume = {110},
Abstract = {Using asymptotic analysis (beyond all orders)in the weak shock limit,
the oscillatory shocks and the solitary saves are constructed. It
shows that the oscillatory shocks can be only be anti-symmetric and
numurical results support the anlytical calculation.}
}
@Article{KT84,
Title = {Escape rate from strange repellers},
Author = {L. Kadanoff and C. Tang},
Journal = {Proc. Natl. Acad. Sci. USA},
Year = {1984},
Pages = {1276},
Volume = {81}
}
@Article{KuAlLe16,
author = {Kuznetsov, N. V. and Alexeeva, T. A. and Leonov, G. A.},
title = {Invariance of {Lyapunov} exponents and {Lyapunov} dimension for regular and irregular linearizations},
journal = {Nonlinear Dyn.},
year = {2016},
pages = {1--7},
doi = {10.1007/s11071-016-2678-4},
url = {http://arXiv.org/abs/1410.2016}
}
@Article{Kudry08,
author = {Kudryashov, N. A.},
title = {Solitary and periodic solutions of the generalized {Kuramoto-Sivashinsky} equation},
journal = {Regul. Chaotic Dyn.},
year = {2008},
volume = {13},
pages = {234--238},
note = {\arXiv{1112.5707}},
doi = {10.1134/S1560354708030088}
}
@Article{Kudry08a,
Title = {Nonlinear evolution equations for description of perturbation in tube},
Author = {N. A. Kudryashov and D. I. Sinelschikov and I. L. Chernyavsky },
Journal = {Nonlinear Dynam.},
Year = {2008},
Pages = {69--86},
Volume = {4}
}
@Inproceedings{KuKuSa05,
author = {Kuznetsov, S. P. and Kuznetsov, A. P. and Sataev, I .R.},
title = {Review and examples of non-{Feigenbaum} critical situations associated with period-doubling},
booktitle = {Physics and Control, 2005 International Conference Proceedings},
year = {2005},
pages = {610--615},
abstract = {We review several critical situations, linked with period-doubling
transition to chaos, which require using at least two-dimensional
maps as models representing the universality classes. Each of them
corresponds to a saddle solution of the two-dimensional generalization
of Feigenbaum-Cvitanovic equation and is characterized by a set of
distinct universal constants analogous to Feigenbaum's alpha; and
delta;. We present a number of examples (driven self-oscillators,
coupled Henon-like maps, coupled driven oscillators, coupled chaotic
self-oscillators), which manifest these types of behavior.},
doi = {10.1109/PHYCON.2005.1514057}
}
@Book{kundu08,
Title = {Fluid Mechanics},
Author = {P. K. Kundu and I.M. Cohen},
Publisher = {Academic},
Year = {2008},
Address = {San Diego, CA}
}
@Unpublished{KuPa09,
Title = {Strict and fussy modes splitting in the tangent space of the {Ginzburg-Landau} equation},
Author = {Kuptsov, P. V. and Parlitz, U.},
Note = {\arXiv{0912.2261}},
Year = {2009}
}
@Article{KuPa12,
author = {Kuptsov, P. V. and Parlitz, U.},
title = {Theory and computation of {Covariant {Lyapunov} Vectors}},
journal = {J. Nonlin. Sci.},
year = {2012},
volume = {22},
pages = {727--762},
note = {\arXiv{1105.5228}},
doi = {10.1007/s00332-012-9126-5}
}
@Article{Kuptsov12,
Title = {Fast numerical test of hyperbolic chaos},
Author = {Kuptsov, P. V.},
Journal = {Phys. Rev. E},
Year = {2012},
Pages = {015203},
Volume = {85},
DOI = {10.1103/PhysRevE.85.015203}
}
@Article{Kuptsov13,
Title = {Violation of hyperbolicity via unstable dimension variability in a chain with local hyperbolic chaotic attractors},
Author = {Kuptsov, P. V.},
Journal = {J. Phys. A},
Year = {2013},
Pages = {254016},
Volume = {46},
DOI = {10.1088/1751-8113/46/25/254016}
}
@Book{kura84tur,
Title = {Chemical Oscillations, Waves and Turbulence},
Author = {Y. Kuramoto},
Publisher = {Springer},
Year = {1984},
Address = {New York}
}
@Article{kuramitsu2012,
Title = {Laboratory investigations on the origins of cosmic rays},
Author = {Kuramitsu, Y. and Sakawa, Y. and Morita, T. and Ide, T. and Nishio, K. and Tanji, H. and Aoki, H. and Dono, S. and Gregory, C. D. and Waugh, J. N. and Woolsey, N. and Dizi?re, A. and Pelka, A. and Ravasio, A. and Loupias, B. and Koenig, M. and Pikuz, S. A. and Li, Y. T. and Zhang, Y. and Liu, X. and Zhong, J. Y. and Zhang, J. and Gregori, G. and Nakanii, N. and Kondo, K. and Mori, Y. and Miura, E. and Kodama, R. and Kitagawa, Y. and Mima, K. and Tanaka, K. A. and Azechi, H. and Moritaka, T. and Matsumoto, Y. and Sano, T. and Mizuta, A. and Ohnishi, N. and Hoshino, M. and Takabe, H.},
Journal = {Plasma Phys. Control. Fusion},
Year = {2012},
Pages = {124049},
Volume = {54},
Abstract = {We report our recent efforts on the experimental investigations related
to the origins of cosmic rays. The origins of cosmic rays are long
standing open issues in astrophysics. The galactic and extragalactic
cosmic rays are considered to be accelerated in non-relativistic
and relativistic collisionless shocks in the universe, respectively.
However, the acceleration and transport processes of the cosmic rays
are not well understood, and how the collisionless shocks are created
is still under investigation. Recent high-power and high-intensity
laser technologies allow us to simulate astrophysical phenomena in
laboratories. We present our experimental results of collisionless
shock formations in laser-produced plasmas.},
DOI = {10.1088/0741-3335/54/12/124049}
}
@Article{KurTsu75,
author = {Kuramoto, Y. and Tsuzuki, T.},
title = {On the formation of dissipative structures in reaction--diffusion systems},
journal = {Prog. Theor. Phys.},
year = {1975},
volume = {54},
pages = {687--699},
doi = {10.1143/PTP.55.356}
}
@Article{KurTsu76,
author = {Y. Kuramoto and T. Tsuzuki},
title = {Persistent propagation of concentration waves in dissipative media far from thermal equilibrium},
journal = {Progr. Theor. Phys.},
year = {1976},
volume = {55},
pages = {365--369},
doi = {10.1143/PTP.55.356},
abstract = {The KSe is derived through the CGLe near the
bifurcation point. Some scaling argument is used to
sort out the domiant terms.}
}
@Article{kuturb78,
Title = {Diffusion-induced chaos in reaction systems},
Author = {Y. Kuramoto},
Journal = {Suppl. Progr. Theor. Phys.},
Year = {1978},
Pages = {346--367},
Volume = {64},
Abstract = {Phase turbulence and amplitude turbulence are named and distinguished
from a dynamical systems point of view. The prototyped equations
are derived.}
}
@Book{Kuzn04,
Title = {Elements of Applied Bifurcation Theory},
Author = {Y. A. Kuznetsov},
Publisher = {Springer},
Year = {2004},
Address = {New York}
}
@Article{Lai97,
Title = {Characterization of the natural measure by unstable periodic orbits in nonhyperbolic chaotic systems},
Author = {Y.-C. Lai}
}
@Article{LaKo89,
Title = {Characterization of an experimental strange attractor by periodic orbits},
Author = {D. P. Lathrop and E. J. Kostelich},
Journal = {Phys. Rev. A},
Year = {1989},
Pages = {4028--4031},
Volume = {40}
}
@Article{lamb_bifurcationperiodic_2003,
Title = {General bifurcation from periodic solutions with spatiotemporal symmetry, including resonances and mode interactions},
Author = {Lamb, J. S. W. and Melbourne, I. and Wulff, C.},
Journal = {J. Diff. Eqn.},
Year = {2003},
Pages = {377--407},
Volume = {191}
}
@Article{lamb98,
Title = {Time reversal symmetry in dynamical systems: {A} survey},
Author = {J. S. W. Lamb and J. A. G. Roberts},
Journal = {Physica D},
Year = {1998},
Pages = {1},
Volume = {112}
}
@Article{LaMeWu06,
author = {Lamb, J. S. W. and Melbourne, I. and Wulff, C. },
title = {Hopf bifurcation from relative periodic solutions; secondary bifurcations from meandering spirals},
journal = {J. Difference Eqn. and Appl.},
year = {2006},
volume = {12},
pages = {1127--1145},
doi = {10.1080/10236190601045747},
abstract = { We consider nonresonant and weakly resonant Hopf
bifurcation from periodic solutions and relative periodic solutions
in dynamical systems with symmetry. In particular, we analyse
phase-locking and irrational torus flows on the bifurcating relative
tori. Results are obtained for systems with compact and noncompact
symmetry groups. In the noncompact case, we distinguish between
bounded and unbounded dynamics. Applications of our results include
secondary Hopf bifurcation from meandering multi-armed spirals. }
}
@Article{LaMi99,
author = {Lahme, B. and Miranda, R.},
title = {{Karhunen-Loeve} decomposition in the presence of symmetry. {I}},
journal = {IEEE Trans. Image Processing},
year = {1999},
volume = {8},
pages = {1183--1190},
abtract = {The Karhunen-Loeve (KL) decomposition is widely used for data which very often exhibit some symmetry, afforded by a group action. For a finite group, we derive an algorithm using group representation theory to reduce the cost of determining the KL basis. We demonstrate the technique on a Lorenz-type ODE system. For a compact group such as tori or SO(3,R) the method also applies, and we extend results to these cases. As a short example, we consider the circle group S1.}
}
@Article{Lan10,
Title = {Cycle expansions: {From} maps to turbulence},
Author = {Y. Lan},
Journal = {Commun. Nonlinear Sci. Numer. Simul.},
Year = {2010},
Pages = {502--526},
Volume = {15},
Abstract = {We present a derivation, a physical explanation and applications of
cycle expansions in different dynamical systems, ranging from simple
one-dimensional maps to turbulence in fluids. Cycle expansion is
a newly devised powerful tool for computing averages of physical
observables in nonlinear chaotic systems which combines many innovative
ideas developed in dynamical systems, such as hyperbolicity, invariant
manifolds, symbolic dynamics, measure theory and thermodynamic formalism.
The concept of cycle expansion has a deep root in theoretical physics,
bearing a close analogy to cumulant expansion in statistical physics
and effective action functional in quantum field theory, the essence
of which is to represent a physical system in a hierarchical way
by utilizing certain multiplicative structures such that the dominant
parts of physical observables are captured by compact, maneuverable
objects while minor detailed variations are described by objects
with a larger space and time scale. The technique has been successfully
applied to many low-dimensional dynamical systems and much effort
has recently been made to extend its use to spatially extended systems.
For one-dimensional systems such as the Kuramoto-Sivashinsky equation,
the method turns out to be very effective while for more complex
real-world systems including the Navier-Stokes equation, the method
is only starting to yield its first fruits and much more work is
needed to enable practical computations. However, the experience
and knowledge accumulated so far is already very useful to a large
set of research problems. Several such applications are briefly described
in what follows. As more research effort is devoted to the study
of complex dynamics of nonlinear systems, cycle expansion will undergo
a fast development and find wide applications.},
DOI = {10.1016/j.cnsns.2009.04.022}
}
@Article{Lan11,
author = {Lan, Y.},
title = {Novel computation of the growth rate of generalized random {Fibonacci} sequences},
journal = {J. Stat. Phys.},
year = {2011},
volume = {142},
pages = {847--861},
doi = {10.1007/s10955-011-0132-z}
}
@Article{lanCvit07,
Title = {Unstable recurrent patterns in {Kuramoto-Sivashinsky} dynamics},
Author = {Y. Lan and P. Cvitanovi{\'c}},
Journal = {Phys. Rev. E},
Year = {2008},
Note = {\arXiv{0804.2474}},
Pages = {026208},
Volume = {78}
}
@Article{landau44,
Title = {On the problem of turbulence},
Author = {L. D. Landau},
Journal = {Dokl. Akad. Nauk SSSR},
Year = {1944},
Pages = {339},
Volume = {44}
}
@Book{Landau59a,
Title = {Fluid Mechanics},
Author = {L.D. Landau and E.M. Lifshitz},
Publisher = {Pergamon Press},
Year = {1959},
Address = {Oxford}
}
@Book{Landau59b,
Title = {Mechanics},
Author = {L.D. Landau and E.M. Lifshitz},
Publisher = {Pergamon Press},
Year = {1959},
Address = {Oxford}
}
@Book{Landau59c,
Title = {Quantum Mechanics},
Author = {L.D. Landau and E.M. Lifshitz},
Publisher = {Pergamon Press},
Year = {1959},
Address = {Oxford}
}
@Book{Landau60,
Title = {Electrodynamics of Continuous Media},
Author = {L.D. Landau and E.M. Lifshitz},
Publisher = {Pergamon Press},
Year = {1960},
Address = {Oxford}
}
@Book{Landau80a,
Title = {Statistical Physics, Part 1},
Author = {L.D. Landau and E.M. Lifshitz},
Publisher = {Pergamon Press},
Year = {1980},
Address = {Oxford}
}
@Book{Landau80b,
Title = {Statistical Physics, Part 2},
Author = {L.D. Landau and E.M. Lifshitz},
Publisher = {Pergamon Press},
Year = {1980},
Address = {Oxford}
}
@Book{langford_normal_1995,
Title = {Normal Forms and Homoclinic Chaos},
Author = {W. F. Langford and W. Nagata},
Publisher = {AMS},
Year = {1995},
Pages = {294}
}
@Article{LaNgKuTa13,
Title = {Large deviations of {Lyapunov} exponents},
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Journal = {J. Phys. A},
Year = {2013},
Pages = {254002},
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DOI = {10.1088/1751-8113/46/25/254002}
}
@Article{lanmaw03,
Title = {Stationary modulated-amplitude waves in the 1{D} complex {Ginzburg-Landau} equation},
Author = {Y. Lan and N. Garnier and P. Cvitanovi\'{c}},
Journal = {Physica D},
Year = {2004},
Pages = {193--212},
Volume = {188}
}
@PhdThesis{LanThesis,
author = {Y. Lan},
title = {Dynamical Systems Approach to {$1-d$} Spatiotemporal Chaos -- {A} Cyclist's View},
school = {School of Physics, Georgia Inst. of Technology},
year = {2004},
address = {Atlanta},
url = {http://ChaosBook.org/projects/theses.html}
}
@Article{lanVar1,
Title = {Variational method for finding periodic orbits in a general flow},
Author = {Y. Lan and P. Cvitanovi{\'c}},
Journal = {Phys. Rev. E},
Year = {2004},
Note = {\arXiv{nlin.CD/0308008}},
Pages = {016217},
Volume = {69}
}
@Article{laquey74,
Title = {Nonlinear saturation of the trapped-ion mode},
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@Article{LaSa97,
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Journal = {J. Dynam. Diff. Eq.},
Year = {1997},
Pages = {535--560},
Volume = {9},
DOI = {10.1007/BF02219397}
}
@Book{latt,
Title = {Quantum Fields on the Computer},
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Publisher = {World Scientific},
Year = {1992},
Address = {Singapore}
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Year = {2000},
Pages = {C735--C751},
Volume = {42E},
Abstract = {A lattice refinement scheme is proposed based on the linear approximation
of a given map. Jacobian matrix is not calculated directly but inferred
from the neighboring points.}
}
@Book{Lau05,
Title = {Physics of Continuous Matter: Exotic and Everyday Phenomena in the Macroscopic World},
Author = {Lautrup, B.},
Publisher = {{CRC Press}},
Year = {2011},
Address = {{Boca Raton, FL}}
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Author = {Y. -T. Lau},
Journal = {Int. J. Bifur. Chaos},
Year = {1992},
Pages = {543--558},
Volume = {2}
}
@Unpublished{LauOrl12,
Title = {The geometry of {Yang-Mills} orbit space on the lattice},
Author = {Laufer, M. and Orland, P.},
Note = {\arXiv{1203.5134}},
Year = {2012}
}
@Article{Laur91,
Title = {Discrete symmetries and the periodic-orbit expansions},
Author = {B. Lauritzen },
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Volume = {43}
}
@Unpublished{Laurent-Polz04,
Title = {Relative periodic orbits in point vortex systems},
Author = {Frederic Laurent-Polz},
Note = {\arXiv{math/0401022}},
Year = {2004}
}
@Incollection{lauw86,
Author = {H. A. Lauwerier},
Booktitle = {Chaos},
Publisher = {Princeton Univ. Press},
Year = {1986},
Address = {Princeton, NJ},
Editor = {V. Holden},
Pages = {39}
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author = {Lamb, J. S. W. and Wulff, C.},
title = {Reversible relative periodic orbits},
journal = {J. Diff. Eqn.},
year = {2002},
volume = {178},
pages = {60--100},
doi = {10.1006/jdeq.2001.4004}
}
@Article{LCC06,
Title = {Variational method for locating invariant tori},
Author = {Y. Lan and C. Chandre and P. Cvitanovi{\'c}},
Journal = {Phys. Rev. E},
Year = {2006},
Note = {\arXiv{nlin.CD/0508026}},
Pages = {046206},
Volume = {74}
}
@Article{LeAllHa00,
Title = {Sensitivity analysis of the climate of a chaotic system},
Author = {Lea, D. J. and Allen, M. R. and Haine, T. W. N.},
Journal = {Tellus A},
Year = {2000},
Pages = {523--532},
Volume = {52},
Abstract = {This paper addresses some fundamental methodological issues concerning
the sensitivity analysis of chaotic geophysical systems. We show,
using the Lorenz system as an example, that a naive approach to variational
(``adjoint'') sensitivity analysis is of limited utility. Applied
to trajectories which are long relative to the predictability time
scales of the system, cumulative error growth means that adjoint
results diverge exponentially from the ``macroscopic climate sensitivity''(that
is, the sensitivity of time-averaged properties of the system to
finite-amplitude perturbations). This problem occurs even for time-averaged
quantities and given infinite computing resources. Alternatively,
applied to very short trajectories, the adjoint provides an incorrect
estimate of the sensitivity, even if averaged over large numbers
of initial conditions, because a finite time scale is required for
the model climate to respond fully to certain perturbations. In the
Lorenz (1963) system, an intermediate time scale is found on which
an ensemble of adjoint gradients can give a reasonably accurate (O(10%))
estimate of the macroscopic climate sensitivity. While this ensemble-adjoint
approach is unlikely to be reliable for more complex systems, it
may provide useful guidance in identifying important parameter-combinations
to be explored further through direct finite-amplitude perturbations.},
DOI = {10.1034/j.1600-0870.2000.01137.x}
}
@Article{LeBlanc00,
Title = {Translational symmetry-breaking for spiral waves},
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Pages = {569--601},
Volume = {10}
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Title = {Periodic orbit spectrum in terms of {Ruelle-Pollicott} resonances},
Author = {Leboeuf, P.},
Journal = {Phys. Rev. E},
Year = {2004},
Note = {\arXiv{nlin/0406042}},
Pages = {026204},
Volume = {69},
DOI = {10.1103/PhysRevE.69.026204},
Numpages = {13}
}
@Article{Lega:holes,
Title = {Traveling hole solutions of the complex {Ginzburg-Landau} equation: a review},
Author = {J. Lega},
Journal = {Physica D},
Year = {2001},
Pages = {152--153},
Volume = {269}
}
@article{Lehoucq1996,
author = {Lehoucq, R. B. and Sorensen, D. C.},
doi = {10.1137/S0895479895281484},
issn = {0895-4798},
journal = {SIAM J. Matrix Anal. Appl.},
pages = {789--821},
title = {{Deflation Techniques for an Implicitly Restarted Arnoldi Iteration}},
volume = {17},
year = {1996}
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@Article{Leis97,
author = {Leis, C.},
title = {Hopf-bifurcation in systems with spherical symmetry. {I: Invariant} tori},
journal = {Documenta Mathematica},
year = {1997},
volume = {2},
pages = {61--113},
url = {http://www.emis.ams.org/journals/DMJDMV/vol-02/04.pdf}
}
@Misc{LeoKuz15,
Title = {A short survey on {Lyapunov} dimension for finite dimensional dynamical systems in {Euclidean} space},
Author = {{Leonov}, G. A. and {Kuznetsov}, N. V.},
Note = {\arXiv{1510.03835}},
Year = {2015}
}
@Article{LePoTo96,
author = {S. Lepri and A. Politi and A. Torcini},
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journal = {J. Stat. Phys.},
year = {1996},
volume = {82},
pages = {1429--1452},
doi = {10.1007/BF02183390},
url = {http://arxiv.org/abs/chao-dyn/9504005}
}
@Article{LePoTo97,
author = {S. Lepri and A. Politi and A. Torcini},
title = {Chronotopic {Lyapunov} analysis. {II. Towards} a unified approach},
journal = {J. Stat. Phys.},
year = {1997},
volume = {88},
pages = {31--45},
doi = {10.1007/BF02508463},
url = {http://arxiv.org/abs/chao-dyn/9602012}
}
@Article{Lesne07,
author = {A. Lesne},
title = {The discrete versus continuous controversy in physics},
journal = {Math. Struct. Computer Sci.},
year = {2007},
volume = {17},
pages = {185--223},
doi = {10.1017/S0960129507005944}
}
@Article{Letellier03,
Title = {Modding out a continuous rotation symmetry for disentangling a laser dynamics},
Author = {C. Letellier},
Journal = {Int. J. Bifur. Chaos},
Year = {2003},
Pages = {1573--1577},
Volume = {13}
}
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Year = {1944},
Pages = {164--168},
Volume = {2}
}
@Misc{LevnMezi08,
Title = {Ergodic theory and visualization {I}: {Visualization} of ergodic partition and invariant sets},
Author = {Levnaji\'c, Z. and Mezi\'c, I.},
Note = {\arXiv{0805.4221}},
Year = {2008},
Abstract = {We present a computational study of the invariant sets visualization
method based on ergodic partition theory. The algorithms for computation
of the time averages of many L1 functions are developed and employed
producing the approximation of the phase space ergodic partitioning.
The method is exposed in the context of discrete-time dynamical systems
producing a graphical representation of the phase space in terms
of the invariant set structure that gives a substantial insight into
the global and local properties of the dynamics. We use the Chirikov
standard map in order to show the implementation of the method, followed
by applications to other multi-dimensional maps. We examine the visualization
of periodic sets using harmonic time averages.}
}
@Article{lfind,
Title = {Systematic computation of the least unstable periodic orbits in chaotic attractors},
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Note = {\arXiv{chao-dyn/9810022}},
Pages = {4349},
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Author = {Liao, S. J.},
Journal = {Tellus-A},
Year = {2009},
Note = {\arXiv{0901.2986}},
Pages = {550--564},
Volume = {61}
}
@Misc{Liao11,
Title = {Chaos: a bridge from microscopic uncertainty to macroscopic randomness},
Author = {Liao, S. J.},
Note = {\arXiv{1108.4472}},
Year = {2011}
}
@Misc{Liao11a,
Title = {On the numerical simulation of propagation of micro-level uncertainty for chaotic dynamic systems},
Author = {Liao, S. J.},
Note = {\arXiv{1109.0130}},
Year = {2011}
}
@Article{Lichtner2007,
Title = {Well-posedness, smooth dependence and center manifold reduction for a semilinear hyperbolic system from laser dynamics},
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Year = {2007},
Pages = {931--960},
Volume = {30}
}
@Book{liede,
Title = {Applications of Lie Groups to Differential Equations},
Author = {P. J. Olver},
Publisher = {Springer},
Year = {1998},
Address = {New York}
}
@Article{LiMaFeCh10,
Title = {Locating unstable periodic orbits: {When} adaptation integrates into delayed feedback control},
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Journal = {Phys. Rev. E},
Year = {2010},
Pages = {046214},
Volume = {82},
DOI = {10.1103/PhysRevE.82.046214}
}
@Book{LindMar95,
Title = {An Introduction to Symbolic Dynamics and Coding},
Author = {D.A. Lind and B. Marcus},
Publisher = {Cambridge Univ. Press},
Year = {1995},
Address = {Cambridge}
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author = {Lin, S. P.},
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year = {1974},
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Title = {Optimal resolution of the state space of a chaotic flow in presence of noise},
Author = {D. Lippolis and P. Cvitanovi\'c},
Note = {In preparation},
Year = {2012}
}
@Article{LipCvi08,
Title = {How well can one resolve the state space of a chaotic map?},
Author = {D. Lippolis and P. Cvitanovi\'c},
Journal = {Phys. Rev. Lett.},
Year = {2010},
Note = {\arXiv{0902.4269}},
Pages = {014101},
Volume = {104},
DOI = {10.1103/PhysRevLett.104.014101}
}
@Book{Lipkin66,
Title = {Lie Groups for Pedestrians},
Author = {Lipkin , H. J.},
Publisher = {North-Holland},
Year = {1966},
Address = {Amsterdam}
}
@Phdthesis{LippolisThesis,
Title = {How well can one resolve the state space of a chaotic map?},
Author = {Lippolis, D.},
School = {School of Physics, Georgia Inst. of Technology},
Year = {2010},
Address = {Atlanta},
URL = {http://ChaosBook.org/projects/theses.html}
}
@Article{LiRe97,
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Journal = {Rev. Mod. Phys.},
Year = {1997},
Pages = {213--275},
Volume = {69},
DOI = {10.1103/RevModPhys.69.213}
}
@Unpublished{liuliupego05,
Title = {Divorcing pressure from viscosity in incompressible {N}avier-{S}tokes dynamics},
Author = {J.-G. Liu and J. Liu and R. L. Pego},
Note = {\arXiv{math.AP/0502549}},
Year = {2005}
}
@Incollection{Llibre15,
author = {Llibre, J.},
title = {The averaging theory for computing periodic orbits},
booktitle = {Central Configurations, Periodic Orbits, and Hamiltonian Systems},
publisher = {Springer},
year = {2015},
isbn = {978-3-0348-0932-0},
series = {Advanced Courses in Mathematics - CRM Barcelona},
pages = {1--104},
address = {Basel},
doi = {10.1007/978-3-0348-0933-7_1}
}
@Book{lls,
Title = {Systems of Quasilinear Equations and Their Applications to Gas Dynamics},
Author = {B. L. Ro\v{z}destvenski\v{i} and N. N. Janenko},
Publisher = {AMS},
Year = {1983},
Address = {Providence, RI}
}
@Book{LM94,
Title = {Chaos, Fractals, and Noise; Stochastic Aspects of Dynamics},
Author = {A. Lasota and M. MacKey},
Publisher = {Springer},
Year = {1994},
Address = {New York}
}
@Article{LoCaCoPeGo11,
author = {Lombardi, M. and Caulfield, C.P. and Cossu, C. and Pesci, A.I. and Goldstein, R.E.},
title = {Growth and instability of a laminar plume in a strongly stratified environment},
journal = {J.\ Fluid Mech.},
year = {2011},
volume = {671},
pages = {184--206},
doi = {10.1017/S0022112010005574}
}
@Article{lop05rel,
Title = {Relative periodic solutions of the complex {Ginzburg-Landau} equation},
Author = {V. L{\'o}pez and P. Boyland and M. T. Heath and R. D. Moser},
Journal = {SIAM J. Appl. Dyn. Syst.},
Year = {2006},
Note = {\arXiv{nlin/0408018}},
Pages = {1042--1075},
Volume = {4},
Abstract = {Define relative periodic orbits and use Fourier modes to find them.},
DOI = {10.1137/040618977}
}
@Unpublished{Lopez2015,
Title = {Numerical continuation of invariant solutions of the complex {Ginzburg--Landau} equation},
Author = {V. L\'{o}pez},
Note = {\arXiv{1502.03862}},
Year = {2015}
}
@Unpublished{lopezLink,
Author = {Vanessa L{\'o}pez},
Note = {private communication}
}
@Article{Lorentz1905,
author = {Lorentz, H. A.},
title = {The motion of electrons in metallic bodies},
journal = {K. Ned. Akad. van Wet. B},
year = {1905},
volume = {7},
pages = {438--453},
url = {http://www.dwc.knaw.nl/DL/publications/PU00013989.pdf}
}
@Article{LorentzDiff,
Title = {Transport properties of the {Lorentz} gas in terms of periodic orbits},
Author = {Cvitanovi\'c, P. and Eckmann, J.-P. and Gaspard, P.},
Journal = {Chaos Solit. Fract.},
Year = {1995},
Note = {\arXiv{chao-dyn/9312003}},
Pages = {113--120},
Volume = {6},
DOI = {10.1016/0960-0779(95)80018-C}
}
@Article{lorenz,
Title = {Deterministic nonperiodic flow},
Author = {E. N. Lorenz},
Journal = {J. Atmos. Sci.},
Year = {1963},
Pages = {130--141},
Volume = {20}
}
@Article{Lorenz60,
author = {E. N. Lorenz},
title = {Maximum simplification of the dynamics equations},
journal = {Quaterly J. Geophys.},
year = {1960},
volume = {12},
pages = {243--254}
}
@Article{Lorenz63a,
Title = {The Mechanics of vacillation},
Author = {E. N. Lorenz},
Journal = {J. Meteorology},
Year = {1963},
Pages = {448--465},
Volume = {20},
DOI = {10.1175/1520-0469(1963)020<0448:TMOV>2.0.CO;2}
}
@Article{Lorenz65,
Title = {A study of the predictability of a 28-variable atmospheric model},
Author = {Lorenz, E. N.},
Journal = {Tellus},
Year = {1965},
Pages = {321--333},
Volume = {17},
Abstract = {A 28-variable model of the atmosphere is constructed by expanding
the equations of a two-level geostrophic model in truncated double-Fourier
series. The model includes the nonlinear interactions among disturbances
of three different wave lengths. Nonperiodic time-dependent solutions
are determined by numerical integration.By comparing separate solutions
with slightly different initial conditions, the growth rate of small
initial errors is studied. The time required for errors comparable
to observational errors in the atmosphere to grow to intolerable
errors is strongly dependent upon the current circulation pattern,
and varies from a few days to a few weeks.Some statistical predictability
of certain quantities seems to be present even after errors in the
complete circulation pattern are no longer small. The feasibility
of performing similar studies with much larger atmospheric models
is considered.},
DOI = {10.1111/j.2153-3490.1965.tb01424.x}
}
@Article{lorenz69,
Title = {The predictability of a flow which possessesmany scales of motion},
Author = {E. N. Lorenz},
Journal = {Tellus},
Year = {1969},
Pages = {289--307},
Volume = {21}
}
@Article{Lorenz84,
Title = {Irregularity: a fundamental property of the atmosphere},
Author = {Lorenz, E. N.},
Journal = {Tellus A},
Year = {1984},
Pages = {98--110},
Volume = {36},
Abstract = {Some early ideas concerning the general circulation of the atmosphere
are reviewed. A model of the general circulation, consisting of three
ordinary differential equations, is introduced. For different intensities
of the axially symmetric and asymmetric thermal forcing, the equations
may possess one or two stable steady-state solutions, one or two
stable periodic solutions, or irregular (aperiodic) solutions. Qualitative
reasoning which has been applied to the real atmosphere may sometimes
be applied to the model, and checked for soundness by comparing the
conclusions with numerical solutions. The implications of irregularity
for the atmosphere and for atmospheric science are discussed.},
DOI = {10.1111/j.1600-0870.1984.tb00230.x}
}
@Article{LoTh10,
Title = {Computing stochastic travelling waves},
Author = {G. J. Lord and V. Th\"{u}mmler},
Journal = {SIAM J. Sci. Comput.},
Year = {2012},
Note = {\arXiv{1006.0428}},
Pages = {B24--B43},
Volume = {34}
}
@Article{Low58,
Title = {A {{Lagrangian}} formulation of the {{Boltzmann-Vlasov}} equation for plasmas},
Author = {F. E. Low},
Journal = {Proc. R. Soc. London A},
Year = {1958},
Pages = {282--287},
Volume = {248},
Abstract = {A variational principle is found for the {Boltzmann-Vlasov} equation
for an ionized gas in an electromagnetic field. The principle leads
to a new formulation of the problem of small oscillations about equilibrium.}
}
@Article{lozi2,
Title = {Un attracteur {\'e}trange (?) du type attracteur de {H\'enon}},
Author = {R. Lozi},
Journal = {J. Phys. (Paris) Colloq.},
Year = {1978},
Pages = {C5--9},
Volume = {39},
DOI = {10.1051/jphyscol:1978505}
}
@Article{LPKDS09,
Title = {Evolutionary phase space in driven elliptical billiards},
Author = {F. Lenz and C. Petri and F. N. R. Koch and F. K. Diakonos and P. Schmelcher},
Journal = {New J. Physics},
Year = {2009},
Note = {\arXiv{0904.3636}},
Pages = {083035},
Volume = {11}
}
@Article{LucDoe1993,
author = {{Luce}, B. P. and {Doering}, C. R.},
title = {Scaling of turbulent spike amplitudes in the complex {Ginzburg-Landau} equation},
journal = {Physics Lett. A},
year = {1993},
volume = {178},
pages = {92--98}
}
@Article{Luce95,
Title = {Homoclinic explosions in the complex {Ginzburg-Landau} equation},
Author = {B. P. Luce},
Journal = {Physica D},
Year = {1995},
Pages = {553--581},
Volume = {84}
}
@Article{LundJohan91,
Title = {Direct simulation of turbulent spots in plane {Couette} flow},
Author = {A. Lundbladh and A. Johansson},
Journal = {J. Fluid Mech.},
Year = {1991},
Pages = {499--516},
Volume = {229}
}
@Book{LuoGuo12,
Title = {Vibro-impact Dynamics},
Author = {Luo, A.C.J. and Guo, Y.},
Publisher = {Wiley},
Year = {2012},
Address = {New York}
}
@Article{Lust01,
author = {K. Lust},
title = {Improved numerical {F}loquet multipliers},
journal = {Int. J. Bifur. Chaos},
year = {2001},
volume = {11},
pages = {2389--2410}
}
@Article{LXSH08,
author = {X. Li and Y. Xue and P. Shi and G. Hu},
title = {Lyapunov spectra of coupled chaotic maps},
journal = {Int. J. Bifur. Chaos},
year = {2008},
volume = {18},
pages = {3759--3770},
doi = {10.1142/S0218127408022718}
}
@Article{LY75,
Title = {Period 3 implies chaos},
Author = {T. Li and J. A. Yorke},
Journal = {Amer. Math. Monthly},
Year = {1975},
Pages = {985},
Volume = {82}
}
@Article{lyaos,
author = {V. I. Oseledec},
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journal = {Trans. Moscow Math. Soc.},
year = {1968},
volume = {19},
pages = {197--221},
abstract = {The Liapunov characteristic numbers are defined by investigating the
limiting volume of infinitesimal size. Their existence is proved
by construction and application of the ergodic theorem. At a.e. every
point, the tangent space is divided into stable and unstable subspace
corresponding the positive and negative Liapunov numbers. If the
eigenvalues are non-degenerate, the Liapunov vectors are well defined
a.e.},
url = {http://mi.mathnet.ru/eng/mmo214}
}
@Article{Lyap1892,
Title = {Probl\`eme g\'en\'eral de la stabilit\'e du mouvement},
Author = {A. Lyapunov},
Journal = {Ann. of Math. Studies},
Year = {1977},
Note = {Russian original Kharkow, 1892},
Volume = {17}
}
@Book{lyschm,
Title = {{Lyapunov-Schmidt} Methods in Nonlinear Analysis and Applications},
Author = {N. Sidorov and B. Loginov and A. Sinitsyn and M. Falaleev},
Publisher = {Kluwer},
Year = {2002},
Address = {Dordrecht}
}
@Article{M86,
Title = {Critical layers in shear flows},
Author = {S. A. Maslowe},
Journal = {Ann. Rev. Fluid Mech.},
Year = {1986},
Pages = {405--432},
Volume = {18}
}
@Article{Maas11se,
author = {Maas, A.},
title = {Describing gauge bosons at zero and finite temperature},
journal = {Phys. Rept.},
year = {2013},
volume = {524},
pages = {203--300},
note = {\arXiv{1106.3942}},
doi = {10.1016/j.physrep.2012.11.002}
}
@Article{macchi2013,
Title = {Ion acceleration by superintense laser-plasma interaction},
Author = {Macchi, A. and Borghesi, M. and Passoni, M.},
Journal = {Rev. Mod. Phys.},
Year = {2013},
Pages = {751--793},
Volume = {85},
Abstract = {Ion acceleration driven by superintense laser pulses is attracting
an impressive and steadily increasing effort. Motivations can be
found in the applicative potential and in the perspective to investigate
novel regimes as available laser intensities will be increasing.
Experiments have demonstrated, over a wide range of laser and target
parameters, the generation of multi-{MeV} proton and ion beams with
unique properties such as ultrashort duration, high brilliance, and
low emittance. An overview is given of the state of the art of ion
acceleration by laser pulses as well as an outlook on its future
development and perspectives. The main features observed in the experiments,
the observed scaling with laser and plasma parameters, and the main
models used both to interpret experimental data and to suggest new
research directions are described.},
DOI = {10.1103/RevModPhys.85.751}
}
@Article{mackmeiss84,
Title = {Transport in {Hamiltonian} systems},
Author = {MacKay, R. S. and Meiss, J. D. and I. C. Percival},
Journal = {Physica D},
Year = {1984},
Pages = {55},
Volume = {13}
}
@Article{mackmeiss87,
Title = {Resonances in area preserving maps},
Author = {MacKay, R. S. and Meiss, J. D. and I. C. Percival},
Journal = {Physica D},
Year = {1987},
Pages = {1},
Volume = {27}
}
@Article{MacSch12,
Title = {Measuring shape with topology},
Author = {R. MacPherson and B. Schweinhart},
Journal = {J. Math. Phys.},
Year = {2012},
Pages = {073516},
Volume = {53},
DOI = {10.1063/1.4737391}
}
@Article{MacZwa83,
Title = {Diffusion in a periodic {Lorentz} gas},
Author = {Machta, J. and Zwanzig, R.},
Journal = {Phys. Rev. Lett.},
Year = {1983},
Pages = {1959--1962},
Volume = {50},
DOI = {10.1103/PhysRevLett.50.1959}
}
@Article{mahmoud08a,
Title = {Analysis of hyperchaotic complex {Lorenz} systems},
Author = {Mahmoud, G. M. and Ahmed, M. E. and Mahmoud, E. E.},
Journal = {Int. J. Modern Phys. C},
Year = {2008},
Pages = {1477--1494},
Volume = {19}
}
@Article{mahmoud08b,
Title = {Chaotic and hyperchaotic attractors of a complex nonlinear system},
Author = {Mahmoud, G.l M. and Al-Kashif, M. A. and Farghaly, A. A.},
Journal = {J. Phys. A},
Year = {2008},
Volume = {41}
}
@Article{mahmoud09,
Title = {Dynamical properties and synchronization of complex non-linear equations for detuned lasers},
Author = {Mahmoud, G. M. and Bountis, T. and Al-Kashif, M. A. and Aly, S. A.},
Journal = {Dyn. Sys.},
Year = {2009},
Pages = {63--79},
Volume = {24}
}
@Unpublished{Mainieri04,
Title = {Can averaged orbits be used to extract scaling functions?},
Author = {R. Mainieri},
Note = {\arXiv{chao-dyn/9302004}},
Year = {1993}
}
@Article{Mainieri92a,
author = {R. Mainieri},
title = {Thermodynamic zeta functions for {Ising} models with long-range interactions},
journal = {Phys. Rev. A},
year = {1992},
volume = {45},
pages = {3580--3591}
}
@Article{Mainieri92b,
author = {R. Mainieri},
title = {Zeta-Function for the {Lyapunov} exponent of a product of random matrices},
journal = {Phys. Rev. Lett.},
year = {1992},
volume = {68},
pages = {1965--1968}
}
@Book{Majda03,
Title = {Introduction to PDEs and Waves for the Atmosphere and Ocean},
Author = {A. Majda},
Publisher = {American Mathematical Society},
Year = {2003},
Address = {New York}
}
@Article{MajSte11,
author = {Majda, A. J. and Stechmann, S. N.},
title = {Nonlinear dynamics and regional variations in the {MJO} skeleton},
journal = {J. Atmos. Sci.},
year = {2011},
volume = {68},
pages = {3053--3071},
doi = {10.1175/JAS-D-11-053.1}
}
@Article{Mak87,
Title = {Dissipative structure of a nonlinear baroclinic system: {Effect} of asymmetric friction},
Author = {Mak, M.},
Journal = {J. Atmos. Sci.},
Year = {1987},
Pages = {2613--2627},
Volume = {44},
DOI = {10.1175/1520-0469(1987)044<2613:DSOANB>2.0.CO;2}
}
@Book{man90b,
Title = {Dissipative Structures and Weak Turbulence},
Author = {Manneville, P.},
Publisher = {Academic},
Year = {1990},
Address = {New York}
}
@Book{mandel,
Title = {The Fractal Geometry of Nature},
Author = {B. B. Mandelbrot},
Publisher = {WH Freeman Co.},
Year = {1982},
Address = {New York}
}
@Article{mangles04,
Title = {Monoenergetic beams of relativistic electrons from intense laser-plasma interactions},
Author = {Mangles, S. P. D. and Murphy, C. D. and Najmudin, Z. and Thomas, A. G. R. and Collier, J. L. and Dangor, A. E. and Divall, E. J. and Foster, P. S. and Gallacher, J. G. and Hooker, C. J. and Jaroszynski, D. A. and Langley, A. J. and Mori, W. B. and Norreys, P. A. and Tsung, F. S. and Viskup, R. and Walton, B. R. and Krushelnick, K.},
Journal = {Nature},
Year = {2004},
Month = sep,
Number = {7008},
Pages = {535--538},
Volume = {431},
Abstract = {High-power lasers that fit into a university-scale laboratory can
now reach focused intensities of more than 1019 W cm-2 at high repetition
rates. Such lasers are capable of producing beams of energetic electrons,
protons and -rays. Relativistic electrons are generated through the
breaking of large-amplitude relativistic plasma waves created in
the wake of the laser pulse as it propagates through a plasma, or
through a direct interaction between the laser field and the electrons
in the plasma. However, the electron beams produced from previous
laser?plasma experiments have a large energy spread, limiting their
use for potential applications. Here we report high-resolution energy
measurements of the electron beams produced from intense laser?plasma
interactions, showing that?under particular plasma conditions?it
is possible to generate beams of relativistic electrons with low
divergence and a small energy spread (less than three per cent).
The monoenergetic features were observed in the electron energy spectrum
for plasma densities just above a threshold required for breaking
of the plasma wave. These features were observed consistently in
the electron spectrum, although the energy of the beam was observed
to vary from shot to shot. If the issue of energy reproducibility
can be addressed, it should be possible to generate ultrashort monoenergetic
electron bunches of tunable energy, holding great promise for the
future development of 'table-top' particle accelerators.},
DOI = {10.1038/nature02939}
}
@Article{Mann04,
Title = {Spots and turbulent domains in a model of transitional plane {Couette} flow},
Author = {P. {Manneville}},
Journal = {Theor. Comp. Fluid Dyn.},
Year = {2004},
Pages = {169--181},
Volume = {18},
DOI = {10.1007/s00162-004-0142-4}
}
@Book{manneville_instabilities_2004,
Title = {Instabilities, Chaos and Turbulence: An Introduction to Nonlinear Dynamics and Complex Systems},
Author = {Manneville, P.},
Publisher = {Imperial College Press},
Year = {2004},
Address = {London}
}
@Book{Mansfield10,
Title = {A practical guide to the invariant calculus},
Author = {Mansfield, E. L.},
Publisher = {Cambridge Univ. Press},
Year = {2010},
Address = {Cambridge}
}
@Misc{marcio03,
Title = {Topological characterization of spatial-temporal chaos},
Author = {M. Gameiro and W. Kalies and K. Mischaikow},
Year = {2003},
Abstract = {Use the Betti number to characterize the spatiotemporal dynamics of
a system. A Lyapunov exponent is suggested to measure the space as
well as temporal complexity of the evolution.}
}
@Article{MaRi83,
Title = {Equation of motion for a small rigid sphere in a nonuniform flow},
Author = {{Maxey}, M.~R. and {Riley}, J.~J.},
Journal = {Phys. Fluids},
Year = {1983},
Pages = {883--889},
Volume = {26}
}
@Article{Marion1989,
author = {Marion, M. and Temam, R.},
title = {Nonlinear {Galerkin} methods},
journal = {SIAM J. Numer. Anal.},
year = {1989},
volume = {26},
pages = {1139--1157},
doi = {10.1137/0726063}
}
@Article{MarLopBla04,
author = {F. Marques and J. M. Lopez and H. M. Blackburn},
title = {Bifurcations in systems with {Z2} spatio-temporal and {O(2)} spatial symmetry},
journal = {Physica D},
year = {2004},
volume = {189},
pages = {247--276},
doi = {10.1016/j.physd.2003.09.041}
}
@Article{MaRoLa02,
Title = {Partial reduction in the {N}-body planetary problem using the angular momentum integral},
Author = {Malige, F. and Robutel, P. and Laskar, J.},
Journal = {Celestial Mech. Dynam. Astronom.},
Year = {2002},
Pages = {283--316},
Volume = {84}
}
@Article{Marques90,
author = {Marqu\'es, F.},
title = {On boundary conditions for velocity potentials in confined flows: {Application} to {Couette} flow},
journal = {Phys. Fluids A},
year = {1990},
volume = {2},
pages = {729--737},
doi = {10.1063/1.857726}
}
@Book{MarRat99,
title = {Introduction to Mechanics and Symmetry},
publisher = {Springer},
year = {1999},
author = {Marsden, J. E. and Ratiu, T. S.},
address = {New York}
}
@Book{Marsd92,
Title = {Lectures on Mechanics},
Author = {Marsden, J. E.},
Publisher = {Cambridge Univ. Press},
Year = {1992},
Address = {Cambridge}
}
@Book{marsden_hamiltonian_2007,
Title = {Hamiltonian Reduction by Stages},
Author = {Marsden, J. E. and G. Misiolek and {J.-P.} Ortega and M. Perlmutter and T. S. Ratiu},
Publisher = {Springer},
Year = {1964},
Address = {New York}
}
@Book{marsdenbb,
Title = {The {Hopf} Bifurcation and its Applications},
Author = {J. E. Marsden and M. McCracken},
Publisher = {Springer},
Address = {New York}
}
@Article{mart87lfa,
Title = {Local Frequency analysis of chaotic motion in multidimensional systems: energy transport and bottlenecks in planar {OCS}},
Author = {C. C. Martens and M. J. Davis and G. S. Ezra},
Journal = {Chem. Phys. Lett.},
Year = {1987},
Pages = {519},
Volume = {142},
Abstract = {local frequency analysis is used and action-angle coordinate is parametrized
by the frequency ratio. The transport zones and partial barriers
are identified according to the ``noble ratio'' consideration.}
}
@Article{Marw08,
Title = {A historical review of recurrence plots},
Author = {N. Marwan},
Journal = {European Phys. J.},
Year = {2008},
Pages = {3--12},
Volume = {164}
}
@Article{MaSaTAS81,
Author = {S. G. Matanyan and G. K. Savvidy and N. G. Ter-Arutyunyan-Savvidy},
Journal = {Sov. Phys. JETP},
Year = {1981},
Pages = {421},
Volume = {53}
}
@Article{MaScha94,
Title = {Landau gauge within the Gribov horizon},
Author = {Maggiore, N. and Schaden, M.},
Journal = {Phys. Rev. D},
Year = {1994},
Pages = {6616--6625},
Volume = {50},
DOI = {10.1103/PhysRevD.50.6616}
}
@Article{MaSiRo73,
Title = {Statistical mechanics of classical systems},
Author = {P. C. Martin and E. D. Siggia and H. A. Rose},
Journal = {Phys. Rev. A},
Year = {1973},
Pages = {423--437},
Volume = {8}
}
@Unpublished{MaSkAn11,
Title = {Probing the local dynamics of periodic orbits by the generalized alignment index {(GALI)} method},
Author = {{Manos}, T. and {Skokos}, C. and {Antonopoulos}, C.},
Note = {\arXiv{1103.0700}},
Year = {2011}
}
@Article{Mat82,
Title = {Existence of quasiperiodic orbits for twist homeomorphisms of the annulus},
Author = {Mather, J. N.},
Journal = {Topology},
Year = {1982},
Pages = {457--467},
Volume = {21},
Annote = {Original paper abour Aubry-Mather sets}
}
@Article{Mather91,
Title = {Variational construction of orbits of twist difeomorphisms},
Author = {J. N. Mather},
Journal = {J. Am. Math. Soc.},
Year = {1991},
Pages = {207--263},
Volume = {4}
}
@Book{MathWalk73,
Title = {Mathematical Methods of Physics},
Author = {J. Mathews and R. L. Walker},
Publisher = {Addison-Wesley},
Year = {1973},
Address = {Reading, MA}
}
@Article{maucher2013,
Title = {Quasiperiodic oscillations and homoclinic orbits in the nonlinear nonlocal {Schr\"odinger} equation},
Author = {Maucher, F. and Siminos, E. and Krolikowski, W. and Skupin, S.},
Journal = {New J. Phys.},
Year = {2013},
Pages = {083055},
Volume = {15},
Abstract = {Quasiperiodic oscillations and shape-transformations of
higher-order bright solitons in nonlinear nonlocal media have been
frequently observed numerically in recent years, however, the origin of
these phenomena was never completely elucidated. In this paper, we
perform a linear stability analysis of these higher-order solitons by
solving the Bogoliubov-de Gennes equations. This enables us to understand
the emergence of a new oscillatory state as a growing unstable mode of a
higher-order soliton. Using dynamically important states as a basis, we
provide low-dimensional visualizations of the dynamics and identify
quasiperiodic and homoclinic orbits, linking the latter to
shape-transformations.},
DOI = {10.1088/1367-2630/15/8/083055}
}
@Article{MaWaRe89,
Author = {C. C. Martens and R. L. Waterland and W. P. Reinhardt},
Journal = {J. Chem. Phys.},
Year = {1989},
Pages = {2328},
Volume = {90}
}
@Article{MaWe74,
Title = {Reduction of symplectic manifolds with symmetry},
Author = {Marsden, J. E. and Weinstein, A.},
Journal = {Rep. Math. Phys.},
Year = {1974},
Pages = {121--30},
Volume = {5}
}
@Article{MC96pha,
Title = {Phase turbulence in the two-dimensional complex {Ginzburg-Landau} equation},
Author = {Manneville, P. and Chat\'{e}, H.},
Journal = {Physica D},
Year = {1996},
Pages = {30--46},
Volume = {96},
Abtract = {instability of plane waves}
}
@InProceedings{McCordMontaldi,
Title = {Relative periodic orbits of symmetric {Lagrangian} systems},
Author = {C. McCord AND J. Montaldi AND M. Roberts AND L. Sbano},
Booktitle = {Proceedings of {Equadiff} 2003},
Year = {2004},
Editor = {Dumortier, F. and Broer, H. W. and Mawhin, J. and Vanderbauwhede, A. and Verduyn Lunel, S.},
Pages = {482--493}
}
@Unpublished{McICvi15,
Title = {Periodic orbit theory of linear response},
Author = {B. McInroe and P. Cvitanovi\'c},
Note = {In preparation},
Year = {2015}
}
@Article{McKean74,
Title = {Selberg's trace formula as applied to a compact {Riemann} surface},
Author = {McKean, H. P.},
Journal = {Comm. Pure Appl. Math.},
Year = {1972},
Pages = {225--246},
Volume = {25}
}
@Article{McKeon04,
Title = {Further observations on the mean velocity in fully-developed pipe flow},
Author = {B. J. McKeon and J. Li and W. Jiang and J. F. Morrison and A. J. Smits},
Journal = {J. Fluid Mech.},
Year = {2004},
Pages = {135--147},
Volume = {501}
}
@Phdthesis{McLellan12,
Title = {Periodic Coefficients and Random {Fibonacci} Sequences},
Author = {McLellan, K. A.},
School = {Dalhousie University},
Year = {2012},
Address = {Halifax, Nova Scotia},
URL = {http://hdl.handle.net/10222/15381}
}
@Article{mclellan2012periodic,
Title = {Periodic coefficients and random {Fibonacci} sequences},
Author = {McLellan, Karyn Anne},
Year = {2012}
}
@Article{McLPerlQui03,
author = {R.I. McLachlan and M. Perlmutter and G.R.W. Quispel},
title = {Lie group foliations: dynamical systems and integrators},
journal = {Future Generation Computer Systems},
year = {2003},
volume = {19},
pages = {1207--1219},
doi = {10.1016/S0167-739X(03)00046-3},
abstract = {Foliate systems are those which preserve some (possibly singular)
foliation of phase space, such as systems with integrals, systems
with continuous symmetries, and skew product systems. We study numerical
integrators which also preserve the foliation. The case in which
the foliation is given by the orbits of an action of a Lie group
has a particularly nice structure, which we study in detail, giving
conditions under which all foliate vector fields can be written as
the sum of a vector field tangent to the orbits and a vector field
invariant under the group action. This allows the application of
many techniques of geometric integration, including splitting methods
and Lie group integrators.}
}
@Article{McNMar01,
Title = {Origin of the hydrodynamic {Lyapunov modes}},
Author = {McNamara, S. and Mareschal, M.},
Journal = {Phys. Rev. E},
Year = {2001},
Pages = {051103},
Volume = {64},
DOI = {10.1103/PhysRevE.64.051103},
Numpages = {14}
}
@Article{Mcoh03,
author = {M. van Hecke},
title = {Coherent and incoherent structures in systems described by the 1{D CGLE}: {Experiments} and identification},
journal = {Physica D},
year = {2003},
volume = {174},
pages = {134},
abstract = {Give an overview of the most important properties of the coherent
structures and address the question how such structures may be identified
in the experimental space-time data set.}
}
@Article{mcord86,
author = {C. K. McCord},
title = {Uniqueness of connection orbits in the equation {$y^{(3)}=y^2-1$}},
journal = {J. Math. Anal. Appl.},
year = {1986},
volume = {114},
pages = {584--592},
doi = {10.1016/0022-247X(86)90110-1}
}
@Book{meer_hamiltonian_1994,
Title = {Hamiltonian Structure of the Reversible Nonsemisimple 1:1 Resonance},
Author = {J.C. van der Meer and J.A. Sanders and Vanderbauwhede, A.},
Publisher = {Eindhoven Univ. of Technology, Dept. of Math. and Comp. Sci.},
Year = {1994},
Address = {Eindhoven},
Series = {Reports on applied and numerical analysis, RANA 94-02}
}
@Book{Mees81,
Title = {Dynamics of Feedback Systems},
Author = {Mees, A. I.},
Publisher = {Wiley},
Year = {1981},
Address = {New York}
}
@Article{MEF04,
Title = {Fractal lifetimes in the transition to turbulence},
Author = {J. Moehlis and B. Eckhardt and H. Faisst},
Journal = {Chaos},
Year = {2004},
Pages = {S11},
Volume = {14}
}
@Book{mehta,
Title = {Random Matrices},
Author = {M. L. Mehta},
Publisher = {Academic},
Year = {1991},
Address = {New York}
}
@Article{Mei08,
author = {J. D. Meiss},
title = {Visual explorations of dynamics: {The} standard map},
journal = {Pramana, Indian Acad. Sci.},
year = {2008},
volume = {70},
pages = {965--988},
note = {\arXiv{0801.0883}}
}
@Article{meis92,
Title = {Symplectic maps, variational principles, and transport},
Author = {Meiss, J. D.},
Journal = {Rev. Mod. Phys.},
Year = {1992},
Pages = {795--848},
Volume = {64}
}
@Book{Meisso7,
Title = {Differential Dynamical Systems},
Author = {J. D. Meiss},
Publisher = {SIAM},
Year = {2007},
Address = {Philadelphia}
}
@Article{Mel99,
Title = {Steady-state bifurcation with {Euclidean} symmetry},
Author = {Melbourne, I.},
Journal = {Trans. Amer. Math. Soc.},
Year = {1999},
Pages = {1575--1603},
Volume = {351}
}
@Article{mellibovsky11,
Title = {{Takens--Bogdanov} bifurcation of travelling-wave solutions in pipe flow},
Author = {Mellibovsky, F. and Eckhardt, B.},
Journal = {J. Fluid Mech.},
Year = {2011},
Pages = {96--129},
Volume = {670}
}
@Article{mellibovsky12,
Title = {From travelling waves to mild chaos: {A} supercritical bifurcation cascade in pipe flow},
Author = {F. Mellibovsky and B. Eckhardt},
Journal = {J. Fluid Mech.},
Year = {2012},
Pages = {149--190},
Volume = {709}
}
@Article{Mendoza13,
author = {Mendoza, V.},
title = {Proof of the {Pruning Front Conjecture} for certain {H\'enon} parameters},
journal = {Nonlinearity},
year = {2013},
volume = {26},
pages = {679--690},
note = {\arXiv{1112.0705}},
doi = {10.1088/0951-7715/26/3/679},
abstract = {The Pruning Front Conjecture is proved for an open set of H\'enon
parameters far from unimodal. More specifically, for an open subset
of H\'enon parameter space, consisting of two connected components
one of which intersects the area-preserving locus, it is shown that
the associated H\'enon maps are prunings of the horseshoe. In particular,
their dynamics is a subshift of the two-sided two-shift.}
}
@Article{MePrKn01,
author = {Mercader, I. and Prat, J. and Knobloch, E.},
title = {The {1:2} mode interaction in {Rayleigh B\'enard} convection with weakly broken midplane symmetry},
journal = {Int. J. Bifur. Chaos},
year = {2001},
volume = {11},
pages = {27--41},
timestamp = {2012.03.28}
}
@Article{MePrKn02,
Title = {Robust heteroclinic cycles in two-dimensional {Rayleigh-B\'enard} convection without {Boussinesq} symmetry},
Author = {Mercader, I. and Prat, J. and Knobloch, E.},
Journal = {Int. J. Bifur. Chaos},
Year = {2002},
Pages = {2501--2522},
Volume = {12},
DOI = {10.1142/S0218127402006047}
}
@Article{Mermin92,
Title = {The space groups of icosahedral quasicrystals and cubic, orthorhombic, monoclinic, and triclinic crystals},
Author = {Mermin, N. D.},
Journal = {Rev. Mod. Phys.},
Year = {1992},
Pages = {3--49},
Volume = {64},
DOI = {10.1103/RevModPhys.64.3}
}
@Article{Mertz79,
Title = {Speckle imaging, photon by photon},
Author = {L. N. Mertz},
Journal = {Appl. Opt.},
Year = {1979},
Pages = {611--614},
Volume = {18},
Abstract = {A speckle processing prescription is described that
should yield diffraction-limited performance for large telescopes
in spite of atmospheric turbulence and at light levels as low as
100 photons/sec in the picture. The prescription involves
rearranging the spatial frequency components of running glimpses of
the scene according to the complex information (entropy) of those
components. The imaginary part (phase) of the information is
rendered unambiguous by maintaining track of the phase. A 2-D
photon counter furnishes the raw observations. Comparisons and
ramifications of the procedure are discussed.},
DOI = {10.1364/AO.18.000611}
}
@Article{Meseguer03,
Title = {Streak breakdown instability in pipe {Poiseuille} flow},
Author = {A. Meseguer},
Journal = {Phys. Fluids},
Year = {2003},
Pages = {1203--1213},
Volume = {15},
DOI = {10.1063/1.1564093}
}
@Article{MeseguerANM07,
Title = {On a solenoidal {Fourier-Chebyshev} spectral method for stability analysis of the {Hagen--Poiseuille} flow},
Author = {A. Meseguer and F. Mellibovsky},
Journal = {Appl. Num. Math.},
Year = {2007},
Pages = {920--938},
Volume = {57}
}
@Article{MesTre03,
Title = {Linearized pipe flow to {Reynolds} number {$10^7$}},
Author = {Meseguer, A. and Trefethen, L. N.},
Journal = {J. Comput. Phys.},
Year = {2003},
Pages = {178--197},
Volume = {186}
}
@Article{MeWi94,
Title = {On the integrability and perturbation of three-dimensional fluid flows with symmetry},
Author = {I. Mezi\'c and S. Wiggins},
Journal = {J. Nonlin. Sci.},
Year = {1994},
Pages = {157--194},
Volume = {4},
Abstract = {The purpose of this paper is to develop analytical methods for studyingparticle
paths in a class of three-dimensional incompressible fluid flows.
In this paper we study three-dimensionalvolume preserving vector
fields that are invariant under the action of a one-parameter symmetry
group whose infinitesimal generator is autonomous and volume-preserving.
We show that there exists a coordinate system in which the vector
field assumes a simple form. In particular, the evolution of two
of the coordinates is governed by a time-dependent, one-degree-of-freedom
Hamiltonian system with the evolution of the remaining coordinate
being governed by a first-order differential equation that depends
only on the other two coordinates and time. The new coordinates depend
only on the symmetry group of the vector field. Therefore they arefield-independent.
The coordinate transformation is constructive. If the vector field
is time-independent, then it possesses an integral of motion. Moreover,
we show that the system can be further reduced toaction-angle-angle
coordinates. These are analogous to the familiar action-angle variables
from Hamiltonian mechanics and are quite useful for perturbative
studies of the class of systems we consider. In fact, we show how
our coordinate transformation puts us in a position to apply recent
extensions of the {Kolmogorov-Arnold-Moser} {(KAM)} theorem for three-dimensional,
volume-preserving maps as well as three-dimensional versions of Melnikov's
method. We discuss the integrability of the class of flows considered,
and draw an analogy with Clebsch variables in fluid mechanics.}
}
@Book{Meyer00,
Title = {Matrix Analysis and Applied Linear Algebra},
Author = {Meyer, C.},
Publisher = {SIAM},
Year = {2000},
Address = {Philadelphia}
}
@Book{MeyerHall09,
Title = {Introduction to Dynamical Systems and the {N}-body Problem},
Author = {K. R. Meyer and G. R. Hall and D. Offin},
Publisher = {Springer},
Year = {2009},
Address = {New York}
}
@Book{MeyerHall92,
Title = {Introduction to {Hamiltonian} Dynamical Systems},
Author = {K. R. Meyer and G. R. Hall},
Publisher = {Springer},
Year = {1992},
Address = {New York}
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@Article{MFE04,
Title = {A low-dimensional model for turbulent shear flows},
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Journal = {New J. Physics},
Year = {2004},
Pages = {56},
Volume = {6}
}
@Article{MFE04b,
Title = {Periodic orbits and chaotic sets in a low-dimensional model for shear flows},
Author = {J. Moehlis and H. Faisst and B. Eckhardt},
Journal = {SIAM J. Appl. Dyn. Syst.},
Year = {2004},
Pages = {352--376},
Volume = {4}
}
@Article{mfind,
Title = {Efficient algorithm for detecting unstable periodic orbits in chaotic systems},
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Journal = {Phys. Rev. E},
Year = {1999},
Pages = {6172},
Volume = {60}
}
@Article{MFKM10,
Title = {Scytale decodes chaos: {A} method for estimating unstable symmetric solutions},
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Journal = {Chaos},
Year = {2010},
Pages = {013126},
Volume = {20},
Abstract = {A method for estimating a period of unstable periodic solutions is
suggested in continuous dissipative chaotic dynamical systems. The
measurement of a minimum distance between a reference state and an
image of transformation of it exhibits a characteristic structure
of the system, and the local minima of the structure give candidates
of period and state of corresponding symmetric solutions. Appropriate
periods and initial states for the Newton method are chosen efficiently
by setting a threshold to the range of the minimum distance and the
period.}
}
@Article{MHMwind96,
Title = {Winding number instability in the phase-turbulence regime of the complex {Ginzburg-Landau} equation},
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Journal = {Phys. Rev. Lett.},
Year = {1996},
Pages = {267},
Volume = {77},
Abstract = {There exist a band of winding numbers associated with stable states
in the phase turbulent regime. The states with large winding numbers
decay to the ones with smaller winding numbers. Defect chaos happens
when the range of stable winding numbers vanishes.}
}
@Article{MHPRS07,
Title = {Uncovering the {Lagrangian} skeleton of turbulence},
Author = {Mathur, M. and Haller, G. and Peacock, T. and Ruppert-Felsot, J. E. and Swinney, H. L.},
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Volume = {98}
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@Article{Michel90,
Title = {Elementary particles as solutions of the {Sivashinsky} equation},
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Year = {1990},
Pages = {502--556},
Volume = {44}
}
@Article{michsiv77,
Title = {Nonlinear analysis of hydrodynamic instability in laminar flames - {II}. Numerical experiments},
Author = {D. M. Michelson and G. I. Sivashinsky},
Journal = {Acta Astronaut.},
Year = {1977},
Pages = {1207--1221},
Volume = {4},
Abstract = {Consider the effect of hydrodynamics instability and diffusion thermal
instability through the numerical calculation of the KSe.}
}
@Article{MicZhi01,
author = {L. Michel and B.I. Zhilinski\'i},
title = {Symmetry, invariants, topology. Basic tools},
journal = {Physics Reports },
year = {2001},
volume = {341},
pages = {11--84},
doi = {10.1016/S0370-1573(00)00088-0},
abstract = {Elementary concepts of group actions: orbits and their
stabilizers, orbit types and their strata are introduced and illustrated
by simple examples. We give the unified description of these notions
which are often used in the different domains of physics under different
names. We also explain some basic facts about rings of invariant
functions and their module structure. This leads to a geometrical study
of the orbit space and of the level surfaces of invariant functions (e.g.
energy levels of Hamiltonians). Combining these tools with Morse theory
we study the extrema of invariant functions. Some physical applications
(not studied in other chapters) are sketched. }
}
@Incollection{Mielke02,
Title = {The {Ginzburg-Landau} equation in its role as a modulation equation},
Author = {A. Mielke},
Booktitle = {Handbook of Dynamical Systems, Vol. 2},
Publisher = {Elsevier},
Year = {2002},
Editor = {B. Fiedler},
Pages = {759--834}
}
@Book{Mielke91,
Title = {{Hamiltonian and Lagrangian} Flows on Center Manifolds},
Author = {A. Mielke},
Publisher = {Springer},
Year = {1991},
Address = {New York}
}
@Incollection{MilThu88,
Title = {Iterated maps of the interval},
Author = {Milnor, J. and Thurston, W.},
Booktitle = {Dynamical {Systems} ({M}aryland 1986-87)},
Publisher = {Springer},
Year = {1988},
Address = {New York},
Editor = {A. Dold and B. Eckmann},
Pages = {465--563},
Series = {Lect. Notes Math.},
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@Techreport{minpack,
Title = {{User Guide for MINPACK-1}},
Author = {Mor\'{e}, J. J. and Garbow, B. S. and Hillstrom, K. E. },
Institution = {Argonne National Laboratory},
Year = {1980},
Number = {ANL-80-74},
URL = {http://www.mcs.anl.gov/\~{}more/ANL8074a.pdf}
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@Unpublished{MiPlSt11,
Title = {On the steady state distributions for turbulence},
Author = {Mini\'c, Dj. and Pleimling, M. and Staples, A. E.},
Note = {\arXiv{1105.2941}},
Year = {2011}
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@Book{Mira87,
Title = {Chaotic dynamics -- {From} one dimensional endomorphism to two dimen\-sional diffeo\-morphism},
Author = {C. Mira},
Publisher = {World Scientific},
Year = {1987},
Address = {Singapore}
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@Book{MiraGum80,
Title = {Recurrances and Discrete Dynamical Systems},
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Publisher = {Springer},
Year = {1980},
Address = {Berlin}
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Year = {1999},
Number = {6},
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@Article{mislor1,
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Journal = {Math. Comp.},
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Pages = {1023--1046},
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@Article{Mitchell12,
author = {K. A. Mitchell},
title = {Partitioning two-dimensional mixed phase spaces},
journal = {Physica D},
year = {2012},
volume = {241},
pages = {1718--1734},
doi = {10.1016/j.physd.2012.07.004}
}
@Book{MitLop95,
Title = {Nonlinear Mechanics, Groups and Symmetry},
Author = {Mitropolsky, Yu. A. and Lopatin, A. K.},
Publisher = {Kluwer},
Year = {1995},
Address = {Dordrecht}
}
@Article{MiZg01,
Title = {Topological entropy for multidimensional perturbations of one dimensional maps},
Author = {M. Misiurewicz and P. Zgliczynski},
Journal = {Int. J. Bifur. Chaos},
Year = {2001},
Pages = {1443--1446},
Volume = {5},
Abstract = {Numerical experiments in [Christiansen97] show that a suitably chosen
Poincar e map P is essentially one dimensional and can be modeled
by a one dimensional map p. Now we build a homotopy by F ( x; w)
P (x; w) 1 ) p(x) 0) This homotopy is compact. We cannot claim that
we can use Theorem 1. 1 to estimate rigorously topological entropy
for the Poincar e map P , because even there Entropy for multidimensional
perturbations 3 is no rigorous proof that the Poincar e map P studied
numerically in [Christiansen97] exists. However, the ideas used in
the proof of Theorem 1.1, continuation of topological horseshoes
and relation between topological horseshoes and entropy, combined
with recently developed rigorous numerics for KS equations (see [ZgRi01])}
}
@Article{Mks86,
Title = {Steady solutions of the {Kuramoto-Sivashinsky} equation},
Author = {D. Michelson},
Journal = {Physica D},
Year = {1986},
Pages = {89--111},
Volume = {19},
Abstract = {The variety of steady solution of the KSe is discussed. For large
c, there is only one odd front-like bounded solution. In decreasing
c, odd solutions with more zeros are born until finally a periodic
solution is born. Associated with the periodic solution, infinite
many tori will appear in the elliptic case and Cantor-type set of
chaotic solutions, with infinite many homoclinic odd solutions.}
}
@Article{Mo86b,
Title = {Recent developments in the theory of {Hamiltonian} systems},
Author = {Moser, J.},
Journal = {SIAM Rev.},
Year = {1986},
Pages = {459--485},
Volume = {28},
Annote = {overview over KAM theory and Aubry-Mather; stability of fixed points}
}
@Article{Moore90,
Title = {Unpredictability and undecidability in dynamical systems},
Author = {Moore, C.},
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Year = {1990},
Pages = {2354--2357},
Volume = {64}
}
@Article{Moore91,
Title = {Generalized shifts: unpredictability and undecidability in dynamical systems},
Author = {Moore, C.},
Journal = {Nonlinearity},
Year = {1991},
Pages = {199--230},
Volume = {4}
}
@Article{MoPaSp03,
author = {P. Montagnier and C. C. Paige and R. J. Spiteri},
title = {Real {Floquet} factors of linear time-periodic systems},
journal = {Systems {\&} Control Lett.},
year = {2003},
volume = {50},
pages = {251--262},
doi = {10.1016/S0167-6911(03)00158-0},
url = {http://www.cs.mcgill.ca/~chris/pub/MonPS03.pdf}
}
@Article{Morena2014,
Title = {On the potential for entangled states between chaotic systems},
Author = {Morena, M.A. and Short, K.M.},
Journal = {Int. J. Bifur. Chaos},
Year = {2014},
Volume = {24},
Abstract = {We report on the tendency of chaotic systems to be controlled
onto their unstable periodic orbits in such a way that these orbits are
stabilized. The resulting orbits are known as cupolets and collectively
provide a rich source of qualitative information on the associated
chaotic dynamical system. We show that pairs of interacting cupolets may
be induced into a state of mutually sustained stabilization that requires
no external intervention in order to be maintained and is thus considered
bound or entangled. A number of properties of this sort of entanglement
are discussed. For instance, should the interaction be disturbed, then
the chaotic entanglement would be broken. Based on certain properties of
chaotic systems and on examples which we present, there is further
potential for chaotic entanglement to be naturally occurring. A
discussion of this and of the implications of chaotic entanglement in
future research investigations is also presented.},
DOI = {10.1142/S0218127414500771}
}
@InProceedings{Morkovin69,
Title = {On the many faces of transition},
Author = {M. V. Morkovin},
Booktitle = {Viscous Drag Reduction},
Year = {1969},
Address = {New York},
Editor = {C. S. Wells},
Pages = {1--31},
Publisher = {Plenum}
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@Article{Morr80,
Title = {The {Maxwell-Vlasov} equations as a continuous {Hamiltonian} system},
Author = {Morrison, P. J.},
Journal = {Phys. Lett. A},
Year = {1980},
Pages = {383--386},
Volume = {80}
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@Article{Morr98,
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Pages = {467--521},
Volume = {70}
}
@Article{MorrGree80,
Title = {Noncanonical {Hamiltonian} density formulation of hydrodynamics and ideal magnetohydrodynamics},
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Journal = {Phys. Rev. Lett.},
Year = {1980},
Note = {see also Phys. Rev. Lett. 48, 569 (1982)},
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@Article{Morrison02,
Title = {Reynolds number dependence of streamwise velocity spectra in turbulent pipe flow},
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Year = {2002},
Pages = {214501},
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@Article{Morrison04,
Title = {Scaling of the streamwise velocity component in turbulent pipe flow},
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Journal = {J. Fluid Mech.},
Year = {2004},
Pages = {99--131},
Volume = {508}
}
@Article{moser96,
Title = {A rapidly converging iteration method and nonlinear partial differential equations},
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Journal = {Ann. Scuola Norm. Super. Pisa},
Year = {1966},
Pages = {265--315},
Volume = {20}
}
@Article{moser96a,
Author = {J. Moser},
Journal = {Ann. Scuola Norm. Super. Pisa},
Year = {1966},
Pages = {499--535},
Volume = {20}
}
@Book{mosersr,
Title = {Stable and Random Motions in Dynamical Systems},
Author = {J. Moser},
Publisher = {Princeton Univ. Press},
Year = {1973},
Address = {Princeton, NJ}
}
@Article{MoSiSo07,
Title = {Amplification in the auditory periphery: {The} effect of coupling tuning mechanisms},
Author = {Montgomery, K. A. and Silber, M. and Solla, S. A.},
Journal = {Phys. Rev. E},
Year = {2007},
Pages = {051924},
Volume = {75},
Abstract = {A mathematical model describing the coupling between two independent
amplification mechanisms in auditory hair cells is proposed and analyzed.
Hair cells are cells in the inner ear responsible for translating
sound-induced mechanical stimuli into an electrical signal that can
then be recorded by the auditory nerve. In nonmammals, two separate
mechanisms have been postulated to contribute to the amplification
and tuning properties of the hair cells. Models of each of these
mechanisms have been shown to be poised near a Hopf bifurcation.
Through a weakly nonlinear analysis that assumes weak periodic forcing,
weak damping, and weak coupling, the physiologically based models
of the two mechanisms are reduced to a system of two coupled amplitude
equations describing the resonant response. The predictions that
follow from an analysis of the reduced equations, as well as performance
benefits due to the coupling of the two mechanisms, are discussed
and compared with published experimental auditory nerve data.},
DOI = {10.1103/PhysRevE.75.051924}
}
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@Article{motter03,
Title = {Relativistic chaos is coordinate invariant},
Author = {Motter, A. E.},
Journal = {Phys. Rev. Lett.},
Year = {2003},
Pages = {231101},
Volume = {91},
Abstract = {The noninvariance of Lyapunov exponents in general relativity has
led to the conclusion that chaos depends on the choice of the space-time
coordinates. Strikingly, we uncover the transformation laws of Lyapunov
exponents under general space-time transformations and we find that
chaos, as characterized by positive Lyapunov exponents, is coordinate
invariant. As a result, the previous conclusion regarding the noninvariance
of chaos in cosmology, a major claim about chaos in general relativity,
necessarily involves the violation of hypotheses required for a proper
definition of the Lyapunov exponents.},
DOI = {10.1103/PhysRevLett.91.231101}
}
@Article{Motter09,
Title = {Time-metric equivalence and dimension change under time reparameterizations},
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Journal = {Phys. Rev. E},
Year = {2009},
Note = {\arXiv{nlin.CD/0905.3416}},
Pages = {065202},
Volume = {79},
Numpages = {4}
}
@Article{motter09-1,
Title = {Relativistic invariance of {Lyapunov} exponents in bounded and unbounded systems},
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Journal = {Phys. Rev. Lett.},
Year = {2009},
Month = may,
Number = {18},
Pages = {184101},
Volume = {102},
Abstract = {The study of chaos in relativistic systems has been hampered by the
observer dependence of Lyapunov exponents ({LEs)} and of conditions,
such as orbit boundedness, invoked in the interpretation of {LEs}
as indicators of chaos. Here we establish a general framework that
overcomes both difficulties and apply the resulting approach to address
three fundamental questions: how {LEs} transform under Lorentz and
Rindler transformations and under transformations to uniformly rotating
frames. The answers to the first and third questions show that inertial
and uniformly rotating observers agree on a characterization of chaos
based on {LEs.} The second question, on the other hand, is an ill-posed
problem due to the event horizons inherent to uniformly accelerated
observers.},
DOI = {10.1103/PhysRevLett.102.184101}
}
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@Book{mta,
title = {Manifolds, Tensor Analysis, and Applications},
publisher = {Springer},
year = {1988},
author = {R. Abraham and J. E. Marsden and T. S. Ratiu},
address = {New York},
edition = {2\textsuperscript{nd}}
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@Article{muloz-almaraz_continuation_2007,
Title = {Continuation of normal doubly symmetric orbits in conservative reversible systems},
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Pages = {17--47},
Volume = {97}
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@Article{Mummert08,
Title = {Holomorphic shadowing for {H\'enon} maps},
Author = {P. P. Mummert},
Journal = {Nonlinearity},
Year = {2008},
Pages = {2887},
Volume = {21},
Abstract = {We describe a correcting operator for those pseudo-orbits of the quadratic
complex {H\'enon} map that remain bounded away from the origin by
an amount that depends on the map. The iterates of the operator converge
to a holomorphic motion that respects the dynamics. This computational
procedure is then applied in the contexts of horseshoes and solenoids.}
}
@Article{munoz-almaraz_continuation_2003,
Title = {Continuation of periodic orbits in conservative and {Hamiltonian} systems},
Author = {Munoz-Almaraz, F. J. and Freire, E. and Galan, J. and Doedel, E. J. and Vanderbauwhede, A.},
Journal = {Physica D},
Year = {2003},
Pages = {1--38},
Volume = {181}
}
@Incollection{MuPe06,
author = {T. Mullin and J. Peixinho},
title = {Recent observations of the transition to turbulence in a pipe},
booktitle = {IUTAM Symposium on Laminar-Turbulent Transition},
publisher = {Springer},
year = {2006},
editor = {Govindarajan, R.},
volume = {78},
series = {Fluid Mechanics and Its Applications},
pages = {45--55},
address = {New York},
doi = {10.1007/1-4020-4159-4_5}
}
@Article{Murat03,
Title = {Path integration over closed loops and {Gutzwiller}'s trace formula},
Author = {Muratore-Ginanneschi, P.},
Journal = {Phys. Rep.},
Year = {2002},
Note = {\arXiv{nlin/0210047}},
Pages = {299},
Volume = {383}
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year = {1998},
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pages = {1896}
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@Article{MvHhole01,
Title = {Ordered and self-disordered dynamics of holes and defects in the one-dimensional complex {Ginzburg-Landau} equation},
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Year = {2001},
Pages = {2018},
Volume = {86},
Abtract = {Ordered hole-defect dynamics happens when an unstable hole invade a plane wave state and periodically nucleates defects from which new holes are born. The interaction between the holes and a self-disordered background are essential for the occurrence of spatiotemporal chaos in hole-defect states.}
}
@Article{n00bs,
Title = {Equilibrium and traveling-wave solutions of plane {Couette} flow},
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Year = {2009},
Note = {\arXiv{0808.3375}},
Pages = {243--266},
Volume = {638}
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@Book{Naka90,
Title = {Geometry, Topology and Physics},
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Publisher = {Inst. of Physics Publ.},
Year = {1990},
Address = {Bristol}
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Pages = {49--58},
Volume = {XLIV}
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@Article{newcgl2,
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Year = {1986},
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@Article{nhouse74,
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year = {1974},
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@Article{nhouse79,
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year = {1979},
volume = {50},
pages = {101--152},
doi = {10.1007/BF02684771}
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@Article{Nichkawde13,
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Year = {2013},
Pages = {022905},
Volume = {87},
DOI = {10.1103/PhysRevE.87.022905},
Numpages = {8}
}
@Article{Nickel07,
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year = {2007},
volume = {33},
pages = {1376--1382},
doi = {10.1016/j.chaos.2006.01.087}
}
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}
@Article{NiHa92,
Title = {An invariance property of the geometrical phase and its consequence in detuned lasers},
Author = {Ning, C. Z. and Haken, H.},
Journal = {Z. Phys. B},
Year = {1992},
Pages = {261--262},
Volume = {89},
Abstract = {We show that the geometrical phase defined for the dissipative systems
{[}1] is invariant under a unitary transformation. This implies for
lasers an invariance of the geometrical phase for different choices
of the reference frequency.}
}
@Article{NiHa92a,
author = {C. Z. Ning and H. Haken},
title = {Geometrical phase and amplitude accumulations in dissipative systems with cyclic attractors},
journal = {Phys. Rev. Lett.},
year = {1992},
volume = {68},
pages = {2109--2112},
abstract = {We show that the geometrical {(Berry)} phases discovered in Hamiltonian
systems can also be defined as resulting from parallel transportation
of vectors for nonlinear dissipative systems with cyclic attractors.
If the nonlinear dissipative systems possess a certain kind of asymptotic
solution defined in this Letter, the phase and amplitude accumulation
of a geometrical type can be defined. Detuned one- and two-photon
lasers showing periodic intensity pulsations are taken as examples
of such systems.}
}
@Article{NingHakenCLE90,
Title = {Detuned lasers and the complex {Lorenz} equations: {Subcritical} and supercritical {Hopf} bifurcations},
Author = {Ning, C. Z. and Haken, H.},
Journal = {Phys. Rev.},
Year = {1990},
Pages = {3826},
Volume = {41}
}
@Article{Nizette2001,
Title = {Stability and bifurcations of periodically modulated, optically injected laser diodes},
Author = {M. Nizette and T. Erneux and A. Gavrielides and V. Kovanis},
Journal = {Phys. Rev. E},
Year = {2001},
Pages = {026212},
Volume = {63}
}
@Article{NLA:NLA198,
Title = {Almost block diagonal linear systems: sequential and parallel solution techniques, and applications},
Author = {Amodio, P. and Cash, J. R. and Roussos, G. and Wright, R. W. and Fairweather, G. and Gladwell, I. and Kraut, G. L. and Paprzycki, M.},
Journal = {Numer. Linear Algebra Appl.},
Year = {2000},
Pages = {275--317},
Volume = {7}
}
@Article{NLSmal91,
Title = {Modulational instabilities and soliton solutions of a generalized nonlinear Schr{\"{o}}dinger equation},
Author = {B. A. Malomed and L. Stenflo},
Journal = {J. Phys. A},
Year = {1991},
Pages = {L1149--L1153},
Volume = {24},
Abtract = {A {H}amiltonian equation is proposed for the generalized nonlinear Schr{\"{o}}dinger equation. Soliton solutions are found and the stability of the plane waves is investigated.}
}
@Article{NLSper77,
Title = {Nonlinear Schr{\"{o}}dinger equation including growth and damping},
Author = {N. R. Pereira and L. Stenflo},
Journal = {Phys. Fluid},
Year = {1977},
Number = {10},
Pages = {1733--1734},
Volume = {20},
Abstract = {A nonlinear Schr{\"{o}}dinger equation is found with growth and damping
terms in the weakly nonlinear regime. An exact solitary solution
is found for the equation.}
}
@Article{NLSstn03,
Title = {Oscillons at a plasma surface},
Author = {L. Stenflo and M. Y. Yu},
Journal = {Phys. Plasmas},
Year = {2003},
Pages = {912--913},
Volume = {10},
Abstract = {Under the external driving force, an oscillon solutions is found.}
}
@Article{Noether15,
author = {Noether, E.},
title = {{Der Endlichkeitssatz der Invarianten endlicher Gruppen}},
journal = {Math. Ann.},
year = {1915},
volume = {77},
pages = {89--92},
doi = {10.1007/BF01456821}
}
@Article{nongal89,
Title = {Nonlinear {G\"{a}lerkin} methods},
Author = {M. Martine and R. Temam},
Journal = {SIAM J. Numer. Anal.},
Year = {1989},
Number = {5},
Pages = {1139--1157},
Volume = {26},
Abstract = {nonlinear {G}{\"{a}}lerkin methods}
}
@Article{NoTaDauTu04,
Title = {Survey of instability thresholds of flow between exactly counter-rotating disks},
Author = {C. Nore and M. Tartar and O. Daube and L. S. Tuckerman},
Journal = {J. Fluid Mech.},
Year = {2004},
Pages = {45--65},
Volume = {511},
DOI = {10.1017/S0022112004008559}
}
@Article{NP,
Title = {An adaptive {Newton-Picard} algorithm with subspace iteration for computing periodic solutions},
Author = {K. Lust and D. Roose},
Journal = {SIAM J. Sci. Comput.},
Year = {1998},
Pages = {1188--1209},
Volume = {19}
}
@Book{nr,
Title = {Numerical Recipes in {C}},
Author = {W. H. Press and S. A. Teukolsky and W. T. Vetterling and B. P. Flannery},
Publisher = {Cambridge Univ. Press},
Year = {1992},
Address = {Cambridge}
}
@Book{ns,
Title = {Numerical Partial Differential Equations},
Author = {J. W. Thomas},
Publisher = {Springer},
Year = {1995},
Address = {New York}
}
@Article{NSTks85,
author = {B. Nicolaenko and B. Scheurer and R. Temam},
title = {Some global dynamical properties of the {Kuramoto-Sivashinsky} equations: nonlinear stability and attractors},
journal = {Physica D},
year = {1985},
volume = {16},
pages = {155--183}
}
@Article{NTOD92,
Title = {Search for regular orbits in the {$x^2 y^2$} potential problem},
Author = {M. L. A. Nip and J. A. Tuszynski and M. Otwinowski and J. M. Dixon},
Journal = {J. Phys. A},
Year = {1992},
Pages = {5553},
Volume = {25},
Abstract = {Analytical and numerical investigation is provided into the motion
of a particle close to unstable straight-interval y=+or-x orbits
in the system H=1/2(p 2 x +p 2 y +x 2 y 2 ). Physical interpretation
is provided which involves a perturbation analysis leading to doubly
periodic solutions for x(t) and y(t). Second-order ordinary differential
equations are derived to describe the trajectory of the particle
in the x-y plane and the functional form of x(t).},
DOI = {10.1088/0305-4470/25/21/014}
}
@Article{OhlRav13,
author = {M. Ohlberger and S. Rave},
title = {Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing},
journal = {Comptes Rendus Mathematique},
year = {2013},
volume = {351},
pages = {901--906},
doi = {10.1016/j.crma.2013.10.028},
abstract = {Abstract We present a new method for the nonlinear
approximation of the solution manifolds of parameterized nonlinear
evolution problems, in particular in hyperbolic regimes with moving
discontinuities. Given the action of a Lie group on the solution space,
the original problem is reformulated as a partial differential algebraic
equation system by decomposing the solution into a group component and a
spatial shape component, imposing appropriate algebraic constraints on
the decomposition. The system is then projected onto a reduced basis
space. We show that efficient online evaluation of the scheme is possible
and study a numerical example showing its strongly improved performance
in comparison to a scheme without freezing. }
}
@Article{Olmsted94,
author = {Olmsted, P. D.},
title = {Rotational invariance and {Goldstone modes} in nematic elastomers and gels},
journal = {J. Phys. II France},
year = {1994},
volume = {4},
pages = {2215--2230},
doi = {10.1051/jp2:1994257}
}
@Article{OlVa94,
Title = {A continuation method to study periodic orbits of the {Froeschl\'e} map},
Author = {A. Olvera and C. Vargas},
Journal = {Physica D},
Year = {1994},
Pages = {351--371},
Volume = {72},
Abstract = {The dynamics of many Hamiltonian systems with three degrees of freedom
is represented by the Froeschl\'e map which is symplectic and four-dimensional.
In this paper we study sequences of periodic orbits approaching the
invariant tori in order to obtain information about its stability.
A homotopic method is used to continue the branches of periodic orbits.
We found that the existence of turning points is related to the linear
stability of the periodic orbit and its rotation vector.},
DOI = {10.1016/0167-2789(94)90238-0}
}
@Book{OlverInv,
Title = {Classical Invariant Theory},
Author = {P. J. Olver},
Publisher = {Cambridge Univ. Press},
Year = {1999},
Address = {Cambridge}
}
@Article{Onsager53,
Title = {Fluctuations and irreversible processes},
Author = {L. Onsager and S. Machlup},
Journal = {Phys. Rev.},
Year = {1953},
Pages = {1505, 1512},
Volume = {91}
}
@Article{optp1,
Title = {Optimal Periodic Orbits of Chaotic Systems},
Author = {B. R. Hunt and E. Ott},
Journal = {Phys. Rev. Lett.},
Year = {1996},
Pages = {2254},
Volume = {76}
}
@Article{optp2,
Title = {Optimal periodic orbits of continuous time chaotic systems},
Author = {T.-H. Yang and B. R. Hunt and E. Ott},
Journal = {Phys. Rev. E},
Year = {2000},
Pages = {1950},
Volume = {62}
}
@Article{Orazio2011,
Title = {Exploding dissipative solitons: {The} analog of the {Ruelle-Takens} route for spatially localized solutions},
Author = {Descalzi, O. and Cartes, C. and Cisternas, J.e and Brand, H. R.},
Journal = {Phys. Rev. E},
Year = {2011},
Pages = {056214},
Volume = {83},
DOI = {10.1103/PhysRevE.83.056214},
Numpages = {6}
}
@Techreport{orbit,
Title = {On locating homoclinic and heteroclinic orbits},
Author = {Doedel, E. J. and Friedman, M. J. and Monteiro, A. C.},
Institution = {Center for Applied Mathematics, Cornell University},
Year = {1993}
}
@Article{OrRa04,
Title = {Symmetry reduction in symplectic and {Poisson} geometry},
Author = {J.-P. Ortega and T. S. Ratiu},
Journal = {Lett. Math. Phys.},
Year = {2004},
Pages = {11--60},
Volume = {69},
DOI = {10.1007/s11005-004-0898-x}
}
@Article{Orszag1,
Title = {Boundary conditions for incompressible flows},
Author = {S. A. Orszag and M. Israeli and M. Deville},
Journal = {J. Sci. Comput.},
Year = {1986},
Pages = {75--111},
Volume = {1}
}
@Article{Orszag71,
Title = {Accurate solution of the {Orr-Sommerfeld} stability equation},
Author = {S. A. Orszag},
Journal = {J. Fluid Mech.},
Year = {1971},
Pages = {689--703},
Volume = {50}
}
@Book{ottbook,
Title = {Chaos and Dynamical Systems},
Author = {E. Ott},
Publisher = {Cambridge Univ. Press},
Year = {2002},
Address = {Cambridge}
}
@Article{OttYo08:3503096,
Title = {When {Lyapunov} exponents fail to exist},
Author = {Ott, W. and Yorke, J. A.},
Journal = {Phys. Rev. E},
Year = {2008},
Volume = {78}
}
@Unpublished{OzHaRoBu13,
Title = {Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization},
Author = {Oza, A. U. and Harris, D. M. and Rosales, R. R. and Bush, J. W. M.},
Note = {submitted to J. Fluid Mech.},
Year = {2013}
}
@Article{OzRoBu13,
Title = {A trajectory equation for walking droplets: hydrodynamic pilot-wave theory},
Author = {Oza, A. U. and Rosales, R. R. and Bush, J. W. M.},
Journal = {J. Fluid Mech.},
Year = {2013},
Pages = {552--570},
Volume = {737},
DOI = {10.1017/jfm.2013.581}
}
@Article{PaHe88,
Title = {Baroclinic eddy fluxes in a one-dimensional model of quasi-geostrophic turbulence},
Author = {Panetta, R. L. and Held, I. M.},
Journal = {J. Atmos. Sci.},
Year = {1988},
Pages = {3354--3365},
Volume = {45},
DOI = {10.1175/1520-0469(1988)045<3354:BEFIAO>2.0.CO;2}
}
@Article{PaHePi87,
Title = {External {Rossby} waves in the two-layer model},
Author = {Panetta, R. L. and Held, I. M. and Pierrehumbert, R. T.},
Journal = {J. Atmos. Sci.},
Year = {1987},
Pages = {2924--2933},
Volume = {44},
DOI = {10.1175/1520-0469(1987)044<2924:ERWITT>2.0.CO;2}
}
@Article{PaiL81,
Title = {A {S}chur Decomposition for {H}amiltonian Matrices},
Author = {C.C. Paige and Van Loan, C.},
Journal = {Linear Algebra and its Applications},
Year = {1981},
Pages = {11--32},
Volume = {41}
}
@Article{Pal61,
Title = {On the existence of slices for actions of non-compact {Lie} groups},
Author = {R. S. Palais},
Journal = {Ann. Math.},
Year = {1961},
Pages = {295--323},
Volume = {73}
}
@Article{palaniyappan2012,
Title = {Dynamics of relativistic transparency and optical shuttering in expanding overdense plasmas},
Author = {Palaniyappan, Sasi and Hegelich, B. Manuel and Wu, Hui-Chun and Jung, Daniel and Gautier, Donald C. and Yin, Lin and Albright, Brian J. and Johnson, Randall P. and Shimada, Tsutomu and Letzring, Samuel and Offermann, Dustin T. and Ren, Jun and Huang, Chengkun and H\"orlein, Rainer and Dromey, Brendan and Fernandez, Juan C. and Shah, Rahul C.},
Journal = {Nature Phys.},
Year = {2012},
Pages = {763--769},
Volume = {8},
Abstract = {Overdense plasmas are usually opaque to laser light. However, when
the light is of sufficient intensity to drive electrons in the plasma
to near light speeds, the plasma becomes transparent. This process?known
as relativistic transparency?takes just a tenth of a picosecond.
Yet all studies of relativistic transparency so far have been restricted
to measurements collected over timescales much longer than this,
limiting our understanding of the dynamics of this process. Here
we present time-resolved electric field measurements (with a temporal
resolution of {\textasciitilde} 50 fs) of the light, initially reflected
from, and subsequently transmitted through, an expanding overdense
plasma. Our result provides insight into the dynamics of the transparent-overdense
regime of relativistic plasmas, which should be useful in the development
of laser-driven particle accelerators, X-ray sources and techniques
for controlling the shape and contrast of intense laser pulses.},
DOI = {10.1038/nphys2390}
}
@Inbook{Palmer06,
chapter = {Predictability of weather and climate: {From} theory to practice},
pages = {1--30},
title = {Predictability of Weather and Climate},
publisher = {Cambridge U. Press},
year = {2006},
author = {T. Palmer},
editor = {T. Palmer and E. Hagedorn}
}
@Article{PaLo10,
Title = {Characteristic {Lyapunov} vectors in chaotic time-delayed systems},
Author = {Paz{\'o}, D. and L{\'o}pez, J. M.},
Journal = {Phys. Rev. E},
Year = {2010},
Note = {\arXiv{1101.2779}},
Pages = {056201},
Volume = {82}
}
@Article{PaLoPo13,
Title = {Universal scaling of {Lyapunov}-exponent fluctuations in space-time chaos},
Author = {Paz{\'o}, D. and L{\'o}pez, J. M. and Politi, APaz{\'o}, D. and L{\'o}pez, J. M.},
Journal = {Phys. Rev. E},
Year = {2013},
Note = {\arXiv{1311.7599}},
Pages = {062909},
Volume = {87}
}
@Article{PaLoRo13,
Title = {On the angle between the first and second {Lyapunov} vectors in spatio-temporal chaos},
Author = {D. Paz\'o and J. M. L\'opez and M. A. Rodr\'iguez},
Journal = {J. Phys. A},
Year = {2013},
Pages = {254014},
Volume = {46},
Abstract = {[...] the first Lyapunov vector (LV) is associated with the largest
Lyapunov exponent and indicates at any point on the attractor the
direction of maximal growth in tangent space. The LV corresponding
to the second largest Lyapunov exponent generally points in a different
direction, but tangencies between both vectors can in principle occur.
Here we find that the probability density function (PDF) of the angle
... spanned by the first and second LVs should be expected to be
approximately symmetric around p/4 and to peak at 0 and p/2. Moreover,
for small angles we uncover a scaling law for the PDF Q of ? l =
ln?? with the system size L : Q (? l ) = L -1/2 f (? l L -1/2 ).
We give a theoretical argument that justifies this scaling form and
also explains why it should be universal (irrespective of the system
details) for spatio-temporal chaos in one spatial dimension. This
article is part of a special issue of Journal of Physics A: Mathematical
and Theoretical devoted to `Lyapunov analysis: from dynamical systems
theory to applications'.},
DOI = {10.1088/1751-8113/46/25/254014}
}
@Article{PalZan13,
Title = {Singular vectors, predictability and ensemble forecasting for weather and climate},
Author = {T. N. Palmer and L. Zanna},
Journal = {J. Phys. A},
Year = {2013},
Pages = {254018},
Volume = {46},
URL = {http://stacks.iop.org/1751-8121/46/i=25/a=254018}
}
@Article{Panetta93,
Title = {Zonal jets in wide baroclinically unstable regions: {Persistence} and scale selection},
Author = {Panetta, R. L.},
Journal = {J. Atmos. Sci.},
Year = {1993},
Pages = {2073--2106},
Volume = {50},
DOI = {10.1175/1520-0469(1993)050<2073:ZJIWBU>2.0.CO;2}
}
@Book{ParChu89,
Title = {Practical Numerical Algorithms for Chaotic Systems},
Author = {Parker, T. S. and Chua, L. O.},
Publisher = {Springer},
Year = {1989},
Address = {New York}
}
@Article{paskauskas09,
author = {Pa\v{s}kauskas, R. and De Ninno, G.},
title = {{Lyapunov} stability of {Vlasov} equilibria using {Fourier-{Hermit}e} modes},
journal = {Phys. Rev. E},
year = {2009},
volume = {80},
pages = {036402}
}
@Article{PaSzLoRo09,
Title = {Structure of characteristic {Lyapunov} vectors in spatiotemporal chaos},
Author = {D. Paz\'o and I. G. Szendro and J. M. L\'opez and M. A. Rodr\'iguez},
Journal = {Phys. Rev. E},
Year = {2008},
Pages = {016209},
Volume = {78},
Abstract = {We study Lyapunov vectors (LVs) corresponding to the largest Lyapunov
exponents in systems with spatiotemporal chaos. We focus on characteristic
LVs and compare the results with backward LVs obtained via successive
Gram-Schmidt orthonormalizations. Systems of a very different nature
such as coupled-map lattices and the (continuous-time) Lorenz `96
model exhibit the same features in quantitative and qualitative terms.
Additionally, we propose a minimal stochastic model that reproduces
the results for chaotic systems. Our work supports the claims about
universality of our earlier results [I. G. Szendro et al., Phys.
Rev. E 76, 025202(R) (2007)] for a specific coupled-map lattice.}
}
@Article{PawSchu91,
Title = {Unstable periodic orbits and prediction},
Author = {K. Pawelzik and H. G. Schuster},
Journal = {Phys. Rev. A},
Year = {1991},
Pages = {1808--1812},
Volume = {43},
DOI = {10.1103/PhysRevA.43.1808}
}
@Article{PCar,
Title = {Group theory for {Feynman} diagrams in non-{Abelian} gauge theories},
Author = {P. Cvitanovi\'c},
Journal = {Phys. Rev. D},
Year = {1976},
Pages = {1536--1553},
Volume = {14}
}
@Book{PCgr,
Title = {{Group Theory} - {Birdtracks}, {Lie}'s, and {Exceptional Groups}},
Author = {P. Cvitanovi\'c},
Publisher = {Princeton Univ. Press},
Year = {2008},
Address = {Princeton, NJ},
URL = {http://birdtracks.eu}
}
@Article{pchaot,
Title = {Exploring chaotic motion through periodic orbits},
Author = {D. Auerbach and P. Cvitanovi\'{c} and J.-P. Eckmann and G. Gunaratne and I. Procaccia},
Journal = {Phys. Rev. Lett.},
Year = {1987},
Pages = {2387--2389},
Volume = {58},
Abstract = {Use near recurrence to extract periodic orbits and their stability
eigenvalues.}
}
@Article{peckham90,
Title = {The necessity of the {Hopf} bifurcation for periodically forced oscillators},
Author = {Peckham, B. B.},
Journal = {Nonlinearity},
Year = {1990},
Pages = {261},
Volume = {3}
}
@Article{PeKl91,
Title = {The nonlinear dynamics of slightly supercritical baroclinic jets},
Author = {Pedlosky, J. and Klein, P.},
Journal = {J. Atmos. Sci.},
Year = {1991},
Pages = {1276--1286},
Volume = {48},
DOI = {10.1175/1520-0469(1991)048<1276:TNDOSS>2.0.CO;2}
}
@Article{PeLiAh11,
author = {Peters, D. A. and Lieb, S. M. and Ahaus, L. A.},
title = {Interpretation of {Floquet} eigenvalues and eigenvectors for periodic systems},
journal = {J. Am. Helicopter Soc.},
year = {2011},
volume = {56},
pages = {1--11},
doi = {10.4050/JAHS.56.032001}
}
@Book{Penr04,
Title = {The Road to Reality: A Complete Guide to the Laws of the Universe},
Author = {R. Penrose},
Publisher = {A. A. Knopf},
Year = {2005},
Address = {New York}
}
@Article{PePaVr08,
Title = {Stochastic optimization for detecting periodic orbits of nonlinear mappings},
Author = {Petalas, Y. G. and Parsopoulos, K. E. and Vrahatis, M. N.},
Journal = {Nonlin. Phenom. Complex Sys.},
Year = {2008},
Pages = {285 -- 291},
Volume = {11}
}
@Article{PePo87,
Title = {Wave-wave interaction of unstable baroclinic waves},
Author = {Pedlosky, J. and Polvani, L. M.},
Journal = {J. Atmos. Sci.},
Year = {1987},
Pages = {631--647},
Volume = {44},
DOI = {10.1175/1520-0469(1987)044<0631:WIOUBW>2.0.CO;2}
}
@Article{percgl,
Title = {Exact periodic solutions of the complex {Ginzburg-Landau} equation},
Author = {A. V. Porubov and M. G. Velarde},
Journal = {J. Math. Phys.},
Year = {1999},
Pages = {884},
Volume = {40}
}
@Book{perg,
Title = {An Introduction to Ergodic Theory},
Author = {P. Walters},
Publisher = {Springer},
Year = {1981},
Address = {New York}
}
@Book{perkob,
Title = {Differential Equations and Dynamical Systems},
Author = {L. Perko},
Publisher = {Springer},
Year = {1991},
Address = {New York}
}
@Book{PerRich82N,
Title = {Introduction to Dynamics},
Author = {I. Percival and D. Richards},
Publisher = {Cambridge Univ. Press},
Year = {1996},
Address = {Cambridge}
}
@Book{Pesin2004,
Title = {Lectures on partial hyperbolicity and stable ergodicity},
Author = {Pesin, Ya. B.},
Publisher = {European Mathematical Society},
Year = {2004},
Address = {Z\"urich},
DOI = {10.4171/003}
}
@Book{PesSch95,
Title = {An Introduction to Quantum Field Theory},
Author = {Peskin, M. E. and Schroeder, D. V.},
Publisher = {Perseus Books},
Year = {1995},
Address = {Cambridge, Massachusetts}
}
@Article{PetCorBol06,
author = {Pethel, S. D. and Corron, N. J. and Bollt, E.},
title = {Symbolic dynamics of coupled map lattices},
journal = {Phys. Rev. Lett.},
year = {2006},
volume = {96},
pages = {034105},
doi = {10.1103/PhysRevLett.96.034105}
}
@Article{PetCorBol07,
author = {Pethel, S. D. and Corron, N. J. and Bollt, E.},
title = {Deconstructing spatiotemporal chaos using local symbolic dynamics},
journal = {Phys. Rev. Lett.},
year = {2007},
volume = {99},
pages = {214101},
doi = {10.1103/PhysRevLett.99.214101}
}
@Article{Peterhof1999,
Title = {All-optical clock recovery using multisection distributed-feedback lasers},
Author = {D. Peterhof and B. Sandstede},
Journal = {J. Nonlinear Sci.},
Year = {1999},
Pages = {575--613},
Volume = {9}
}
@Book{PetTri09,
title = {Applications of Group Theory in Quantum Mechanics},
publisher = {Dover},
year = {2009},
author = {Petrashen, M. I. and Trifonov, E. D.},
isbn = {9780486472232},
address = {New York}
}
@Book{Peyret02,
Title = {Spectral Methods for Incompressible Flows},
Author = {R. Peyret},
Publisher = {Springer},
Year = {2002},
Address = {New York}
}
@Incollection{Pfenninger61,
author = {W. Pfenninger},
title = {Transition in the inlet length of tubes at high Reynolds numbers},
booktitle = {Boundary Layer and Flow Control},
publisher = {Pergamon},
year = {1961},
editor = {G. V. Lachmann},
pages = {970--980},
address = {Oxford, UK},
doi = {10.1016/B978-1-4832-1323-1.50013-0}
}
@Book{PG97,
Title = {Chaos, Scattering and Statistical Mechanics},
Author = {P. Gaspard},
Publisher = {Cambridge Univ. Press},
Year = {1997},
Address = {Cambridge}
}
@Misc{PGER13,
Title = {Groebner basis methods for stationary solutions of a shear flow},
Author = {Pausch, M. and Grossmann, F. and Eckhardt, B. and Romanovski, V. G.},
Note = {J. Nonlinear Sci., to appear.},
Year = {2013}
}
@Article{phillips51,
author = {N. A. Phillips},
title = {A simple three-dimensional model for the study of large-scale extratropical flow patterns},
journal = {J. Meteorology},
year = {1951},
volume = {8},
pages = {381--394}
}
@Article{PhysToday04,
Title = {New experiments set the scale for the onset of turbulence in pipe flow},
Author = {R. Fitzgerald},
Journal = {Physics Today},
Year = {2004},
Pages = {21--23},
Volume = {57}
}
@Article{PhysWorld04,
Title = {Turbulent transition for fluids},
Author = {C. Barenghi},
Journal = {Physics World},
Year = {2004},
Volume = {17}
}
@Article{pimsimp,
Title = {The {PIM}-simplex method: an extension of the {PIM}-triple method to saddles with an arbitrary number of expanding directions},
Author = {P. Moresco and S. P. Dawson},
Journal = {Physica D},
Year = {1999},
Pages = {38},
Volume = {126},
Abstract = {They generilized the PIM method to treat the chaotic saddle with more
than one unstable direction by introducing a simplex to cross the
stable manifolds. Combining with one method for finding local extremum
in a simplex, the authors do essentially the same job as in the simple
PIM method.}
}
@Article{pimstag,
Title = {Stagger-and-Step Method: Detecting and Computing Chaotic Saddles in Higher Dimensions},
Author = {D. Sweet and H. E. Nusse. and J. A. Yorke},
Journal = {Phys. Rev. Lett.},
Year = {2001},
Pages = {2261},
Volume = {86},
Abstract = {This is a statistical extension of the PIM method. Here, the representative
points are selected according to the so-called Exponential Stagger
Distribution about the current point, instead of the points on the
line or vertices of a simplex. This is a brutal force method while
the authors claim that it is quite efficient.}
}
@Article{pimyk,
Title = {A procedure for finding numerical trajectories on chaotic saddles},
Author = {H. E. Nusse and J. A. York},
Journal = {Physica D},
Year = {1989},
Pages = {137},
Volume = {36},
Abstract = {{PIM} (proper interior maximum) method is proposed to find chaotic
trajectories on a nonattracting set with one-d unstable manifold.
By partitioning a straight line straddling the stable manifolds,
the point with longer and longer escaping time is found in a sequel
until a sufficiently small interval is reached when the point is
considered to be very close to the stable manifolds. After that,
evolving and restraining the line element results in a NUMERICAL
orbit is obtained.}
}
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Title = {Planar perturbation expansion},
Author = {Cvitanovi\'c, P.},
Journal = {Phys. Lett. B},
Year = {1981},
Pages = {49},
Volume = {99}
}
@Article{PlaSirFit91,
Title = {An investigation of chaotic {Kolmogorov} flows},
Author = {Platt, N. and Sirovich, L. and Fitzmaurice, N.},
Journal = {Phys.\ Fluids A},
Year = {1991},
Pages = {681--696},
Volume = {3},
DOI = {10.1063/1.858074}
}
@Article{PoGiYaMa06,
Title = {From synchronization to {Lyapunov} exponents and back},
Author = {Politi, A. and Ginelli, F. and Yanchuk, S. and Maistrenko, Y.},
Journal = {Physica D},
Year = {2006},
Note = {\arXiv{nlin/0605012}},
Pages = {90},
Volume = {224},
Abstract = {The goal of this paper is twofold. In the first part we discuss a
general approach to determine Lyapunov exponents from ensemble- rather
than time-averages. The approach passes through the identification
of locally stable and unstable manifolds (the Lyapunov vectors),
thereby revealing an analogy with generalized synchronization. The
method is then applied to a periodically forced chaotic oscillator
to show that the modulus of the Lyapunov exponent associated to the
phase dynamics increases quadratically with the coupling strength
and it is therefore different from zero already below the onset of
phase-synchronization. The analytical calculations are carried out
for a model, the generalized special flow, that we construct as a
simplified version of the periodically forced Rossler oscillator.}
}
@Incollection{PoHi00,
Title = {Simulation of billiards and of hard body fluids},
Author = {Posch, H. A. and Hirschl, R.},
Booktitle = {Hard ball systems and the {Lorentz gas}},
Publisher = {Springer},
Year = {2000},
Address = {Berlin},
Editor = {Sz{\'a}sz, D. and Bunimovich, L. A.},
Pages = {279--314}
}
@Article{Poinc1896,
Title = {Sur les solutions p\'eriodiques et le principe de moindre action},
Author = {H. Poincar\'e},
Journal = {C. R. Acad. Sci. Paris},
Year = {1896},
Pages = {915--918},
Volume = {123}
}
@Book{poincare,
Title = {Les M\'ethodes Nouvelles de la M\'echanique C\'eleste},
Author = {H. Poincar\'e},
Publisher = {Guthier-Villars},
Year = {1899},
Address = {Paris},
Note = {For a very readable exposition of {Poincar\'e}'s work and the development of the dynamical systems theory up to 1920's see \refreff{JBG97}.}
}
@Book{poinper,
Title = {New Methods in Celestial Mechanics},
Author = {H. Poincar\'{e}},
Publisher = {Springer},
Year = {1992},
Address = {New York}
}
@Article{Poiseuille1844,
Title = {Recherches exp{\'e}rimentales sur le mouvement des liquides dans les tubes de tr{\`e}s-petits diam{\`e}tres},
Author = {J. L. Poiseuille},
Journal = {C. R. Acad. Sci.},
Year = {1840},
Pages = {961},
Volume = {11}
}
@Article{PoKno05,
author = {J. Porter and E. Knobloch},
title = {Dynamics in the 1:2~{s}patial resonance with broken reflection symmetry},
journal = {Physica D},
year = {2005},
volume = {201},
pages = {318--344},
doi = {10.1016/j.physd.2005.01.001}
}
@Article{Polizzi09,
Title = {Density-matrix-based algorithm for solving eigenvalue problems},
Author = {Polizzi, E.},
Journal = {Phys. Rev. B},
Year = {2009},
Pages = {115112},
Volume = {79},
DOI = {10.1103/PhysRevB.79.115112}
}
@Incollection{Pollicott02,
Title = {Chapter 5 Periodic orbits and zeta functions},
Author = {M. Pollicott},
Booktitle = {Handbook of Dynamical Systems},
Publisher = {Elsevier},
Year = {2002},
Editor = {B. Hasselblatt and A. Katok},
Pages = {409--452},
Volume = {1, Part A},
Abstract = {The study of periodic orbits for dynamical systems dates back to the
very origins of the subject. This chapter provides an overview of
some of the main results without any claims of being exhaustive.
The fundamental problem in the study of zeta functions is to understand
the analytic domain of such functions or more generally the meromorphic
domain. In many interesting cases, these domains extend far beyond
the initial domain of convergence of the series. The chapter discusses
dynamical variant on the study of the Lefschetz zeta function. This
serves to illustrate an important theme in the study of dynamical
zeta functions, namely that local data (e.g., periodic orbits) can
have a bearing on the global properties of the transformation f.
In particular, this involves the idea of characterizing closed orbits
as either twisted or untwisted. The Lefschetz zeta function then
reflects the balance between these two types of orbits.},
DOI = {10.1016/S1874-575X(02)80007-8}
}
@Article{Pollicott91,
author = {Pollicott, M.},
title = {A note on the {Artuso-Aurell-Cvitanovic} approach to the {Feigenbaum} tangent operator},
journal = {J. Stat. Phys.},
year = {1991},
volume = {62},
pages = {257--267},
doi = {10.1007/BF01020869}
}
@Book{Pollicott93,
Title = {Lectures on ergodic theory and Pesin theory on compact manifolds},
Author = {Pollicott, M.},
Publisher = {Cambridge Univ. Press},
Year = {1993},
Address = {Cambridge}
}
@Article{pomeau80,
Title = {Intermittent transition to turbulence in dissipative dynamical systems},
Author = {Y. Pomeau and Manneville, P.},
Journal = {Commun. Math. Phys.},
Year = {1980},
Pages = {189},
Volume = {74}
}
@Book{popebook,
Title = {Turbulent Flows},
Author = {S. B. Pope},
Publisher = {Cambridge Univ. Press},
Year = {2000},
Address = {Cambridge}
}
@Article{PorCvi04,
author = {Porter, M. A. and Cvitanovi\'c, P.},
title = {A perturbative analysis of modulated amplitude waves in {Bose-Einstein} condensates},
journal = {Chaos},
year = {2004},
volume = {14},
pages = {739--755},
doi = {10.1063/1.1779991}
}
@Article{porter2004modulated,
Title = {Modulated amplitude waves in {Bose-Einstein} condensates},
Author = {Porter, M. A. and Cvitanovi\'c, P.},
Journal = {Phys. Rev. E},
Year = {2004},
Pages = {047201},
Volume = {69}
}
@Article{Posch13,
Title = {Symmetry properties of orthogonal and covariant {Lyapunov} vectors and their exponents},
Author = {H. A. Posch},
Journal = {J. Phys. A},
Year = {2013},
Pages = {254006},
Volume = {46},
DOI = {10.1088/1751-8113/46/25/254006}
}
@Article{postgal89,
Title = {Postprocessing the {G\"{a}lerkin} method: {A} novel approach to approximate inertial manifolds},
Author = {B. Garc\'{i}a-Archilla and J. Novo and E. S. Titi},
Journal = {SIAM J. Numer. Anal.},
Year = {1998},
Pages = {941--972},
Volume = {35},
Abstract = {{G}{\"{a}}lerkin method}
}
@Article{Postle09,
Title = {Stabilization of long-period periodic orbits using time-delayed feedback control},
Author = {Postlethwaite, C. M.},
Journal = {SIAM J. Appl. Dyn. Sys.},
Year = {2009},
Note = {\arXiv{0804.2903}},
Pages = {21--39},
Volume = {8}
}
@Article{PoToLe98,
author = {A. Politi and A. Torcini and S. Lepri},
title = {{Lyapunov} exponents from node-counting arguments},
journal = {J. Phys. IV},
year = {1998},
volume = {8},
pages = {263},
doi = {10.1051/jp4:1998636}
}
@Article{PP83,
author = {W. Parry and M. Pollicott},
title = {An analogue of the prime number theorem for closed orbits of {Axiom A} flows},
journal = {Ann. Math.},
year = {1983},
volume = {118},
pages = {573--591}
}
@Article{pre88top,
Title = {Topological and metric properties of {H\'{e}non}-type strange attractors},
Author = {P. Cvitanovi\'{c} and G. H. Gunaratne and I. Procaccia},
Journal = {Phys. Rev. A},
Year = {1988},
Pages = {1503},
Volume = {38}
}
@Book{Press96,
Title = {Numerical Recipes in Fortran},
Author = {W. H. Press and B. P. Flannery and S. A. Teukolsky and W. T. Vetterling},
Publisher = {Cambridge Univ. Press},
Year = {1996},
Address = {Cambridge}
}
@Article{Pringle07,
Title = {Asymmetric, helical, and mirror-symmetric traveling waves in pipe flow},
Author = {Pringle, C. C. T. and Kerswell, R. R.},
Journal = {Phys.\ Rev.\ Lett.},
Year = {2007},
Pages = {074502},
Volume = {99}
}
@Article{Pringle2009,
Title = {Highly symmetric travelling waves in pipe flow},
Author = {Pringle, C. C. T. and Duguet, Y. and Kerswell, R.},
Journal = {Philos. Trans. R. Soc. A},
Year = {2009},
Pages = {457--472},
Volume = {367},
Owner = {root},
Timestamp = {2012.10.23}
}
@Article{Proctor88,
Title = {Odd symplectic groups},
Author = {Proctor, R. A.},
Journal = {Invent. Math.},
Year = {1988},
Pages = {307--332},
Volume = {92}
}
@Article{Proctor93,
author = {Proctor, R. A.},
title = {Reflection and algorithm proofs of some more {Lie} group dual pair identities},
journal = {J. Combin. Theory A},
year = {1993},
volume = {62},
number = {1},
pages = {107--127}
}
@Article{PSD01,
Title = {Detecting unstable periodic orbits in chaotic continuous-time dynamical systems},
Author = {D. Pingel and P. Schmelcher and F. K. Diakonos},
Journal = {Phys. Rev. E},
Year = {2001},
Number = {2},
Pages = {026214},
Volume = {64},
Abstract = {Change the stability of the fixed points on the Poincar\'{e} section
to find periodic orbits.}
}
@Article{PSDB00,
Title = {Theory and applications of the systematic detection of unstable periodic orbits in dynamical systems},
Author = {D. Pingel and P. Schmelcher and F. K. Diakonos and O. Biham},
Journal = {Phys. Rev. E},
Year = {2000},
Note = {\arXiv{nlin.CD/0006011}},
Pages = {2119},
Volume = {62}
}
@Article{PSHMR14,
Title = {Cluster synchronization and isolated desynchronization in complex networks with symmetries},
Author = {Pecora, L. M. and Sorrentino, F. and Hagerstrom, A. M. and Murphy, T. E. and Roy, R.},
Journal = {Nature communications},
Year = {2014},
Volume = {5},
DOI = {10.1038/ncomms5079}
}
@Article{pugh67cl,
Title = {An improved closing lemma and a general density theorem},
Author = {C. Pugh},
Journal = {Amer. J. Math.},
Year = {1967},
Volume = {89}
}
@Article{PugShu89,
Title = {Ergodic attractors},
Author = {Pugh, C. and Shub, M.},
Journal = {Trans. Amer. Math. Soc.},
Year = {1989},
Pages = {1--54},
Volume = {312},
DOI = {10.1090/S0002-9947-1989-0983869-1}
}
@Article{PuShSt04,
author = {C. Pugh and M. Shub and A. Starkov},
title = {Stable ergodicity},
journal = {Bull. Am. Math. Soc.},
year = {2004},
volume = {41},
pages = {1--41},
doi = {10.1090/S0273-0979-03-00998-4}
}
@Phdthesis{PutkaradzeThesis,
Title = {Local structures in extended systems},
Author = {Putkaradze, V.},
School = {Copenhagen Univ.},
Year = {1997},
Address = {Copenhagen},
URL = {http://ChaosBook.org/projects/theses.html}
}
@Article{Pyragas92,
author = {K. Pyragas},
title = {Continuous control of chaos by self-controlling feedback},
journal = {Phys. Lett. A },
year = {1992},
volume = {170},
pages = {421--428},
doi = {10.1016/0375-9601(92)90745-8}
}
@Article{Qin03,
Title = {Bifurcations of steady states in a class of lattices of nonlinear discrete {Klein-Gordon} type with double-quadratic on-site potential},
Author = {Qin, Wen-Xin},
Journal = {J. Phys. A},
Year = {2003},
Pages = {9865},
Volume = {36},
Abstract = {By using the method of symbolic dynamics, we study the bifurcations
of steady states in a class of lattices of nonlinear discrete Klein-Gordon
type with double-quadratic on-site potential. We derive by virtue
of the admissible condition the critical value of the coupling strength,
below which the steady states persist without bifurcations. If the
coupling coefficient passes through the critical value, some of the
steady states disappear. Meanwhile there are no new steady states
created as varies. We obtain bifurcation values of some lower-order
spatially periodic steady states by introducing the concept 'characteristic
polynomial' of periodic sequences.}
}
@Article{R1883,
Title = {An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and the law of resistance in parallel channels},
Author = {O. Reynolds},
Journal = {Proc. Roy. Soc. Lond. Ser. A},
Year = {1883},
Pages = {935--982},
Volume = {174}
}
@Article{R1894,
Title = {On the dynamical theory of incompressible viscous flows and the determination of the criterion},
Author = {O. Reynolds},
Journal = {Proc. Roy. Soc. Lond. Ser. A},
Year = {1894},
Pages = {123--161},
Volume = {186}
}
@Article{Radziunas2006,
Title = {Numerical bifurcation analysis of the traveling wave model of multisection semiconductor lasers},
Author = {M. Radziunas},
Journal = {Physica D},
Year = {2006},
Pages = {98--112},
Volume = {213}
}
@Article{RaHaAb96,
Title = {Global stability properties of the complex {Lorenz} model},
Author = {Rauh, A. and Hannibal, L. and Abraham, N. B.},
Journal = {Physica D},
Year = {1996},
Pages = {45--58},
Volume = {99}
}
@Article{RamSri00,
Title = {A comparative study of computation of {Lyapunov} spectra with different algorithms},
Author = {Ramasubramanian, K. and Sriram, M. S.},
Journal = {Physica D},
Year = {2000},
Pages = {72--86},
Volume = {139}
}
@Article{Rand82,
Title = {Dynamics and symmetry - predictions for modulated waves in rotating fluids},
Author = {Rand, D.},
Journal = {Arch. Rational Mech. Anal.},
Year = {1982},
Pages = {1--3},
Volume = {79}
}
@Article{RCDL93,
Title = {A practical method for calculating largest {Lyapunov} exponents from small data sets},
Author = {Rosenstein, M. T. and Collins, J. J. and De Luca, C. J. },
Journal = {Physica D},
Year = {1993},
Pages = {117--134},
Volume = {65},
Abstract = {Detecting the presence of chaos in a dynamical system is an important
problem that is solved by measuring the largest Lyapunov exponent.
Lyapunov exponents quantify the exponential divergence of initially
close state-space trajectories and estimate the amount of chaos in
a system. We present a new method for calculating the largest Lyapunov
exponent from an experimental time series. The method follows directly
from the definition of the largest Lyapunov exponent and is accurate
because it takes ...}
}
@Article{RCMR04,
author = {Rempel, E. L. and Chian, A. C. and Macau, E. E. and Rosa, R. R.},
title = {Analysis of chaotic saddles in high-dimensional dynamical systems: the {Kuramoto-Sivashinsky} equation},
journal = {Chaos},
year = {2004},
volume = {14},
pages = {545--56},
doi = {10.1063/1.1759297}
}
@Article{ReCh07,
author = {Rempel, E. L. and Chian, A. C.},
title = {Origin of transient and intermittent dynamics in spatiotemporal chaotic systems},
journal = {Phys. Rev. Lett.},
year = {2007},
volume = {98},
pages = {014101}
}
@Article{ReChMi07,
Title = {Chaotic saddles at the onset of intermittent spatiotemporal chaos},
Author = {Rempel, E. L. and Chian, A. C. and Miranda, R. A.},
Journal = {Phys. Rev. E},
Year = {2007},
Pages = {056217},
Volume = {76},
Abstract = {In a recent study [Rempel and Chian, Phys. Rev. Lett. 98, 014101 (2007)],
it has been shown that nonattracting chaotic sets (chaotic saddles)
are responsible for intermittency in the regularized long-wave equation
that undergoes a transition to spatiotemporal chaos (STC) via quasiperiodicity
and temporal chaos. In the present paper, it is demonstrated that
a similar mechanism is present in the damped Kuramoto-Sivashinsky
equation. Prior to the onset of STC, a spatiotemporally chaotic saddle
coexists with a spatially regular attractor. After the transition
to STC, the chaotic saddle merges with the attractor, generating
intermittent bursts of STC that dominate the post-transition dynamics.}
}
@Article{ReCi05,
Title = {Intermittency induced by attractor-merging crisis in the {Kuramoto-Sivashinsky} equation},
Author = {Rempel, E. L. and Chian, A. C.},
Journal = {Phys. Rev. E},
Year = {2005},
Pages = {016203},
Volume = {71},
Abstract = {We characterize an attractor-merging crisis in a spatially extended
system exemplified by the Kuramoto-Sivashinsky equation. The simultaneous
collision of two coexisting chaotic attractors with an unstable periodic
orbit and its associated stable manifold occurs in the high-dimensional
phase space of the system, giving rise to a single merged chaotic
attractor. The time series of the post-crisis regime displays intermittent
behavior. The origin of this crisis-induced intermittency is elucidated
in terms of alternate switching between two chaotic saddles embedded
in the merged chaotic attractor.}
}
@Article{Recke1998,
Title = {Abstract forced symmetry breaking and forced frequency locking of modulated waves},
Author = {L. Recke and D. Peterhof},
Journal = {J. Diff. Eqn.},
Year = {1998},
Pages = {233--262},
Volume = {144}
}
@Article{Recke1998a,
Title = {Forced frequency locking of rotating waves},
Author = {L. Recke},
Journal = {Ukrain. Math. J.},
Year = {1998},
Pages = {94--101},
Volume = {50}
}
@Article{Recke2010,
Title = {Frequency locking of modulated waves},
Author = {L. Recke and A. Samoilenko and A. Teplinsky and V. Tkachenko and S. Yanchuk},
Journal = {Discrete Contin. Dynam. Systems},
Year = {2011},
Note = {\arXiv{1108.5990}},
Pages = {847--875},
Volume = {31}
}
@Article{reimann1,
author = {P. Reimann},
title = {Noisy one-dimensional maps near a crisis 1: {Weak {Gauss}ian} white and colored noise},
journal = {J. Stat. Phys.},
year = {1996},
volume = {82},
pages = {1467--1501}
}
@Article{reimann2,
Title = {Noisy one-dimensional maps near a crisis 2: {General} uncorrelated weak noise},
Author = {P. Reimann},
Journal = {J. Stat. Phys.},
Year = {1996},
Pages = {403--425},
Volume = {85}
}
@Misc{ReSaTkYa11,
Title = {Frequency locking by external forcing in systems with rotational symmetry},
Author = {{Recke}, L. and {Samoilenko}, A. and {Tkachenko}, V. and {Yanchuk}, S. },
Note = {\arXiv{1108.5990}},
Year = {2011}
}
@Book{Richter10,
title = {Semiclassical Theory of Mesoscopic Quantum Systems},
publisher = {Springer},
year = {2000},
author = {Richter, K.},
isbn = {9783662156506},
address = {Berlin},
doi = {10.1007/BFb0109634},
url = {http://www.physik.uni-regensburg.de/forschung/richter/richter/pages/research/springer-tracts-161.pdf}
}
@Article{RiKl97,
Title = {Effects of an asymmetric friction on the nonlinear equilibration of a baroclinic system},
Author = {Rivi\'ere, P. and Klein, P.},
Journal = {J. Atmos. Sci.},
Year = {1997},
Pages = {1610--1627},
Volume = {54},
DOI = {10.1175/1520-0469(1997)054<1610:EOAAFO>2.0.CO;2}
}
@Article{Rink200331,
author = {B. Rink},
title = {Symmetric invariant manifolds in the {Fermi-Pasta-Ulam} lattice},
journal = {Physica D},
year = {2003},
volume = {175},
pages = {31--42},
abstract = {The Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions
and n particles admits a large group of discrete symmetries. The
fixed point sets of these symmetries naturally form invariant symplectic
manifolds that are investigated in this short note. For each k dividing
n we find k degree of freedom invariant manifolds. They represent
short wavelength solutions composed of k Fourier modes and can be
interpreted as embedded lattices with periodic boundary conditions
and only k particles. Inside these invariant manifolds other invariant
structures and exact solutions are found which represent for instance
periodic and quasi-periodic solutions and standing and travelling
waves. Similar invariant manifolds exist also in the Klein-Gordon
(KG) lattice and in the thermodynamic and continuum limits.}
}
@Book{Risken96,
title = {The {Fokker-Planck} Equation},
publisher = {Springer},
year = {1996},
author = {H. Risken and H. Haken},
address = {New York}
}
@Article{Rivas13,
Title = {Semiclassical matrix elements for a chaotic propagator in the scar function basis},
Author = {A. M. F. Rivas},
Journal = {J. Phys. A},
Year = {2013},
Volume = {46},
Abstract = {A semiclassical approximation for the matrix elements of a quantum
chaotic propagator in the scar function basis has been derived. The
obtained expression is solely expressed in terms of canonical invariant
objects. For our purpose, we have used the recently developed, semiclassical
matrix elements of the propagator in coherent states, together with
the linearization of the flux in the neighborhood of a classically
unstable periodic orbit of chaotic two-dimensional systems. The expression
derived here is successfully verified to be exact for a (linear)
cat map, after the theory is adapted to a discrete phase space appropriate
to a quantized torus.},
DOI = {10.1088/1751-8113/46/14/145101}
}
@Incollection{RMChaosHist,
Author = {R. Mainieri},
Booktitle = {{Chaos: Classical and Quantum}},
Publisher = {Niels Bohr Inst.},
Year = {2015},
Address = {Copenhagen},
Chapter = {{A} brief history of chaos},
URL = {http://ChaosBook.org/paper.shtml#appendHist}
}
@Article{Ro88,
Title = {Painleve property and constants of the motion of the complex {Lorenz} model},
Author = {Roekaerts, D},
Journal = {J. Phys. A},
Year = {1988},
Pages = {L495--L498},
Volume = {21}
}
@Article{robb,
Title = {Discrete symmetries in periodic-orbit theory},
Author = {Robbins, J. M.},
Journal = {Phys. Rev. A},
Year = {1989},
Pages = {2128--2136},
Volume = {40},
DOI = {10.1103/PhysRevA.40.2128}
}
@Article{Robinson1995,
author = {Robinson, J. C.},
title = {Finite-dimensional behavior in dissipative partial differential equations},
journal = {Chaos},
year = {1995},
volume = {5},
pages = {330--345},
doi = {10.1063/1.166081}
}
@Article{Robinson91,
Title = {Coherent motions in the turbulent boundary layer},
Author = {S. K. Robinson},
Journal = {Ann. Rev. Fluid Mech.},
Year = {1991},
Pages = {601--639},
Volume = {23}
}
@Article{Robinson-PLA1994,
Title = {Inertial manifolds for the {Kuramoto-Sivashinsky} equation},
Author = {Robinson, J. C.},
Journal = {Phys. Lett. A},
Year = {1994},
Pages = {190--193},
Volume = {184},
Abstract = {A new theorem is applied to the Kuramoto-Sivashinsky equation with
L-periodic boundary conditions, proving the existence of an asymptotically
complete inertial manifold attracting all initial data. Combining
this result with a new estimate of the size of the globally absorbing
set yields an improved estimate of the dimension, $N \sim L^{2.46}$.}
}
@Article{RoCaRe12,
Title = {Dual pairing of symmetry and dynamical groups in physics},
Author = {Rowe, D. J. and Carvalho, M. J. and Repka, J.},
Journal = {Rev. Mod. Phys.},
Year = {2012},
Pages = {711--757},
Volume = {84},
DOI = {10.1103/RevModPhys.84.711},
Issue = {2}
}
@Article{Rodriguez1990,
author = {Rodr\'iguez, J. D. and Schell M.},
title = {Global bifurcations into chaos in systems with {SO(2)} symmetry},
journal = {Phys. Lett. A},
year = {1990},
volume = {146},
pages = {25--31},
timestamp = {2012.03.28}
}
@Book{Roman12,
Title = {Fundamentals of Group Theory: An Advanced Approach},
Author = {Roman, S.},
Publisher = {Birkh\"auser},
Year = {2012},
Address = {Boston},
ISBN = {0817683003}
}
@Article{Romanov73,
Title = {Stability of plane-parallel {Couette} flow},
Author = {V. A. Romanov},
Journal = {Funct. Anal. Appl.},
Year = {1973},
Pages = {137--146},
Volume = {7}
}
@Article{RoMeBaSchHe09,
Title = {Spectral analysis of nonlinear flows},
Author = {Rowley, C. W. and Mezi\'c, I. and Bagheri, S. and Schlatter, P. and Henningson, D. S.},
Journal = {J. Fluid M.},
Year = {2009},
Pages = {115},
Volume = {641}
}
@Article{romero-bastida2012,
Title = {Covariant hydrodynamic {Lyapunov} modes and strong stochasticity threshold in {Hamiltonian} lattices},
Author = {Romero-Bastida, M. and Paz\'o, Diego and L\'opez, Juan M.},
Journal = {Phys. Rev. E},
Year = {2012},
Note = {\arXiv{1202.3476}},
Pages = {026210},
Volume = {85},
Abstract = {We scrutinize the reliability of covariant and Gram-Schmidt Lyapunov
vectors for capturing hydrodynamic Lyapunov modes ({HLMs)} in one-dimensional
Hamiltonian lattices. We show that, in contrast with previous claims,
{HLMs} do exist for any energy density, so that strong chaos is not
essential for the appearance of genuine (covariant) {HLMs.} In contrast,
Gram-Schmidt Lyapunov vectors lead to misleading results concerning
the existence of {HLMs} in the case of weak chaos.},
DOI = {10.1103/PhysRevE.85.026210}
}
@Article{RoMo08,
author = {Robinson, D. J. and Morriss, G. P.},
title = {{Lyapunov} mode dynamics in hard-disk systems},
journal = {J. Stat. Phys.},
year = {2008},
volume = {131},
pages = {1--31},
note = {\arXiv{0709.3143}},
doi = {10.1007/s10955-007-9473-z}
}
@Article{RoSa00,
Title = {Nonlinear dimensionality reduction by locally linear embedding},
Author = {Roweis, S. T. and Saul, L. K.},
Journal = {Science},
Year = {2000},
Pages = {2323--2326},
Volume = {290},
Abstract = {Many areas of science depend on exploratory data analysis and visualization.
The need to analyze large amounts of multivariate data raises the
fundamental problem of dimensionality reduction: how to discover
compact representations of high-dimensional data. Here, we introduce
locally linear embedding {(LLE),} an unsupervised learning algorithm
that computes low-dimensional, neighborhood-preserving embeddings
of high-dimensional inputs. Unlike clustering methods for local dimensionality
reduction, {LLE} maps its inputs into a single global coordinate
system of lower dimensionality, and its optimizations do not involve
local minima. By exploiting the local symmetries of linear reconstructions,
{LLE} is able to learn the global structure of nonlinear manifolds,
such as those generated by images of faces or documents of text.},
DOI = {10.1126/science.290.5500.2323}
}
@Phdthesis{RosenqvThesis,
Title = {Periodic orbit theory beyond semiclassics: convergence, diffraction and {$\hbar$} corrections},
Author = {Rosenqvist, P. E.},
School = {Copenhagen Univ.},
Year = {1995},
Address = {Copenhagen},
URL = {http://ChaosBook.org/projects/theses.html}
}
@Article{ross,
Title = {An equation for continuous chaos},
Author = {O. E. R\"ossler},
Journal = {Phys. Lett. A},
Year = {1976},
Pages = {397},
Volume = {57}
}
@Article{roth99,
Title = {Persistent patterns in transient chaotic fluid mixing},
Author = {D. Rothstein and E. Henry and J. P. Gollub},
Journal = {Nature},
Year = {1999},
Pages = {770--772},
Volume = {401},
Abstract = {Experimentally generate a 2-d periodic(or weakly turbulent) velocity
field using glycerol-water on magnets driven by electric current.
Persistent spatial patterns are observed, which decays exponentially.}
}
@Article{Roupas11,
Title = {Phase space geometry and chaotic attractors in dissipative {Nambu} mechanics},
Author = {Z. Roupas},
Journal = {J. Phys. A},
Year = {2012},
Note = {\arXiv{1110.0766}},
Pages = {195101},
Volume = {45}
}
@Article{rowley_reconstruction_2000,
Title = {Reconstruction equations and the {Karhunen-Lo\'eve} expansion for systems with symmetry},
Author = {Rowley, C. W. and Marsden, J. E.},
Journal = {Physica D},
Year = {2000},
Pages = {1--19},
Volume = {142},
Abstract = {We present a method for applying the {Karhunen-Lo\'eve} decomposition
to systems with continuous symmetry. The techniques in this paper
contribute to the general procedure of removing variables associated
with the symmetry of a problem, and related ideas have been used
in previous works both to identify coherent structures in solutions
of {PDEs,} and to derive low-order models via Galerkin projection.
The main result of this paper is to derive a simple and easily implementable
set of reconstruction equations which close the system of {ODEs}
produced by Galerkin projection. The geometric interpretation of
the method closely parallels techniques used in geometric phases
and reconstruction techniques in geometric mechanics. We apply the
method to the {Kuramoto-Sivashinsky} equation and are able to derive
accurate models of considerably lower dimension than are possible
with the traditional {Karhunen-Lo\'eve} expansion.}
}
@Article{rowley_reduction_2003,
Title = {Reduction and reconstruction for self-similar dynamical systems},
Author = {Rowley, C. W. and Kevrekidis, I. G. and Marsden, J. E. and Lust, K.},
Journal = {Nonlinearity},
Year = {2003},
Pages = {1257--1275},
Volume = {16},
Abstract = {We present a general method for analysing and numerically solving
partial differential equations with self-similar solutions. The method
employs ideas from symmetry reduction in geometric mechanics, and
involves separating the dynamics on the shape space (which determines
the overall shape of the solution) from those on the group space
(which determines the size and scale of the solution). The method
is computationally tractable as well, allowing one to compute self-similar
solutions by evolving a dynamical system to a steady state, in a
scaled reference frame where the self-similarity has been factored
out. More generally, bifurcation techniques can be used to find self-similar
solutions, and determine their behaviour as parameters in the equations
are varied. The method is given for an arbitrary Lie group, providing
equations for the dynamics on the reduced space, for reconstructing
the full dynamics and for determining the resulting scaling laws
for self-similar solutions. We illustrate the technique with a numerical
example, computing self-similar solutions of the Burgers equation}
}
@Article{RSBH98,
Title = {On stability of streamwise streaks and transition thresholds in plane channel flows},
Author = {S. C. Reddy and P. J. Schmid and J. S. Baggett and D. S. Henningson},
Journal = {J. Fluid Mech.},
Year = {1998},
Pages = {269--303},
Volume = {365}
}
@Article{RSW90N,
Title = {A breathing chaos},
Author = {P. H. Richter and H.-J. Scholz and A. Wittek},
Journal = {Nonlinearity},
Year = {1990},
Pages = {45},
Volume = {1}
}
@Book{rtb,
Title = {Theory of orbits},
Author = {V. Szebehely},
Publisher = {Academic},
Year = {1967},
Address = {New York}
}
@Article{rue04ne,
Title = {Conversations on nonequilibrium physics with an extraterrestrial},
Author = {D. Ruelle},
Journal = {Physics Today},
Year = {2004},
Pages = {48--53},
Volume = {57},
Abstract = {New ideas of nonequilibrium physics are introduced. Emphasize the
SRB measure approach and energy fluctuation theorem.}
}
@Article{rue87a,
Title = {Resonances for {Axiom A} flows},
Author = {D. Ruelle},
Journal = {J. Diff. Geom.},
Year = {1987},
Pages = {99--116},
Volume = {25},
URL = {http://projecteuclid.org/euclid.jdg/1214440726}
}
@Article{rue87b,
Title = {One-dimensional {Gibbs} states and {Axiom A} diffeomorphisms},
Author = {D. Ruelle},
Journal = {J. Diff. Geom.},
Year = {1987},
Pages = {117--137},
Volume = {25},
URL ={http://projecteuclid.org/euclid.jdg/1214440727}
}
@Article{ruelext,
Title = {Large volume limit of the distribution of characteristic exponents in turbulence},
Author = {D. Ruelle},
Journal = {Commun. Math. Phys.},
Year = {1982},
Pages = {287--302},
Volume = {87}
}
@Article{ruell71,
Title = {On the nature of turbulence},
Author = {D. Ruelle and F. Takens},
Journal = {Commun. Math. Phys.},
Year = {1971},
Pages = {167},
Volume = {20}
}
@Article{ruell73,
author = {D. Ruelle},
title = {Bifurcations in presence of a symmetry group},
journal = {Arch. Rational Mech. Anal.},
year = {1973},
volume = {51},
pages = {136--152}
}
@Book{ruelle,
title = {Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics},
publisher = {Cambridge Univ. Press},
year = {2004},
author = {D. Ruelle},
isbn = {9780521546492},
address = {Cambridge},
edition = {2\textsuperscript{nd}}
}
@Article{Ruelle03,
author = {D. Ruelle},
title = {Differentiation of {SRB} states: {Correction} and complements},
journal = {Comm. Math. Phys.},
year = {2003},
volume = {234},
pages = {185--190},
doi = {10.1007/s00220-002-0779-z}
}
@Article{Ruelle09,
author = {D. Ruelle},
title = {A review of linear response theory for general differentiable dynamical systems},
journal = {Nonlinearity},
year = {2009},
volume = {22},
pages = {855--870},
note = {\arXiv{0901.0484}},
doi = {10.1088/0951-7715/22/4/009},
abstract = {The classical theory of linear response applies to
statistical mechanics close to equilibrium. Away from equilibrium, one
may describe the microscopic time evolution by a general differentiable
dynamical system, identify nonequilibrium steady states (NESS) and
study how these vary under perturbations of the dynamics. Remarkably,
it turns out that for uniformly hyperbolic dynamical systems (those
satisfying the 'chaotic hypothesis'), the linear response away from
equilibrium is very similar to the linear response close to
equilibrium: the Kramers--Kronig dispersion relations hold, and the
fluctuation--dispersion theorem survives in a modified form (which takes
into account the oscillations around the 'attractor' corresponding to
the NESS). If the chaotic hypothesis does not hold, two new phenomena
may arise. The first is a violation of linear response in the sense
that the NESS does not depend differentiably on parameters (but this
nondifferentiability may be hard to see experimentally). The second
phenomenon is a violation of the dispersion relations: the
susceptibility has singularities in the upper half complex plane. These
'acausal' singularities are actually due to 'energy nonconservation':
for a small periodic perturbation of the system, the amplitude of the
linear response is arbitrarily large. This means that the NESS of the
dynamical system under study is not 'inert' but can give energy to the
outside world. An 'active' NESS of this sort is very different from an
equilibrium state, and it would be interesting to see what happens for
active states to the Gallavotti--Cohen fluctuation theorem.}
}
@Article{Ruelle12,
author = {Ruelle, D.},
title = {A mechanical model for {Fourier's} law of heat conduction},
journal = {Comm. Math. Phys.},
year = {2012},
volume = {311},
pages = {755--768},
doi = {10.1007/s00220-011-1304-z}
}
@Article{Ruelle76,
Title = {Zeta-functions for expanding maps and {Anosov} flows},
Author = {D. Ruelle},
Journal = {Invent. Math.},
Year = {1976},
Pages = {231--242},
Volume = {34}
}
@Article{Ruelle76a,
Title = {Generalized zeta-functions for {Axiom A} basic sets},
Author = {Ruelle, D.},
Journal = {Bull. Amer. Math. Soc},
Year = {1976},
Pages = {153--156},
Volume = {82}
}
@Article{ruelle79,
author = {D. Ruelle},
title = {Ergodic theory of differentiable dynamical systems},
journal = {Publ. Math. IHES},
year = {1979},
volume = {50},
pages = {27--58},
doi = {10.1007/BF02684768}
}
@Article{ruelle86,
Title = {Locating resonances for {Axiom A} dynamical systems},
Author = {D. Ruelle},
Journal = {J. Stat. Phys.},
Year = {1986},
Pages = {281--292},
Volume = {44},
Abstract = {For a class of differentiable dynamical systems (called Axiom A systems)
it has been shown by Pollicott and the author that correlation functions
have Fourier transforms which are meromorphic in a strip. The poles
(or resonances) are, however, not easy to locate. This note reviews
the results which are known and discusses a simple model where the
position of resonances can be estimated. The effect of noise is also
discussed.}
}
@Article{ruelle86a,
Title = {Resonances of chaotic dynamical systems},
Author = {D. Ruelle},
Journal = {Phys. Rev. Lett.},
Year = {1986},
Pages = {405--407},
Volume = {56}
}
@Article{ruelle89,
author = {D. Ruelle},
title = {The thermodynamic formalism for expanding maps},
journal = {Commun. Math. Phys.},
year = {1989},
volume = {125},
pages = {239--262},
idoi = {10.1007/BF01217908}
}
@Article{Ruelle96,
author = {Ruelle, D.},
title = {Differentiation of {SRB} states},
journal = {Comm. Math. Phys.},
year = {1997},
volume = {187},
pages = {227--241},
doi = {10.1007/s002200050134}
}
@Article{Ruelle98,
author = {D. Ruelle},
title = {General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium},
journal = {Phys. Lett. A },
year = {1998},
volume = {245},
pages = {220--224},
doi = {10.1016/S0375-9601(98)00419-8},
abstract = {Given a nonequilibrium steady state ϱ we derive formally the
linear response formula for the variation of an expectation value at time
t under a time-dependent infinitesimal perturbation $\delta$$\tau$F of the acting
forces. This leads to a form of the fluctuation-dissipation theorem valid
far from equilibrium: the complex singularities of the susceptibility are
in part those of the spectral density, and in part of a different nature
to be discussed. }
}
@Article{Ruelle98a,
Title = {Nonequilibrium statistical mechanics near equilibrium: computing higher-order terms},
Author = {D. Ruelle},
Journal = {Nonlinearity},
Year = {1998},
Pages = {5},
Volume = {11-18},
Abstract = {Using Sinai - Ruelle - Bowen measures to describe
nonequilibrium steady states, one can in principle compute the
coefficients of expansions around equilibrium. We discuss how this can
be done in practice, and how the results correspond to the zero noise
limit when there is a stochastic perturbation. The approach used is
formal rather than rigorous.},
DOI = {10.1088/0951-7715/11/1/002}
}
@Phdthesis{RughThesis,
Title = {Time Evoluation and Correlations in Chaotic Dynamical Systems},
Author = {Rugh, H. H.},
School = {Univ. of Copenhagen},
Year = {1992},
Address = {Copenhagen}
}
@Article{Ruhe10,
Title = {Rational {Krylov} for real pencils with complex eigenvalues},
Author = {Ruhe, A.},
Journal = {Taiwanese J. Math.},
Year = {2010},
Pages = {795},
Volume = {14}
}
@Article{Rumb00,
author = {M. Rumberger},
title = {{Lyapunov} exponents on the orbit space},
journal = {Discrete Contin. Dynam. Systems},
year = {2000},
volume = {7},
pages = {91--113},
doi = {10.3934/dcds.2001.7.91}
}
@Article{Rumb01,
Title = {On eigenvalues on the orbit space},
Author = {M. Rumberger},
Journal = {J. Pure Appl. Algebra},
Year = {2001},
Pages = {89--99},
Volume = {158},
DOI = {10.1016/S0022-4049(00)00023-2}
}
@Incollection{RumSch01,
Title = {The orbit space method: {Theory} and application},
Author = {M. Rumberger and J. Scheurle},
Booktitle = {Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems},
Publisher = {Springer},
Year = {2001},
Address = {New York},
Editor = {B. Fiedler},
URL = {http://dynamics.mi.fu-berlin.de/danse/}
}
@Article{Saad1986,
author = {Saad, Y. and Schultz, M. H.},
title = {{GM}RES: {A} generalized minimal residual algorithm for solving nonsymmetric linear systems},
journal = {SIAM J. Sci. Stat. Comput.},
year = {1986},
volume = {7},
pages = {856--869}
}
@Article{Sacker65,
Title = {A new approach to the perturbation theory of invariant surfaces},
Author = {Sacker, R. J.},
Journal = {Commun. Pure Appl. Math.},
Year = {1965},
Pages = {717--732},
Volume = {18},
DOI = {10.1002/cpa.3160180409}
}
@InProceedings{SadoEfst05,
Title = {No polar coordinates},
Author = {D. A. Sadovski and K. Efstathiou},
Booktitle = {Geometric Mechanics and Symmetry: the Peyresq Lectures},
Year = {2005},
Address = {Cambridge},
Editor = {J. Montaldi and T. Ratiu},
Pages = {211--302},
Publisher = {Cambridge Univ. Press}
}
@Unpublished{sahai2015,
Title = {A chaotic dynamical system that paints},
Author = {Sahai, T. and Mathew, G. and Surana, A.},
Note = {\arXiv{1504.02010}},
Year = {2015}
}
@Article{Saiki07,
Title = {Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors},
Author = {Saiki, Y.},
Journal = {Nonlin. Proc. Geophys.},
Year = {2007},
Pages = {615--620},
Volume = {14},
DOI = {10.5194/npg-14-615-2007}
}
@Article{SaiYam98,
Title = {Time averaged properties along unstable periodic orbits and chaotic orbits in two map systems},
Author = {Y. Saiki and M. Yamada},
Journal = {Nonlin. Proc. Geophys.},
Year = {2008},
Pages = {675--680},
Volume = {15},
DOI = {10.5194/npg-15-675-2008}
}
@Article{saka90br,
Title = {Breakdown of the phase dynamics},
Author = {H. Sakaguchi},
Journal = {Progr. Theor. Phys.},
Year = {1990},
Pages = {792--800},
Volume = {84}
}
@Article{SakSug03,
Title = {A projection method for generalized eigenvalue problems using numerical integration},
Author = {Sakurai, T. and Sugiura, H.},
Journal = { J. Comput. Appl. Math.},
Year = {2003},
Pages = {119--128},
Volume = {159}
}
@Article{SakWhe03,
Title = {Semiclassical trace formulas for noninteracting identical particles},
Author = {J. Sakhr and N. D. Whelan},
Journal = {Phys. Rev. E},
Year = {2003},
Pages = {066213},
Volume = {67}
}
@Article{SalArt15,
Title = {Oseledets' splitting of standard-like maps},
Author = {Sala, M. and Artuso, R.},
Journal = {Chaos},
Year = {2015},
Volume = {25},
DOI = {10.1063/1.4909524}
}
@Incollection{Sald12,
Author = {L. E. Saldana},
Booktitle = {{ChaosBook.org/projects}},
Publisher = {Georgia Inst. of Technology},
Year = {2012},
Chapter = {A survey of spiral wave studies: {Dynamics} and symmetries},
URL = {http://ChaosBook.org/projects/index.shtml#Saldana}
}
@Book{Salmon98,
Title = {Lectures on Geophysical Fluid Dynamics},
Author = {Salmon, R.},
Publisher = {Oxford Univ. Press},
Year = {1998},
Address = {Oxford}
}
@Article{saltzman1962,
Title = {Finite amplitude free convection as an initial value problem},
Author = {B. Saltzman},
Journal = {J. Atm. Sciences},
Year = {1962},
Pages = {329--341},
Volume = {19},
Abstract = {The {Oberbeck-Boussinesq} equations are reduced to a two-dimensional
form governing convection between two free surfaces maintained at
a constant temperature difference. These equations are then transformed
to a set of ordinary differential equations governing the time variations
of the {double-Fourier} coefficients for the motion and temperature
fields. Non-linear transfer processes are retained and appear as
quadratic interactions between the Fourier coefficients. Energy and
heat transfer relations appropriate to this Fourier resolution, and
a numerical method for solution from arbitrary initial conditions
are given. As examples of the method, numerical solutions for a highly
truncated Fourier representation are presented. These solutions,
which are for a fixed Prandtl number and variable Rayleigh numbers,
show the transient growth of convection from small perturbations,
and in all cases studied approach steady states. The steady states
obtained agree favorably with steady-state solutions obtained by
previous investigators.}
}
@Article{same01,
author = {Samelson, R. M.},
title = {Periodic orbits and disturbance growth for baroclinic waves},
journal = {J. Atmos. Sci.},
year = {2001},
volume = {58},
pages = {436--450}
}
@Article{samelson01,
author = {Samelson, R. M.},
title = {{Lyapunov}, {Floquet}, and singular vectors for baroclinic waves},
journal = {Nonlin. Proc. Geophys.},
year = {2001},
volume = {60},
pages = {439--448}
}
@Article{samelson03,
author = {Samelson, R. M. and Wolfe, C. L.},
title = {A nonlinear baroclinic wave-mean oscillation with multiple normal-mode instabilities},
journal = {J. Atmos. Sci.},
year = {2003},
volume = {60},
pages = {1186--1199}
}
@Article{samelson06,
author = {Wolfe, C. L. and Samelson, R. M.},
title = {Normal-mode analysis of a baroclinic wave-mean oscillation},
journal = {J. Atmos. Sci.},
year = {2006},
volume = {63},
pages = {2795--2812}
}
@Article{samelson08,
Title = {Singular vectors and time-dependent normal modes of a baroclinic wave-mean oscillation},
Author = {Wolfe, C. L. and Samelson, R. M.},
Journal = {J. Atmos. Sci.},
Year = {2008},
Pages = {875--894},
Volume = {65}
}
@Article{Samoilenko2005,
Title = {Conditions for synchronization of one oscillation system},
Author = {A. M. Samoilenko and L. Recke},
Journal = {Ukrain. Math. J.},
Year = {2005},
Pages = {1089--1119},
Volume = {57}
}
@Article{sand-jensen2007,
Title = {How to write consistently boring scientific literature},
Author = {Sand-Jensen, K.},
Journal = {Oikos},
Year = {2007},
Pages = {723--727},
Volume = {116},
DOI = {10.1111/j.0030-1299.2007.15674.x}
}
@Article{SanNet10,
author = {S\'anchez, J. and Net, M.},
title = {On the multiple shooting continuation of periodic orbits by {Newton-Krylov} methods},
journal = {Int. J. Bifur. Chaos},
year = {2010},
volume = {20},
pages = {43--61},
doi = {10.1142/S0218127410025399}
}
@Article{SanNet13,
author = {J. S\'anchez and M. Net},
title = {A parallel algorithm for the computation of invariant tori in large-scale dissipative systems},
journal = {Physica D},
year = {2013},
volume = {252},
pages = {22--33},
doi = {10.1016/j.physd.2013.02.008},
abstract = {A parallelizable algorithm to compute invariant tori
of high-dimensional dissipative systems, obtained upon discretization of
PDEs is presented. The size of the set of equations to be solved is
only a small multiple of the dimension of the original system. The
sequential and parallel implementations are compared with a previous
method (Sanchez et{~}al. (2010)){~}[11], showing that important savings in
wall-clock time can be achieved. In order to test it, a thermal
convection problem of a binary mixture of fluids has been used. The new
method can also be applied to problems with very low rotation numbers,
for which the previous is not suitable. This is tested in two examples of
two-dimensional maps.}
}
@Article{sanz12,
Title = {Model for ultraintense laser-plasma interaction at normal incidence},
Author = {Sanz, J. and Debayle, A. and Mima, K.},
Journal = {Phys. Rev. E},
Year = {2012},
Pages = {046411},
Volume = {85},
Abstract = {An analytical study of the relativistic interaction of a linearly
polarized laser field of ? frequency with highly overdense plasma
is presented. In agreement with one-dimensional particle-in-cell
simulations, the model self-consistently explains the transition
between the sheath inverse bremsstrahlung absorption regime and the
... heating (responsible for the ... electron bunches), as well as
the high harmonic radiations and the mean electron energy.},
DOI = {10.1103/PhysRevE.85.046411}
}
@Article{SaRo02,
Title = {Think Globally, Fit Locally: Unsupervised Learning of Low Dimensional Manifolds},
Author = {Saul, L. K. and Roweis, S. T.},
Journal = {J. Mach. Learn. Res.},
Year = {2003},
Pages = {119--155},
Volume = {4}
}
@Article{sarri10,
Title = {Observation of postsoliton expansion following laser propagation through an underdense plasma},
Author = {Sarri, G. and Singh, D. K. and Davies, J. R.},
Journal = {Phys. Rev. Lett.},
Year = {2010},
Pages = {175007},
Volume = {105}
}
@Article{SarTal98,
author = {Sartori, G. and Talamini, V.},
title = {Orbit spaces of compact coregular simple Lie groups with 2, 3 and 4~{b}asic polynomial invariants: {Effective} tools in the analysis of invariant potentials},
journal = {J. Math. Phys.},
year = {1998},
volume = {39},
pages = {2367--2401},
doi = {10.1063/1.532294}
}
@Article{Sartori91,
author = {Sartori, G.},
title = {Geometric invariant theory: a model-independent approach to spontaneous symmetry and/or supersymmetry breaking},
journal = {La Rivista del Nuovo Cimento},
year = {1991},
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pages = {1--120},
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@Article{Sartorius1998,
Title = {All-optical clock recovery module based on self-pulsating dfb laser},
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@Article{SaScWu97a,
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Year = {1997},
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}
@Incollection{SaScWu99,
Title = {Dynamical behavior of patterns with {Euclidean} symmetry},
Author = {Sandstede, B. and Scheel, A. and Wulff, C.},
Booktitle = {Pattern Formation in Continuous and Coupled Systems},
Publisher = {Springer},
Year = {1999},
Address = {New York},
Pages = {249--264}
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Year = {1999},
Pages = {439--478},
Volume = {9}
}
@Article{saxena2013,
Title = {Interaction of spatially overlapping standing electromagnetic solitons in plasmas},
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Journal = {Phys. Lett. A},
Year = {2013},
Pages = {473--477},
Volume = {377},
Abstract = {Numerical investigations on mutual interactions between two spatially
overlapping standing electromagnetic solitons in a cold unmagnetized
plasma are reported. It is found that an initial state comprising
of two overlapping standing solitons evolves into different end states,
depending on the amplitudes of the two solitons and the phase difference
between them. For small amplitude solitons with zero phase difference,
we observe the formation of an oscillating bound state whose period
depends on their initial separation. These results suggest the existence
of a bound state made of two solitons in the relativistic cold plasma
fluid model.},
DOI = {10.1016/j.physleta.2012.12.010}
}
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@Article{SCD07,
author = {Cvitanovi{\'c}, P. and Davidchack, R. L. and Siminos, E.},
title = {On the state space geometry of the {Kuramoto-Sivashinsky} flow in a periodic domain},
journal = {SIAM J. Appl. Dyn. Syst.},
year = {2010},
volume = {9},
pages = {1--33},
note = {\arXiv{0709.2944}},
doi = {10.1137/070705623}
}
@Unpublished{SCD09b,
Title = {Symmetry reduction: {Geometry} of a {Kuramoto-Sivashinsky} attractor revealed},
Author = {Siminos, E. and Budanur, N. B. and Cvitanovi{\'c}, P. and Davidchack, R. L.},
Note = {In preparation},
Year = {2015}
}
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Year = {2007},
Pages = {034502},
Volume = {99}
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@Unpublished{SchlAvVa11,
Title = {Numerical bifurcation study of superconducting patterns on a square},
Author = {Schl{\"o}mer, N. and Avitabile, D. and Vanroose, W.},
Note = {\arXiv{1102.1212}},
Year = {2011}
}
@Phdthesis{Schmi99,
Title = {Transition to Turbulence in Linearly Stable Shear Flows},
Author = {A. Schmiegel},
School = {Philipps-Universit{\"a}t Marburg},
Year = {1999},
URL = {http://archiv.ub.uni-marburg.de/diss/z2000/0062}
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Volume = {15}
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Title = {Geometrical Methods of Mathematical Physics},
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Publisher = {Cambridge Univ. Press},
Year = {1980},
Address = {Cambridge}
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@Book{Schutz85,
Title = {A First Course in General Relativity},
Author = {Schutz, B. F.},
Publisher = {Cambridge Univ. Press},
Year = {1985},
Address = {Cambridge}
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author = {Schwartzman, S.},
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pages = {786--791}
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Journal = {Science},
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Pages = {1594--1598},
Volume = {305},
Abstract = {Transition to turbulence in pipe flow is one of the most fundamental
and longest-standing problems in fluid dynamics. Stability theory
suggests that the flow remains laminar for all flow rates, but in
practice pipe flow becomes turbulent even at moderate speeds. This
transition drastically affects the transport efficiency of mass,
momentum, and heat. On the basis of the recent discovery of unstable
traveling waves in computational studies of the Navier-Stokes equations
and ideas from dynamical systems theory, a model for the transition
process has been suggested. We report experimental observation of
these traveling waves in pipe flow, confirming the proposed transition
scenario and suggesting that the dynamics associated with these unstable
states may indeed capture the nature of fluid turbulence.}
}
@Misc{scipy,
Title = {{SciPy}: {Open} source scientific tools for {Python}},
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@Article{Selberg56,
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title = {Geometry of the physical phase space in quantum gauge systems},
journal = {Phys. Rept.},
year = {2000},
volume = {326},
pages = {1--163},
note = {\arXiv{hep-th/0002043}},
doi = {10.1016/S0370-1573(99)00085-X}
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@Article{SHcgl90,
Title = {Pulses and fronts in the complex {Ginzburg-Landau} equation near a subcritical bifurcation},
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@Article{shch04fli,
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Year = {2004},
Pages = {1--7},
Volume = {121},
Abstract = {Fast Lyapunov indicator method is used to identify the chaotic and
regular regions in the phase space. Periodic orbits are determined
to characterized the resonance properties of these regions.}
}
@Article{SHcoh92,
author = {van Saarloos, W. and Hohenberg, P. C.},
title = {Fronts, pulses, sources and sinks in generalized complex {Ginzburg-Landau} equations},
journal = {Physica D},
year = {1992},
volume = {56},
pages = {303--367}
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@Article{sheng02,
author = {Sheng, Z.-M. and Mima, K. and Sentoku, Y. and Jovanovi\'c, M. S. and Taguchi, T. and Zhang, J. and Meyer-ter-Vehn, J.},
title = {Stochastic heating and acceleration of electrons in colliding laser fields in plasma},
journal = {Phys. Rev. Lett.},
year = {2002},
volume = {88},
pages = {055004},
doi = {10.1103/PhysRevLett.88.055004},
abstract = {We propose a mechanism that leads to efficient acceleration of electrons
in plasma by two counterpropagating laser pulses. It is triggered
by stochastic motion of electrons when the laser fields exceed some
threshold amplitudes, as found in single-electron dynamics. It is
further confirmed in particle-in-cell simulations. In vacuum or tenuous
plasma, electron acceleration in the case with two colliding laser
pulses can be much more efficient than with one laser pulse only.
In plasma at moderate densities, such as a few percent of the critical
density, the amplitude of the Raman-backscattered wave is high enough
to serve as the second counterpropagating pulse to trigger the electron
stochastic motion. As a result, even with one intense laser pulse
only, electrons can be heated up to a temperature much higher than
the corresponding laser ponderomotive potential.}
}
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Year = {1982},
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Year = {1988},
Pages = {2014--2025},
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author = {Shimada, I.},
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year = {1979},
volume = {62},
pages = {61--69},
doi = {10.1143/PTP.62.61},
abstract = {A representation of the Lorenz attractor by a 1-dimensional Ising
system is constructed. Gibbsian distribution function which describes
the irregular motion of this dissipative dynamical system is investigated
with the help of this representation. Relation between measure-theoretical
entropy and positive Lyapunov characteristic exponent is also investigated.
The following conclusions are obtained: (1) The statistical properties
of the Lorenz system can be reduced to those of 1-dimensional Ising
system with short-range interaction, in other words, the time correlation
function of the Lorenz system shows no singular long-time behaviour.
(2) The positive Lyapunov characteristic exponent of the Lorenz system
is almost equal to its measure-theoretical entropy.}
}
@Article{ShiNag79,
Title = {A numerical approach to ergodic problem of dissipative dynamical systems},
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Year = {1979},
Pages = {1605--1615},
Volume = {61}
}
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Title = {Solitary waves in layer of viscous fluid},
Author = {V. Y. Shkadov},
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Year = {1977},
Pages = {63--66},
Volume = {1},
Address = {New York},
Publisher = {Consultants Bureau}
}
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author = {G. M. Shroff and H. B. Keller},
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volume = {30},
pages = {1099--1120}
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Author = {Shapere, A. and Wilczek, F.},
Journal = {J. Fluid Mech.},
Year = {2006},
Pages = {557},
Volume = {198}
}
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Title = {Self-propulsion at low {Reynolds} number},
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Journal = {Phys. Rev. Lett.},
Year = {1987},
Pages = {2051--2054},
Volume = {58}
}
@Article{ShWi89,
Title = {Efficiencies of self-propulsion at low {Reynolds} number},
Author = {Shapere, A. and Wilczek, F.},
Journal = {J. Fluid Mech.},
Year = {1989},
Pages = {587--599},
Volume = {198}
}
@Article{ShWi89a,
Title = {Gauge kinematics of deformable bodies},
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Volume = {57}
}
@Article{SiCvi10,
Title = {Continuous symmetry reduction and return maps for high-dimensional flows},
Author = {Siminos, E. and Cvitanovi{\'c}, P.},
Journal = {Physica D},
Year = {2011},
Pages = {187--198},
Volume = {240},
DOI = {10.1016/j.physd.2010.07.010}
}
@Article{Sieber2002,
Title = {Numerical bifurcation analysis for multisection semiconductor lasers},
Author = {J. Sieber},
Journal = {SIAM J. Appl. Math.},
Year = {2002},
Pages = {248--270},
Volume = {1}
}
@Article{siegel42,
Title = {Iteration of analytic functions},
Author = {C. L. Siegel},
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Year = {1942},
Pages = {607--612},
Volume = {43}
}
@Article{siminos11,
author = {Siminos, Evangelos and B\'enisti, Didier and Gremillet, Laurent},
title = {Stability of nonlinear {Vlasov-Poisson} equilibria through spectral deformation and {Fourier-{Hermit}e} expansion},
journal = {Phys. Rev. E},
year = {2011},
volume = {83},
pages = {056402},
doi = {10.1103/PhysRevE.83.056402},
abstract = {We study the stability of spatially periodic, nonlinear Vlasov-Poisson
equilibria as an eigenproblem in a Fourier-Hermite basis (in the
space and velocity variables, respectively) of finite dimension,
N. When the advection term in the Vlasov equation is dominant, the
convergence with N of the eigenvalues is rather slow, limiting the
applicability of the method. We use the method of spectral deformation
introduced by Crawford and Hislop Ann. Phys. ({NY)} 189 265 (1989)
to selectively damp the continuum of neutral modes associated with
the advection term, thus accelerating convergence. We validate and
benchmark the performance of our method by reproducing the kinetic
dispersion relation results for linear (spatially homogeneous) equilibria.
Finally, we study the stability of a periodic Bernstein-Greene-Kruskal
mode with multiple phase-space vortices, compare our results with
numerical simulations of the Vlasov-Poisson system, and show that
the initial unstable equilibrium may evolve to different asymptotic
states depending on the way it was perturbed.}
}
@Article{siminos2012,
Title = {Effect of electron heating on self-induced transparency in relativistic-intensity laser-plasma interactions},
Author = {Siminos, E. and Grech, M. and Skupin, S. and Schlegel, T. and Tikhonchuk, V. T.},
Journal = {Phys. Rev. E},
Year = {2012},
Pages = {056404},
Volume = {86},
Abstract = {The effective increase of the critical density associated with the
interaction of relativistically intense laser pulses with overcritical
plasmas, known as self-induced transparency, is revisited for the
case of circular polarization. A comparison of particle-in-cell simulations
to the predictions of a relativistic cold-fluid model for the transparency
threshold demonstrates that kinetic effects, such as electron heating,
can lead to a substantial increase of the effective critical density
compared to cold-fluid theory. These results are interpreted by a
study of separatrices in the single-electron phase space corresponding
to dynamics in the stationary fields predicted by the cold-fluid
model. It is shown that perturbations due to electron heating exceeding
a certain finite threshold can force electrons to escape into the
vacuum, leading to laser pulse propagation. The modification of the
transparency threshold is linked to the temporal pulse profile, through
its effect on electron heating.},
DOI = {10.1103/PhysRevE.86.056404}
}
@Unpublished{siminos2014c,
Title = {Signatures of relativistic chaos in optical lattices, work in progress},
Author = {Siminos, E.},
Year = {2014}
}
@Unpublished{siminos2014p,
Title = {Onset of self-induced transparency in the hole boring regime in relativistic laser-plasma interaction, in preparation},
Author = {Siminos, E. and Grech, M. and Skupin, S.},
Year = {2014}
}
@Unpublished{siminos2014s,
Title = {Describing relativistic soliton interactions through envelope equations, in preparation},
Author = {Siminos, E. and S\'anchez-Arriaga, G. and Saxena, V. and Kourakis, I.},
Year = {2014}
}
@Phdthesis{SiminosThesis,
Title = {Recurrent Spatio-temporal Structures in Presence of Continuous Symmetries},
Author = {Siminos, E.},
School = {School of Physics, Georgia Inst. of Technology},
Year = {2009},
Address = {Atlanta},
URL = {http://ChaosBook.org/projects/theses.html}
}
@Phdthesis{Simonis2006,
Title = {Inexact {Newton} Methods Applied to Under-Determined Systems},
Author = { Simonisk, J. P.},
School = {Polytechnic Inst.},
Year = {2006},
Address = {Worcester, MA}
}
@Article{sinai,
Title = {Gibbs measures in ergodic theory},
Author = {Ya. G. Sinai},
Journal = {Russian Math. Surveys},
Year = {1972},
Pages = {21},
Volume = {27},
DOI = {10.1070/RM1972v027n04ABEH001383}
}
@Article{Sinai04,
Title = {What is ...a billiard},
Author = {Sinai, Ya. G.},
Journal = {Notices Amer. Math. Soc.},
Year = {2004},
Pages = {412--413},
Volume = {51},
URL = {http://www.ams.org/notices/200404/what-is.pdf}
}
@Article{Sinai70,
author = {Sinai, Ya. G.},
title = {Dynamical systems with elastic reflections: {Ergodic} properties of dispersing billiards},
journal = {Uspekhi Mat. Nauk},
year = {1970},
volume = {25},
pages = {141--192},
doi = {10.1070/RM1970v025n02ABEH003794},
abstract = {In this paper we consider dynamical systems resulting from
the motion of a material point in domains with strictly convex
boundary, that is, such that the operator of the second quadratic form
is negative-definite at each point of the boundary, where the boundary
is taken to be equipped with the field of inward normals. We prove that
such systems are ergodic and are K -systems. The basic method of
investigation is the construction of transversal foliations for such
systems and the study of their properties.}
}
@Book{sinai76,
Title = {Introduction to Ergodic Theory},
Author = {Sinai, Ya. G.},
Publisher = {Princeton Univ. Press},
Year = {1976},
Address = {Princeton, NJ}
}
@Article{Singer78,
author = {Singer, I. M.},
title = {Some Remarks on the {Gribov} Ambiguity},
journal = {Commun. Math. Phys.},
year = {1978},
volume = {60},
pages = {7--12},
doi = {10.1007/BF01609471}
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Title = {Coherent structures and chaos: {A} model problem},
Author = {L. Sirovich and J. D. Rodriguez},
Journal = {Phys. Lett. A},
Year = {1987},
Pages = {211--214},
Volume = {120}
}
@Article{sirovcgl90,
Title = {Low-dimensional dynamics for the complex {Ginzburg-Landau} equation},
Author = {J. D. Rodriguez and L. Sirovich},
Journal = {Physica D},
Year = {1990},
Pages = {77--86},
Volume = {43}
}
@Article{Sirovich94,
author = {L. Sirovich and X. Zhou},
title = {Reply to ``{O}bservations regarding '{C}oherence and chaos in a model of turbulent boundary layer' by {X}. {Z}hou and {L}. {S}irovich''},
journal = {Phys. Fluids},
year = {1994},
volume = {6},
pages = {1579--1582}
}
@Article{siv,
author = {G. I. Sivashinsky},
title = {Nonlinear analysis of hydrodynamical instability in laminar flames - {I}. {D}erivation of basic equations},
journal = {Acta Astronaut.},
year = {1977},
volume = {4},
pages = {1177--1206},
doi = {10.1016/0094-5765(77)90096-0}
}
@Article{sivmich80,
Title = {On irregular wavy flow of a liquid down a vertical plane},
Author = {G. I. Sivashinsky and D. M. Michelson},
Journal = {Progr. Theor. Phys. Suppl.},
Year = {1980},
Pages = {2112},
Volume = {66}
}
@Article{skeleton,
Title = {Periodic orbits as the skeleton of classical and quantum chaos},
Author = {P. Cvitanovi\'{c}},
Journal = {Physica D},
Year = {1991},
Pages = {138},
Volume = {51}
}
@Book{SKMacHH09,
Title = {Exact Solutions of {Einstein's} Field Equations},
Author = {Stephani, H and Kramer, D. and MacCallum, M. and Hoenselaers, C. and Herlt, E.},
Publisher = {Cambridge Univ. Press},
Year = {2009},
Address = {Cambridge}
}
@Article{Skokos02,
Title = {Alignment indices: a new, simple method for determining the ordered or chaotic nature of orbits},
Author = {Skokos, C.},
Journal = {J. Phys. A},
Year = {2001},
Pages = {10029},
Volume = {34}
}
@Incollection{Skokos08,
author = {Skokos, C.},
title = {The {Lyapunov} characteristic exponents and their computation},
booktitle = {Dynamics of Small Solar System Bodies and Exoplanets},
publisher = {Springer},
year = {2010},
editor = {Souchay, J. J. and Dvorak, R.},
volume = {790},
series = {Lect. Notes Phys.},
pages = {63--135},
address = {New York},
note = {\arXiv{0811.0882}},
doi = {10.1007/978-3-642-04458-8_2}
}
@Article{SlBaJu13,
author = {J. Slipantschuk and O. F. Bandtlow and W. Just},
title = {On the relation between {Lyapunov} exponents and exponential decay of correlations},
journal = {J. Phys. A},
year = {2013},
volume = {46},
pages = {075101},
doi = {10.1088/1751-8113/46/7/075101},
abstract = {Chaotic dynamics with sensitive dependence on initial conditions may
result in exponential decay of correlation functions. We show that
for one-dimensional interval maps the corresponding quantities, that
is, Lyapunov exponents and exponential decay rates, are related.
More specifically, for piecewise linear expanding Markov maps observed
via piecewise analytic functions, we show that the decay rate is
bounded above by twice the Lyapunov exponent, that is, we establish
lower bounds for the subleading eigenvalue of the corresponding Perron-Frobenius
operator. In addition, we comment on similar relations for general
piecewise smooth expanding maps.}
}
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Title = {Differentiable dynamical systems},
Author = {S. Smale},
Journal = {Bull. Amer. Math. Soc.},
Year = {1967},
Pages = {747--817},
Volume = {73}
}
@Book{Smale00,
Title = {The collected papers of {Stephen Smale}},
Author = {S. Smale and F. Cucker and R. Wong},
Publisher = {World Scientific},
Year = {2000},
Address = {Singapore}
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Author = {J. Scheuer and B. A. Malomed},
Journal = {Physica D},
Year = {2002},
Pages = {102--115},
Volume = {161},
Abstrac = {The behavior of the solutions of the CGLe without diffusion is studied with a periodic boundary condition.}
}
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@Phdthesis{SmithThesis12,
Title = {Point Vortices: Finding Periodic Orbits and their Topological Classification},
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School = {Mathematics Dept., Tufts University},
Year = {2012},
Address = {Medford, Mass},
URL = {http://gradworks.umi.com/3541851.pdf}
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journal = {Physica D},
year = {2005},
volume = {211},
pages = {347--376},
doi = {10.1016/j.physd.2005.09.002}
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@Book{Sniatycki13,
Title = {Differential Geometry of Singular Spaces and Reduction of Symmetry},
Author = {{\'S}niatycki, J.},
Publisher = {Cambridge Univ. Press},
Year = {2013},
Address = {Cambridge}
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author = {J. M. Soto-Crespo and N. Akhmediev and N. Devine and C. Mej\'{i}a-Cort\'{e}s},
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journal = {Opt. Express},
year = {2008},
volume = {16},
pages = {15388--15401},
doi = {10.1364/OE.16.015388},
abstract = {Dissipative media admit the existence of two types of stationary self-organized
beams: continuously self-focused and continuously self-defocused.
Each beam is stable inside of a certain region of its existence.Beyond
these two regions, beams loose their stability, and new dynamical
behaviors appear. We present several types of instabilities related
to each beam configuration and give examples of beam dynamics in
the areas adjacent to the two regions. We observed that, in one case
beams loose the radial symmetry while in the other one the radial
symmetry is conserved during complicated beam transformations.}
}
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Volume = {8},
Abstract = {We discuss experimental and numerical studies of the effects of Lagrangian
chaos (chaotic advection) on the stretching of a drop of an immiscible
impurity in a flow. We argue that the standard capillary number used
to describe this process is inadequate since it does not account
for advection of a drop between regions of the flow with varying
velocity gradient. Consequently, we propose a Lagrangian-generalized
capillary number CL number based on finite-time Lyapunov exponents.
We present preliminary tests of this formalism for the stretching
of a single drop of oil in an oscillating vortex flow, which has
been shown previously to exhibit Lagrangian chaos. Probability distribution
functions (PDFs) of the stretching of this drop have features that
are similar to PDFs of CL. We also discuss on-going experiments that
we have begun on drop stretching in a blinking vortex flow. PACS:
47.52.+j; 47.55.Dz; 94.10.Lf },
DOI = {10.1016/S1007-5704(03)00047-9}
}
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Title = {Chaotic advection in a two-dimensional flow: {L{\'e}vy} flights and anomalous diffusion},
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Title = {Structure of Dynamical Systems},
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Publisher = {Springer},
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Address = {New York}
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Year = {1996},
Address = {New York}
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Title = {Spatiotemporal chaos in the one-dimensional complex {Ginzburg-Landau} equation},
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@Article{SSL10,
Title = {Relativistic solitary waves with phase modulation embedded in long laser pulses in plasmas},
Author = {S\'anchez-Arriaga, G. and Siminos, E. and Lefebvre, E.},
Journal = {Phys. Plasmas},
Year = {2011},
Pages = {082304--082304-10},
Volume = {18},
Abstract = {We investigate the existence of nonlinear phase-modulated relativistic
solitary waves embedded in an infinitely long circularly polarized
electromagnetic wave propagating through a plasma. These states are
exact nonlinear solutions of the 1-dimensional Maxwell-fluid model
for a cold plasma composed of electrons and ions. The solitary wave,
which consists of an electromagnetic wave trapped in a self-generated
Langmuir wave, presents a phase modulation when the group velocity
V and the phase velocity Vph of the long circularly polarized electromagnetic
wave do not match the condition {VVph}?=?c2. The main properties
of the waves as a function of their group velocities, wavevectors,
and frequencies are studied, as well as bifurcations of the dynamical
system that describes the waves when the parameter controlling the
phase modulation changes from zero to a finite value. Such a transition
is illustrated in the limit of small amplitude waves where an analytical
solution for a grey solitary wave exists. The solutions are interpreted
as the stationary state after the collision of a long laser pulse
with an isolated solitary wave.},
DOI = {10.1063/1.3624498}
}
@Article{SSL10-1,
Title = {Relativistic solitary waves modulating long laser pulses in plasmas},
Author = {S\'anchez-Arriaga, G. and Siminos, E. and Lefebvre, E.},
Journal = {Plasma Phys. Control. Fusion},
Year = {2011},
Pages = {045011},
Volume = {53},
Abstract = {This paper discusses the existence of solitary electromagnetic waves
trapped in a self-generated Langmuir wave and embedded in an infinitely
long circularly polarized electromagnetic wave propagating through
a plasma. From a mathematical point of view they are exact solutions
of the one-dimensional relativistic cold fluid plasma model with
nonvanishing boundary conditions. Under the assumption of travelling
wave solutions with velocity V and vector potential frequency ?,
the fluid model is reduced to a Hamiltonian system. The solitary
waves are homoclinic (grey solitons) or heteroclinic (dark solitons)
orbits to fixed points. Using a dynamical systems description of
the Hamiltonian system and a spectral method, we identify a large
variety of solitary waves, including asymmetric ones, discuss their
disappearance for certain parameter values and classify them according
to (i) grey or dark character, (ii) the number of humps of the vector
potential envelope and (iii) their symmetries. The solutions come
in continuous families in the parametric V?? plane and extend up
to velocities that approach the speed of light. The stability of
certain types of grey solitary waves is investigated with the aid
of particle-in-cell simulations that demonstrate their propagation
for a few tens of the inverse of the plasma frequency.},
DOI = {10.1088/0741-3335/53/4/045011},
Language = {en}
}
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Year = {1993},
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Volume = {LI}
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@Unpublished{stadium95,
Title = {Symbolic dynamics and {Markov} partitions for the stadium billiard},
Author = {Cvitanovi\'c, P. and Hansen, K. T.},
Note = {\arXiv{chao-dyn/9502005}; {\em J. Stat. Phys.}, accepted 1996, revised version still not resubmitted}
}
@Article{StaNich12,
Title = {A suspension of the {H\'enon} map by periodic orbits},
Author = {J. Starrett and C. Nicholas},
Journal = {Chaos Solit. Fract.},
Year = {2012},
Pages = {1486--1493},
Volume = {45},
Abstract = {We create polynomial differential equations for a suspension of the
{H\'enon} map embedded in R 3 . By globalizing the local tangent
vectors to suspended periodic orbits of the {H\'enon} map, we are
able to find approximate autonomous differential equations for that
geometric suspension. Using as few as two suspended periodic orbits,
we can generate a robust three dimensional attractor whose Poincar\'e
map has very nearly the dynamics of the original {H\'enon} map on
the attractor.},
DOI = {10.1016/j.chaos.2012.07.013}
}
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title = {Asymptotic estimates and stability analysis of {Kuramoto-Sivashinsky} type models},
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year = {2011},
volume = {11},
pages = {605--635},
doi = {10.1007/s00028-011-0103-5}
}
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Title = {Transport and Diffusion Across Cell Membranes},
Author = {Stein, W.},
Publisher = {Elsevier},
Year = {2012},
Address = {New York}
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Journal = {Phys. Plasmas},
Year = {1994},
Pages = {2804},
Volume = {1},
Abstract = {A nonlinear equation of the plasma surface is derived for the small
deformation. Possible solitary solutions are discussed.}
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Volume = {5}
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Author = {S. Sternberg},
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Volume = {81}
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@Unpublished{stevenj,
Title = {Topics in applied mathematics},
Author = {Johnson, S. G.},
Note = {MIT course 18.325},
Year = {2005}
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author = {G. W. Stewart},
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year = {2002},
volume = {23},
pages = {601--614},
doi = {10.1137/S0895479800371529}
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author = {G. W. Stewart},
title = {Addendum to ``{A Krylov--Schur} algorithm for large eigenproblems"},
journal = {SIAM J. Matrix Anal. Appl.},
year = {2002},
volume = {24},
pages = {599--601},
doi = {10.1137/S0895479802403150}
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Volume = {14}
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@Book{StGo09,
Title = {Mathematics for Physics: A Guided Tour for Graduate Students},
Author = {Stone, M. and Goldbart, P.},
Publisher = {Cambridge Univ. Press},
Year = {2009},
Address = {Cambridge}
}
@Article{StMaKh08,
Title = {Nonlinear dynamics of hydrostatic internal gravity waves},
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Journal = {Theor. Comput. Fluid Dyn.},
Year = {2008},
Pages = {407--432},
Volume = {22},
DOI = {10.1007/s00162-008-0080-7}
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Pages = {46--52},
Volume = {241},
Abstract = {Singular perturbation method. Through scaling transformation, a small
parameter is introduced such that when it is equal to zero, the map
becomes simple and controlable though non-deterministic. The symbolic
dynamics is easy to build and the AI limit orbit can be extended
in a unique way to the small parameter case when the AI limit is
nondegenerate. {H\'enon} map is used to demonstrate the idea.}
}
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Volume = {7}
}
@Book{strogb,
Title = {Nonlinear Dynamics and Chaos},
Author = {S. H. Strogatz},
Publisher = {Perseus Publishing},
Year = {2000},
Address = {Cambridge, Mass},
Series = {Studies in Nonlinearity}
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Year = {1967},
Pages = {420--442},
Volume = {95},
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Year = {1968},
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journal = {Chin. Ann. Math.},
year = {2002},
volume = {23},
pages = {267--276},
doi = {10.1142/S0252959902000250},
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year = {1989},
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@Article{SumRules,
Title = {Beyond the periodic orbit theory},
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Journal = {Nonlinearity},
Year = {1998},
Note = {\arXiv{chao-dyn/9712002}},
Pages = {1209--1232},
Volume = {11}
}
@InProceedings{sunil_ahuja_template-based_2006,
Title = {Template-based stabilization of relative equilibria},
Author = {S. Ahuja and {I.G.} Kevrekidis and {C.W.} Rowley},
Booktitle = {2006 American Control Conference},
Year = {2006},
Address = {Piscataway, NJ},
Pages = {6},
Publisher = {{IEEE}},
Abstract = {We present an approach to the design of feedback control laws that
stabilize the relative equilibria of general nonlinear systems with
continuous symmetry. Using a template-based method, we factor out
the dynamics associated with the symmetry variables and obtain evolution
equations in a reduced frame that evolves in the symmetry direction.
The relative equilibria of the original system are fixed points of
these reduced equations. Our controller design methodology is based
on the linearization of the reduced equations about such fixed points.
Assuming equivariant actuation, we derive feedback laws for the reduced
system that are optimal in the sense that they minimize a quadratic
cost function. We illustrate the method by stabilizing unstable traveling
waves of a dissipative {PDE} possessing translational invariance}
}
@Article{SunKanZha11,
author = {Sun, L.-S. and Kang, X.-Y. and Zhang, Q. and Lin, L.-X.},
title = {A method of recovering the initial vectors of globally coupled map lattices based on symbolic dynamics},
journal = {Chin. Phys. B},
year = {2011},
volume = {20},
pages = {120507},
doi = {10.1088/1674-1056/20/12/120507}
}
@Article{SWG02,
Title = {Toward a structural understanding of turbulent drag reduction: nonlinear coherent states in viscoelastic shear flows},
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Year = {2002},
Pages = {20831},
Volume = {89}
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@Article{SYE05,
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Journal = {Phys. Rev. Lett.},
Year = {2006},
Pages = {174101},
Volume = {96},
Abstract = {identify and follow the edge of chaos and provide evidence
that it is a surface. For low Reynolds numbers we find that the edge of chaos
coincides with the stable manifold of a periodic orbit, whereas at higher
Reynolds numbers it is the stable set of a higher-dimensional chaotic
object.}
}
@Incollection{symb_dyn,
Author = {P. Cvitanovi\'{c}},
Booktitle = {{Chaos: Classical and Quantum}},
Publisher = {Niels Bohr Inst.},
Year = {2015},
Address = {Copenhagen},
Chapter = {{Charting} the state space},
URL = {http://ChaosBook.org/paper.shtml#knead}
}
@Article{symmcgl,
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Journal = {Physica D},
Year = {1996},
Pages = {398--428},
Volume = {97}
}
@Unpublished{szendro-2007,
Title = {Spatiotemporal structure of {Lyapunov} vectors in chaotic coupled-map lattices},
Author = {I. G. Szendro and D. Pazo and M. A. Rodriguez and J. M. Lopez},
Note = {\arXiv{0706.1706}},
Year = {2007},
Abstract = {useful list of references on Lyapunov vectors}
}
@Article{SZN98,
Title = {Direct numerical simulation of transition in pipe flow under the influence of wall disturbances},
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Journal = {Int. J. Heat Fluid Flow},
Year = {1998},
Pages = {320--325},
Volume = {19},
DOI = {10.1016/S0142-727X(98)10008-5}
}
@Book{Tabor89,
Title = {Chaos and Integrability in Nonlinear Dynamics: An Introduction},
Author = {M. Tabor},
Publisher = {Wiley},
Year = {1989},
Address = {New York}
}
@Misc{TaCh11,
Title = {Can the inertial manifold be captured by unstable periodic orbits?},
Author = {Takeuchi, K. A. and Chat\'{e}, H.},
Year = {2011}
}
@Article{TaCh12,
author = {Takeuchi, K. A. and Chat\'e, H. },
title = {Collective {Lyapunov} modes},
journal = {J. Phys. A},
year = {2013},
volume = {46},
pages = {254007},
addendum = {\arXiv{1207.5571}},
doi = {10.1088/1751-8113/46/25/254007}
}
@Article{TaGiCh09,
author = {Takeuchi, K. A. and Ginelli, F. and Chat\'{e}, H.},
title = {{Lyapunov} analysis captures the collective dynamics of large chaotic systems},
journal = {Phys. Rev. Lett.},
year = {2009},
volume = {103},
pages = {154103},
note = {\arXiv{0907.4298}},
abstract = {We show, using generic globally-coupled systems, that the collective
dynamics of large chaotic systems is encoded in their Lyapunov spectra:
most modes are typically localized on a few degrees of freedom, but
some are delocalized, acting collectively on the trajectory. For
globally-coupled maps, we show moreover a quantitative correspondence
between the collective modes and some of the so-called Perron-Frobenius
dynamics. Our results imply that the conventional definition of extensivity
must be changed as soon as collective dynamics sets in.}
}
@Article{TaGiCh11,
author = {Takeuchi, K. A. and Yang, {H.-l.} and Ginelli, F. and Radons, G. and Chat\'{e}, H.},
title = {Hyperbolic decoupling of tangent space and effective dimension of dissipative systems},
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year = {2011},
volume = {84},
pages = {046214},
note = {\arXiv{1107.2567}},
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@Article{tajima02,
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Pages = {017205},
Volume = {66}
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@Article{takac98,
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Year = {1998},
Pages = {241--257},
Volume = {89}
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Year = {1994},
Note = {\arXiv{hep-th/9301111}},
Pages = {295--316},
Volume = {160},
DOI = {10.1007/BF02103278}
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@Article{Talamini06,
Title = {P-matrices in orbit spaces and invariant theory},
Author = {V. Talamini},
Journal = {J. Phys.: Conference Series},
Year = {2006},
Pages = {30},
Volume = {30},
DOI = {10.1088/1742-6596/30/1/005}
}
@Article{Talamini10,
Title = {Flat bases of invariant polynomials and {P-matrices} of {E7} and {E8}},
Author = {Talamini, V.},
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Year = {2010},
Pages = {023520},
Volume = {51},
DOI = {10.1063/1.3272569}
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@Article{Talamini11,
Title = {Invariant theory to study symmetry breakings in the adjoint representation of {E8}},
Author = {Talamini, V.},
Journal = {J. Phys.: Conference Series},
Year = {2011},
Pages = {012057},
Volume = {284},
DOI = {10.1088/1742-6596/284/1/012057}
}
@Article{TaSa07,
Title = {Role of unstable periodic orbits in phase transitions of coupled map lattices},
Author = {Takeuchi, K. and Sano, M.},
Journal = {Phys. Rev. E},
Year = {2007},
Pages = {036201},
Volume = {75},
DOI = {10.1103/PhysRevE.75.036201}
}
@Article{TaWi92,
Title = {Quantization of chaotic systems},
Author = {G. Tanner and D. Wintgen},
Journal = {Chaos},
Year = {1992},
Pages = {53},
Volume = {2}
}
@Article{tbohr,
Title = {The diversity of steady state solutions of the complex {Ginzburg-Landau} equation},
Author = {M. Bazhenov and T. Bohr and K. Gorshkov and M. Rabinovich},
Journal = {Phys. Lett. A},
Year = {1996},
Pages = {104},
Volume = {217}
}
@Article{TeHaHe10,
Title = {Spatial optimal growth in three-dimensional boundary layers},
Author = {Tempelmann, D. and Hanifi, A. and Henningson, D. S.},
Journal = {J.\ Fluid Mech.},
Year = {2010},
Pages = {5--37},
Volume = {646},
DOI = {10.1017/S0022112009993260}
}
@Article{Tel2000,
author = {T. {T\'el} and G. K\'{a}rolyi and {\'{A}}. P\'{e}ntek and I. Scheuring and Z. Toroczkai and C. Grebogi and J. Kadtke},
title = {Chaotic advection, diffusion, and reactions in open flows},
journal = {Chaos},
year = {2000},
volume = {10},
pages = {89--98},
abstract = {scatters tracers off a Von Karman vortex street, where they get stuck
behind the cylinder for a while in the mixing region}
}
@Article{temam90,
author = {Temam, R.},
title = {Inertial manifolds},
journal = {Math. Intell.},
year = {1990},
volume = {12},
pages = {68--74},
doi = {10.1007/BF03024036}
}
@Article{temam91edd,
Title = {Approximation of attractors, large eddy simulations and multiscale methods},
Author = {R. Temam},
Journal = {Proc. R. Soc. Lond. A},
Year = {1991},
Pages = {23--39},
Volume = {434},
Abstract = {Estimates on the dimension of the attractors and the decay rates of
Fourier mode of NSe are given. By approximation of the inertial manifold,
the interaction between the large and small eddies are derived. Such
laws open the way to numerically efficient algorithms via multiscale
methods.}
}
@Article{tenenbaum2000,
author = {Tenenbaum, J. B. and Silva, V. de and Langford, J. C.},
title = {A global geometric framework for nonlinear dimensionality reduction},
journal = {Science},
year = {2000},
volume = {290},
pages = {2319--2323},
doi = {10.1126/science.290.5500.2319},
abstract = {Scientists working with large volumes of high-dimensional data, such
as global climate patterns, stellar spectra, or human gene distributions,
regularly confront the problem of dimensionality reduction: finding
meaningful low-dimensional structures hidden in their high-dimensional
observations. The human brain confronts the same problem in everyday
perception, extracting from its high-dimensional sensory inputs -30,000
auditory nerve fibers or 106 optic nerve fibers- manageably small
number of perceptually relevant features. Here we describe an approach
to solving dimensionality reduction problems that uses easily measured
local metric information to learn the underlying global geometry
of a data set. Unlike classical techniques such as principal component
analysis (PCA) and multidimensional scaling (MDS), our approach is
capable of discovering the nonlinear degrees of freedom that underlie
complex natural observations, such as human handwriting or images
of a face under different viewing conditions. In contrast to previous
algorithms for nonlinear dimensionality reduction, ours efficiently
computes a globally optimal solution, and, for an important class
of data manifolds, is guaranteed to converge asymptotically to the
true structure.}
}
@Article{teubner09,
author = {Teubner, U. and Gibbon, P.},
title = {High-order harmonics from laser-irradiated plasma surfaces},
journal = {Rev. Mod. Phys.},
year = {2009},
volume = {81},
pages = {445--479},
doi = {10.1103/RevModPhys.81.445},
abstract = {The investigation of high-order harmonic generation ({HHG)} of femtosecond
laser pulses by means of laser-produced plasmas is surveyed. This
kind of harmonic generation is an alternative to the {HHG} in gases
and shows significantly higher conversion efficiency. Furthermore,
with plasma targets there is no limitation on applicable laser intensity
and thus the generated harmonics can be much more intense. In principle,
harmonic light may also be generated at relativistic laser intensity,
in which case their harmonic intensities may even exceed that of
the focused laser pulse by many orders of magnitude. This phenomenon
presents new opportunities for applications such as nonlinear optics
in the extreme ultraviolet region, photoelectron spectroscopy, and
opacity measurements of high-density matter with high temporal and
spatial resolution. On the other hand, {HHG} is strongly influenced
by the laser-plasma interaction itself. In particular, recent results
show a strong correlation with high-energy electrons generated during
the interaction process. The harmonics are a promising tool for obtaining
information not only on plasma parameters such as the local electron
density, but also on the presence of large electric and magnetic
fields, plasma waves, and the (electron) transport inside the target.
This paper reviews the theoretical and experimental progress on {HHG}
via laser-plasma interactions and discusses the prospects for applying
{HHG} as a short-wavelength, coherent optical tool.}
}
@Article{TFGphase97,
Title = {Studies of phase turbulence in the one-dimensional complex {Ginzburg-Landau} equation},
Author = {A. Torcini and H. Frauenkron and P. Grassberger},
Journal = {Phys. Rev. E},
Year = {1997},
Pages = {5073},
Volume = {55},
Abstract = {The phase gradient is used as an order parameter. Different states
are identified. In the PT region, a modified KSe rules the phase
dynamics of the CGLe.}
}
@Article{ThiBoo99,
Title = {Geometrical constraints on finite-time {Lyapunov} exponents in two and three dimensions},
Author = {Thiffeault, {J.-L.} and Boozer, A. H.},
Journal = {Chaos},
Year = {2001},
Note = {\arXiv{physics/0009017}},
Pages = {16--28},
Volume = {11},
Abstract = {Constraints are found on the spatial variation of finite-time Lyapunov
exponents of two and three-dimensional systems of ordinary differential
equations. In a chaotic system, finite-time Lyapunov exponents describe
the average rate of separation, along characteristic directions,
of neighboring trajectories. The solution of the equations is a coordinate
transformation that takes initial conditions (the Lagrangian coordinates)
to the state of the system at a later time (the Eulerian coordinates).
This coordinate transformation naturally defines a metric tensor,
from which the Lyapunov exponents and characteristic directions are
obtained. By requiring that the Riemann curvature tensor vanish for
the metric tensor (a basic result of differential geometry in a flat
space), differential constraints relating the finite-time Lyapunov
exponents to the characteristic directions are derived. These constraints
are realized with exponential accuracy in time. A consequence of
the relations is that the finite-time Lyapunov exponents are locally
small in regions where the curvature of the stable manifold is large,
which has implications for the efficiency of chaotic mixing in the
advection-diffusion equation. The constraints also modify previous
estimates of the asymptotic growth rates of quantities in the dynamo
problem, such as the magnitude of the induced current.}
}
@Article{Thiffeault11,
Title = {Using multiscale norms to quantify mixing and transport},
Author = {Thiffeault, J.-L. },
Journal = {Nonlinearity},
Year = {2012},
Note = {\arXiv{1105.1101}},
Pages = {1--44},
Volume = {25},
DOI = {10.1088/0951-7715/25/2/R1}
}
@Article{Thiffeault2001,
Title = {Derivatives and constraints in chaotic flows: asymptotic behaviour and a numerical method},
Author = {Thiffeault, J.-L. },
Journal = {Physica D},
Year = {2002},
Note = {\arXiv{nlin/0101012}},
Pages = {139--161},
Volume = {172},
Abstract = {In a smooth flow, the leading-order response of trajectories to infinitesimal
perturbations in their initial conditions is described by the finite-time
Lyapunov exponents and associated characteristic directions of stretching.
We give a description of the second-order response to perturbations
in terms of Lagrangian derivatives of the exponents and characteristic
directions. These derivatives are related to generalised Lyapunov
exponents, which describe deformations of phase space elements beyond
ellipsoidal. When the flow is chaotic, care must be taken in evaluating
the derivatives because of the exponential discrepancy in scale along
the different characteristic directions. Two matrix decomposition
methods are used to isolate the directions of stretching, the first
appropriate in finding the asymptotic behaviour of the derivatives
analytically, the second better suited to numerical evaluation. The
derivatives are shown to satisfy differential constraints that are
realised with exponential accuracy in time. With a suitable reinterpretation,
the results of the paper are shown to apply to the Eulerian framework
as well.}
}
@Phdthesis{Thum05,
Title = {Numerical Analysis of the Method of Freezing Traveling Waves},
Author = {V. Th\"ummler},
School = {Bielefeld Univ.},
Year = {2005}
}
@Book{Tinkham,
Title = {Group Theory and Quantum Mechanics},
Author = {Tinkham, M.},
Publisher = {Dover},
Year = {2003},
Address = {New York}
}
@Article{ToDe94,
Title = {Geometric phases in lasers and liquid flows},
Author = {Toronov, V. Y. and Derbov, V. L.},
Journal = {Phys. Rev. E},
Year = {1994},
Pages = {1392--1399},
Volume = {49},
Abstract = {Pancharatnam's geometric phase is introduced for such nonlinear dissipative
systems as lasers and liquid flows. Two types of geometric; phases
are shown to arise in these systems: the phase induced by the inner
dynamics of the system and the one caused by the cyclic and adiabatic
variation of the system parameters. A possible generalization of
the geometric-effects theory in other dissipative systems is discussed.}
}
@Article{ToDe94a,
Title = {Geometric-phase effects in laser dynamics},
Author = {Toronov, V. Y. and Derbov, V. L.},
Journal = {Phys. Rev. A},
Year = {1994},
Pages = {878--881},
Volume = {50},
Abstract = {We show that such phenomena of laser dynamics as mean-phase-slope
jumps and temporal phase jumps at resonance between the cavity and
spectral line frequencies are intrinsically connected with the topology
of attractors in the space of rays and can be interpreted as the
manifestations of the geometric-phase properties of the evolution
operator.}
}
@Article{ToDe97,
Title = {Geometric phases in a ring laser},
Author = {Toronov, V. Y. and Derbov, V. L.},
Journal = {Quantum Electronics},
Year = {1997},
Pages = {644--648},
Volume = {27},
Abstract = {An investigation is made of the geometric phases in a ring laser with
counterpropagating waves. It is shown that the frequency splitting
of the counterpropagating waves that appears in the case of radiation
pulsations or is induced by displacement of an external mirror can
be regarded as a manifestation of a geometric phase.}
}
@Article{ToDe97a,
Title = {Boundedness of attractors in the complex {Lorenz} model},
Author = {Toronov, V. Y. and Derbov, V. L.},
Journal = {Phys. Rev. E},
Year = {1997},
Pages = {3689--3692},
Volume = {55}
}
@Article{ToDe98,
Title = {Topological properties of laser phase},
Author = {Toronov, V. Y. and Derbov, V. L.},
Journal = {J. Optical Soc. of America B},
Year = {1998},
Pages = {1282--1290},
Volume = {15}
}
@Article{ToKa78,
Title = {Approximate equation for long nonlinear waves on a viscous fluid},
Author = {J. Topper and T. Kawahara},
Journal = {J. Phys. Soc. Japan},
Year = {1978},
Pages = {663--666},
Volume = {44}
}
@Article{tompaid96,
Title = {Numerical Study of Invariant Sets of a Quasi-periodic Perturbation of a Symplectic Map},
Author = {S. Tompaidis},
Journal = {Experimental Math.},
Year = {1996},
Pages = {211--230},
Volume = {5},
Abstract = {{Newton} method is used to construct periodic orbits of longer and
longer period to approach a invariant torus with specific rotation
vector. Behavior after the break of a torus is described.}
}
@Article{torc96,
Title = {Order Parameter for the Transition from Phase to Amplitude Turbulence},
Author = {A. Torcini},
Journal = {Phys. Rev. Lett.},
Year = {1996},
Pages = {1047},
Volume = {77},
Abstract = {The maximal consersed phase gradient is introduced as an order parameter
to characterize the transition from phase to defect turbulence in
the CGLe. It has a finite value in the PT regime and decreases to
zero when the transition to defect turbulence is approached. A modified
KSe is able to reproduce the main feature of the stable waves and
to explain their origin.}
}
@Article{toricgl,
Title = {Invariant 2-tori in the time-dependent {Ginzburg-Landau} equation},
Author = {P. Tak\'{a}\v{c}},
Journal = {Nonlinearity},
Year = {1992},
Pages = {289--321},
Volume = {5}
}
@Book{Townsend76,
title = {The Structure of Turbulent Shear Flows, 2\textsuperscript{nd} Edn.},
publisher = {Cambridge U. Press},
year = {1976},
author = {A. A. Townsend},
address = {Cambridge}
}
@Article{transgol,
Title = {Transport by capillary waves, part {II}: {Scalar} dispersion and the structure of the concentration field},
Author = {R. Ramshankar and J. P. Gollub},
Journal = {Phys. Fluids A},
Year = {1991},
Pages = {1344},
Volume = {3}
}
@Book{Trefethen97,
Title = {Numerical Linear Algebra},
Author = {L. N. Trefethen and D. Bau},
Publisher = {SIAM},
Year = {1997},
Address = {Philadelphia}
}
@Book{trefethenSpectral,
Title = {Spectral Methods in MATLAB},
Author = {L. N. Trefethen},
Publisher = {SIAM},
Year = {2000},
Address = {Philadelphia}
}
@Article{TrePan98,
Title = {Periodic orbits, {Lyapunov} vectors, and singular vectors in the {L}orenz system},
Author = {Trevisan, A. and Pancotti, F.},
Journal = {J. Atmos. Sci.},
Year = {1998},
Pages = {390},
Volume = {55},
Abstract = {A periodic orbit analysis in the Lorenz system and the study of the
properties of the associated tangent linear equations are performed.
A set of vectors are found that satisfy the Oseledec (1968) theorem
and reduce to Floquet eigenvectors in the particular case of a periodic
orbit. These vectors, called Lyapunov vectors, can be considered
the generalization to aperiodic orbits of the normal modes of the
instability problem and are not necessarily mutually orthogonal.
The relation between singular vectors and Lyapunov vectors is clarified.
The mechanism responsible for super-Lyapunov growth is shown to be
related to the nonorthogonality of Lyapunov vectors. The leading
Lyapunov vectors, as defined here, as well as the asymptotic final
singular vectors, are tangent to the attractor, while the leading
initial singular vectors, in general, point away from it. Perturbations
that are on the attractor can be found in the subspace of the leading
Lyapunov vectors.}
}
@Article{TreWei14,
author = {Trefethen, L. and Weideman, J.},
title = {The exponentially convergent trapezoidal rule},
journal = {SIAM Review},
year = {2014},
volume = {56},
pages = {385--458},
doi = {10.1137/130932132}
}
@Unpublished{TrIsTa09,
author = {A. Trevisan and M. D'Isidoro and O. Talagrand},
title = {Four-dimensional variational assimilation in the unstable subspace {({4D}Var-AUS)} and the optimal subspace dimension},
note = {\arXiv{0902.2714}},
year = {2009},
abstract = { A key a priori information used in 4DVar is the knowledge of the
system's evolution equations. We propose a method for taking full
advantage of the knowledge of the system's dynamical instabilities
in order to improve the quality of the analysis. We present an algorithm,
four-dimensional variational assimilation in the unstable subspace
(4DVar-AUS), that consists in confining in this subspace the increment
of the control variable. The existence of an optimal subspace dimension
for this confinement is hypothesized. Theoretical arguments in favor
of the present approach are supported by numerical experiments in
a simple perfect non-linear model scenario. It is found that the
RMS analysis error is a function of the dimension N of the subspace
where the analysis is confined and is minimum for N approximately
equal to the dimension of the unstable and neutral manifold. For
all assimilation windows, from 1 to 5 days, 4DVar-AUS performs better
than standard 4DVar. In the presence of observational noise, the
4DVar solution, while being closer to the observations, if farther
away from the truth. The implementation of 4DVar-AUS does not require
the adjoint integration. }
}
@Unpublished{TrThNi14,
Title = {Knotted strange attractors and matrix {Lorenz} systems},
Author = {{Tranchida}, J. and {Thibaudeau}, P. and {Nicolis}, S.},
Note = {\arXiv{1404.7774}},
Year = {2014}
}
@Book{Truesdell91,
Title = {A First Course in Rational Continuum Mechanics: {General} concepts},
Author = {Truesdell, C.},
Publisher = {Academic},
Year = {1991},
Address = {New York},
Volume = {1}
}
@Article{TSYTES15,
Title = {Phase-lag synchronization in networks of coupled chemical oscillators},
Author = {Totz, J. F. and Snari, R. and Yengi, D. and Tinsley, M. R. and Engel, H. and Showalter, K.},
Journal = {Phys. Rev. E},
Year = {2015},
Pages = {022819},
Volume = {92},
DOI = {10.1103/PhysRevE.92.022819}
}
@Article{TTRD93,
Title = {Hydrodynamic stability without eigenvalues},
Author = {L. N. Trefethen and A. E. Trefethen and S. C. Reddy and T. A. Driscoll},
Journal = {Science},
Year = {1993},
Pages = {578--584},
Volume = {261}
}
@Article{TuckBar03,
Title = {Symmetry breaking and turbulence in perturbed plane {Couette} flow},
Author = {L. S. Tuckerman and D. Barkley},
Journal = {Theor. Comp. Fluid Dyn.},
Year = {2002},
Note = {\arXiv{physics/0312051}},
Pages = {91--97},
Volume = {16},
Abstract = {Perturbed plane Couette flow containing a thin spanwise-oriented ribbon
undergoes a subcritical bifurcation at Re = 230 to a steady 3D state
containing streamwise vortices. This bifurcation is followed by several
others giving rise to a fascinating series of stable and unstable
steady states of different symmetries and wavelengths. First, the
backwards-bifurcating branch reverses direction and becomes stable
near Re = 200. Then, the spanwise reflection symmetry is broken,
leading to two asymmetric branches which are themselves destabilized
at Re = 420. Above this Reynolds number, time evolution leads first
to a metastable state whose spanwise wavelength is halved and then
to complicated time-dependent behavior. These features are in agreement
with experiments.}
}
@Article{tucker1-2,
Title = {The {Lorenz} attractor exists},
Author = {W. Tucker},
Journal = {C. R. Acad. Sci. Paris S\'er. I},
Year = {1999},
Pages = {1197--1202},
Volume = {328}
}
@Article{tuckerman89,
Title = {Divergence-free velocity fields in nonperiodic geometries},
Author = {L.S. Tuckerman},
Journal = {J. Comp. Phys.},
Year = {1989},
Note = {see also \weblink{www.pmmh.espci.fr/~laurette/highlights/polar/polar.html}},
Pages = {403--441},
Volume = {80}
}
@Book{Tung85,
title = {Group Theory in Physics},
publisher = {World Scientific},
year = {1985},
author = {Tung, W.-K.},
isbn = {978-9971-966-57-7},
address = {Singapore}
}
@Article{umb94pat,
Title = {Transition to parametric wave patterns in a vertically oscillated granular layer},
Author = {F. Melo and P. Umbanhowar and H. L. Swinney},
Journal = {Phys. Rev. Lett.},
Year = {1994},
Pages = {172--175},
Volume = {72}
}
@Book{Vallis06,
Title = {Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation},
Author = {G. K. Vallis},
Publisher = {Cambridge Univ. Press},
Year = {2006},
Address = {Cambridge}
}
@Article{vanBaal91,
author = {van Baal, P.},
title = {More (thoughts on) {Gribov} copies},
journal = {Nucl. Phys.},
year = {1992},
volume = {B369},
pages = {259--275},
doi = {10.1016/0550-3213(92)90386-P}
}
@Incollection{vanBaal97,
author = {van Baal, P.},
title = {Gribov ambiguities and the fundamental domain},
booktitle = {Confinement, Duality, and Non-Perturbative Aspects of QCD},
publisher = {Springer},
year = {2002},
editor = {van Baal, P.},
volume = {368},
series = {NATO Science Series: B},
pages = {161--178},
address = {New York},
note = {\arXiv{hep-th/9711070}},
doi = {10.1007/0-306-47056-X_7}
}
@Article{Vance92,
author = {W. N. Vance},
title = {Unstable periodic-orbits and transport-properties of nonequilibrium steady-states},
journal = {Phys. Rev. Lett.},
year = {1992},
volume = {69},
pages = {1356--1359},
doi = {10.1103/PhysRevLett.69.1356}
}
@Unpublished{Vanderb,
Title = {On the continuation of relative periodic orbits in reversible systems},
Author = {Vanderbauwhede, A.},
URL = {http://cage.ugent.be/~avdb/eng/frameseng.html}
}
@Article{vanderbauwhede_example_2000,
author = {Vanderbauwhede, A.},
title = {An example of symmetry reduction in {Hamiltonian} systems},
journal = {International Conf. on Differential Equations: Berlin, Germany, 1-7 August 1999},
year = {2000}
}
@Book{vanderbauwhede_heteroclinic_1992,
Title = {Heteroclinic Cycles and Periodic Orbits in Reversible Systems},
Author = {Vanderbauwhede, A.},
Publisher = {Pitman},
Year = {1992},
Address = {Boston},
Pages = {250}
}
@Book{vanderbauwhede_local_1982,
Title = {Local Bifurcation and Symmetry},
Author = {Vanderbauwhede, A.},
Publisher = {Pitman},
Year = {1982},
Address = {Boston}
}
@Phdthesis{vanderbauwhede_lokale_1980,
Title = {Lokale Bifurcatietheorie en Symmetrie},
Author = {Vanderbauwhede, A.},
School = {Rijksuniv. Gent, Fac. van de Wetenschappen},
Year = {1980}
}
@Article{vanderbauwhede_normal_1995,
Title = {Normal forms and versal unfoldings of symplectic linear mappings},
Author = {Vanderbauwhede, A.},
Journal = {World Scientific Series in Applicable Analysis},
Year = {1995},
Pages = {685--700},
Volume = {4}
}
@Unpublished{vanderbauwhede_short_1997,
Title = {A short tutorial on {Hamiltonian} systems and their reduction near a periodic orbit},
Author = {Vanderbauwhede, A.},
Year = {1997}
}
@Article{vanderbauwhede_topics_1994,
author = {Vanderbauwhede, A.},
title = {Topics in Bifurcation Theory and Applications {(Gerard Iooss and Moritz Adelmeyer)}},
journal = {SIAM review},
year = {1994},
volume = {36},
pages = {323},
doi = {10.1137/1036090}
}
@InProceedings{Vanderbauwhede1989,
Title = {Centre manifolds, normal forms and elementary bifurcations},
Author = {A. Vanderbauwhede},
Booktitle = {Dynamics Reported},
Year = {1989},
Address = {New York},
Editor = {Kirchgraber, U. and Walther, H.-O.},
Pages = {89--169},
Publisher = {Springer},
Volume = {2},
DOI = {10.1007/978-3-322-96657-5_4}
}
@Article{VanHove54,
Title = {Correlations in space and time and {Born} approximation scattering in systems of interacting particles},
Author = {Van Hove, L.},
Journal = {Phys. Rev.},
Year = {1954},
Pages = {249--262},
Volume = {95},
DOI = {10.1103/PhysRev.95.249}
}
@Article{VaZw12,
Title = {The {Gribov} problem and {QCD} dynamics},
Author = {Vandersickel, N. and Zwanziger, D.},
Journal = {Phys. Rep.},
Year = {2012},
Note = {\arXiv{1202.1491}; To appear.}
}
@Article{VC08,
Title = {Stable manifolds and the transition to turbulence in pipe flow},
Author = {D. Viswanath and P. Cvitanovi\'{c}},
Journal = {J. Fluid Mech.},
Year = {2009},
Note = {\arXiv{0801.1918}},
Pages = {215--233},
Volume = {627}
}
@Article{VeCa00,
Title = {Semiclassical quantization with short periodic orbits},
Author = {Vergini, E. G. and Carlo, G. G.},
Journal = {J. Phys. A},
Year = {2000},
Pages = {4717},
Volume = {33},
Abstract = {We apply a recently developed semiclassical theory of short periodic
orbits to the stadium billiard. We give explicit expressions for
the resonances of periodic orbits and for the application of the
semiclassical Hamiltonian operator to them. Then, by using the three
shortest periodic orbits and two more living in the bouncing-ball
region, we obtain the first 25 odd-odd eigenfunctions with surprising
accuracy.},
DOI = {10.1088/0305-4470/33/25/312}
}
@Article{VeCa01,
Title = {Semiclassical construction of resonances with hyperbolic structure: the scar function},
Author = {Vergini, E. G. and Carlo, G. G.},
Journal = {J. Phys. A},
Year = {2001},
Pages = {4525},
Volume = {34},
Abstract = {The formalism of resonances in quantum chaos is improved by using
conveniently defined creation-annihilation operators. With these
operators at hand, we are able to construct transverse excited resonances
at a given Bohr-quantized energy. Then, by requiring minimum energy
dispersion we obtain solutions in terms of even or odd transverse
excitations. These wavefunctions, which are constructed in the vicinity
of a periodic orbit with maximum energy localization, provide a precise
definition of a scar function . These scar functions acquire, in
the semiclassical limit, the hyperbolic structure characteristic
of unstable periodic orbits.},
DOI = {10.1088/0305-4470/34/21/308}
}
@Article{Vergini00,
Title = {Semiclassical theory of short periodic orbits in quantum chaos},
Author = {E. G. Vergini},
Journal = {J. Phys. A},
Year = {2000},
Pages = {4709},
Volume = {33},
Abstract = {We have developed a semiclassical theory of short periodic orbits
to obtain all the quantum information of a bounded chaotic Hamiltonian
system.},
DOI = {10.1088/0305-4470/33/25/311}
}
@Article{Vergini12,
Title = {Semiclassical approach to long time propagation in quantum chaos: predicting scars},
Author = {Vergini, E. G.},
Journal = {Phys. Rev. Lett.},
Year = {2012},
Pages = {264101},
Volume = {108},
DOI = {10.1103/PhysRevLett.108.264101}
}
@Misc{Vergini13,
Title = {Semiclassical propagation up to the {Heisenberg} time},
Author = {Vergini, E. G.},
Year = {2013}
}
@Article{vHaFrLuPr00,
author = {J. von Hardenberg and K. Fraedrich and F. Lunkeit and A. Provenzale},
title = {Transient chaotic mixing during a baroclinic life cycle},
year = {2000},
volume = {10},
pages = {122--134},
doi = {10.1063/1.166491}
}
@Article{Vierkandt1892,
author = {Vierkandt, A.},
title = {\"Uber gleitende und rollende bewegung},
journal = {Monatshefte f\"ur Math. und Phys.},
year = {1892},
volume = {III},
pages = {31--54}
}
@Article{vincenti12,
Title = {Attosecond lighthouses: {How} to use spatiotemporally coupled light fields to generate isolated attosecond pulses},
Author = {Vincenti, H. and Qu\'er\'e, F.},
Journal = {Phys. Rev. Lett.},
Year = {2012},
Pages = {113904},
Volume = {108},
Abstract = {Under the effect of even simple optical components, the spatial properties
of femtosecond laser beams can vary over the duration of the light
pulse. We show how using such spatiotemporally coupled light fields
in high harmonic generation experiments (e.g., in gases or dense
plasmas) enables the production of attosecond lighthouses, i.e.,
sources emitting a collection of angularly well-separated light beams,
each consisting of an isolated attosecond pulse. This general effect
opens the way to a new generation of light sources, particularly
suitable for attosecond pump-probe experiments, and provides a new
tool for ultrafast metrology, for instance, giving direct access
to fluctuations of the carrier-envelope relative phase of even the
most intense ultrashort lasers.},
DOI = {10.1103/PhysRevLett.108.113904}
}
@Incollection{Visw07a,
Title = {The dynamics of transition to turbulence in plane {Couette} flow},
Author = {D. Viswanath},
Booktitle = {Mathematics and Computation, a Contemporary View. The Abel Symposium 2006},
Publisher = {Springer},
Year = {2008},
Address = {New York},
Note = {\arXiv{physics/0701337}},
Series = {Abel Symposia},
Volume = {3}
}
@Article{Visw07b,
Title = {Recurrent motions within plane {Couette} turbulence},
Author = {D. Viswanath},
Journal = {J. Fluid Mech.},
Year = {2007},
Note = {\arXiv{physics/0604062}},
Pages = {339--358},
Volume = {580}
}
@Article{Visw08,
Title = {The critical layer in pipe flow at high {R}eynolds number},
Author = {D. Viswanath},
Journal = {Philos. Trans. Royal Soc. A},
Year = {2008}
}
@Article{VKK05,
author = {L. van Veen and S. Kida and G. Kawahara},
title = {Periodic motion representing isotropic turbulence},
journal = {Fluid Dynamics Research},
year = {2006},
volume = {38},
pages = {19--46}
}
@Article{VlToDe98,
Title = {The complex {Lorenz} model: {Geometric} structure, homoclinic bifurcation and one-dimensional map},
Author = {Vladimirov, A. G. and Toronov, V. Y. and Derbov, V. L.},
Journal = {Int. J. Bifur. Chaos},
Year = {1998},
Pages = {723--729},
Volume = {8},
Abstract = {It is shown that the phase space of the complex Lorenz model has the
geometric structure associated with a fiber bundle. Using the equations
of motion in the base space of the fiber bundle the surfaces bounding
the attractors in this space are found. The homoclinic ``butterffy{''}
responsible for the Lorenz-like attractor appearance is shown to
correspond to a codimension-two bifurcation. One-dimensional map
describing bifurcation phenomena in the complex Lorenz model is constructed.}
}
@Article{VlToDe98a,
Title = {Properties of the phase space and bifurcations in the complex {Lorenz} model},
Author = {Vladimirov, A. G. and Toronov, V. Y. and Derbov, V. L.},
Journal = {Technical Physics},
Year = {1998},
Pages = {877--884},
Volume = {43}
}
@Article{voth02,
Title = {Experimental measurements of streching fields in fluid mixing},
Author = {G. A. Voth and G. Haller and J. P. Gollub},
Journal = {Phys. Rev. Lett.},
Year = {2002},
Pages = {254501},
Volume = {88},
Abstract = {Based on the experimentally generated velocity field, fixed points
on poincar\'{e} section are identified as well as their stable and
unstable manifolds. These information can be related to the gradient
of the stretching field and the distribution of passive scalar quantities.
It seems that on these manifolds the stretching field gains its maximum.}
}
@Article{vp98,
Title = {Gradient-based approach to solve optimal periodic output feedback control problems},
Author = {A. Varga and S. Pieters},
Journal = {Automatica},
Year = {1998},
Pages = {477--481},
Volume = {34},
Abstract = {A numerical method based on the gradient minimization and Periodic
Schur Decomposition (PSD) is devised to solve the linear-quadratic
(LQ) optimization problem the linear periodic discrete-time control
systems. It can be used as a discretized version of the similar problem
in the continuous regime.}
}
@Article{Vrah95,
Title = {An efficient method for locating and computing periodic orbits of nonlinear mappings},
Author = {Vrahatis, M.},
Journal = {J. Comp. Phys.},
Year = {1995},
Pages = {105--119},
Volume = {119}
}
@Article{VSRBB10,
Title = {Diagonal matrix elements in a scar function basis set},
Author = {E. G. Vergini and E. L. Sibert III and F. Revuelta and R. M. Benito and F. Borondo},
Journal = {Europhysics Lett.},
Year = {2010},
Pages = {40013},
Volume = {89},
Abstract = {We provide canonically invariant expressions to evaluate diagonal
matrix elements of powers of the Hamiltonian in a scar function basis
set. As a function of the energy, each matrix element consists of
a smooth contribution associated with the central periodic orbit,
plus oscillatory contributions given by a finite set of relevant
homoclinic orbits. Each homoclinic contribution depends, in leading
order, on four canonical invariants of the corresponding homoclinic
orbit; a geometrical interpretation of these not well-known invariants
is given. The obtained expressions are verified in a chaotic coupled
quartic oscillator.},
DOI = {10.1209/0295-5075/89/40013}
}
@Article{W01,
Title = {Exact coherent structures in channel flow},
Author = {F. Waleffe},
Journal = {J. Fluid Mech.},
Year = {2001},
Pages = {93--102},
Volume = {435}
}
@InProceedings{W02,
Title = {Exact coherent structures and their instabilities: Toward a dynamical-system theory of shear turbulence},
Author = {F. Waleffe},
Booktitle = {Proceedings of the International Symposium on ``Dynamics and Statistics of Coherent Structures in Turbulence: Roles of Elementary Vortices''},
Year = {2002},
Editor = {S. Kida},
Pages = {115--128},
Publisher = {National Center of Sciences, Tokyo, Japan}
}
@Article{W03,
Title = {Homotopy of exact coherent structures in plane shear flows},
Author = {F. Waleffe},
Journal = {Phys. Fluids},
Year = {2003},
Pages = {1517--1543},
Volume = {15}
}
@InProceedings{W90b,
Title = {On the origin of the streak spacing in turbulent shear flow},
Author = {F. Waleffe},
Booktitle = {Annual Research Briefs},
Year = {1990},
Pages = {159--168},
Publisher = {Center for Turbulence Research, Stanford University}
}
@Article{W95a,
Title = {Hydrodynamic stability and turbulence: {Beyond} transients to a self-sustaining process},
Author = {F. Waleffe},
Journal = {Stud. Applied Math.},
Year = {1995},
Pages = {319--343},
Volume = {95}
}
@Article{W95b,
Title = {Transition in shear flows: {Nonlinear} normality versus non-normal linearity},
Author = {F. Waleffe},
Journal = {Phys. Fluids},
Year = {1995},
Pages = {3060--3066},
Volume = {7}
}
@Article{W98,
Title = {Three-dimensional coherent states in plane shear flows},
Author = {F. Waleffe},
Journal = {Phys. Rev. Lett.},
Year = {1998},
Pages = {4140--4148},
Volume = {81}
}
@Incollection{WaKo12,
Author = {G. M. Wadsworth and D. T. Kovari},
Booktitle = {{ChaosBook.org/projects}},
Publisher = {Georgia Inst. of Technology},
Year = {2012},
Chapter = {Optimal resolution of state space in chaotic hyperbolic 2-dimensional maps},
URL = {http://ChaosBook.org/projects/index.shtml#Kovari}
}
@Unpublished{Waleffe90,
author = {F. Waleffe},
title = {Proposal for a self-sustaining process in shear flows},
note = {Center for Turbulence Research, Stanford University/NASA Ames},
year = {1990},
url = {http://www.math.wisc.edu/~waleffe/ECS/sspctr90.pdf}
}
@Article{Waleffe97,
author = {F. Waleffe},
title = {On a self-sustaining process in shear flows},
journal = {Phys. Fluids},
year = {1997},
volume = {9},
pages = {883--900}
}
@Article{WaLiLi11,
Title = {Computational uncertainty and the application of a high-performance multiple precision scheme to obtaining the correct reference solution of {Lorenz} equations},
Author = {Wang, P. F. and Li, J. P. and Li, Q.},
Journal = {Numerical Algorithms},
Year = {2011}
}
@Incollection{WalPant97,
Title = {How streamwise rolls and streaks self-sustain in a shear flow},
Author = {F. Waleffe and J. Kim},
Booktitle = {Self-Sustaining Mechanisms of Wall Bounded Turbulence},
Publisher = {Computational Mechanics Publications},
Year = {1997},
Address = {Southampton},
Editor = {R. L. Panton},
Pages = {385--422},
URL = {http://www.math.wisc.edu/~waleffe/ECS/WK97.pdf}
}
@Article{Wan11,
Title = {An adaptive high-order minimum action method},
Author = {X. Wan},
Journal = {J. Comp. Phys.},
Year = {2011},
Pages = { - },
DOI = {10.1016/j.jcp.2011.08.006}
}
@Article{Wang13,
author = {Wang, Q.},
title = {Forward and adjoint sensitivity computation of chaotic dynamical systems},
journal = {J. Comp. Phys.},
year = {2013},
volume = {235},
pages = {1--13},
note = {\arXiv{1202.5229}},
doi = {10.1016/j.jcp.2012.09.007},
abstract = {This paper describes a forward algorithm and an adjoint algorithm
for computing sensitivity derivatives in chaotic dynamical systems,
such as the Lorenz attractor. The algorithms compute the derivative
of long time averaged statistical quantities to infinitesimal perturbations
of the system parameters. The algorithms are demonstrated on the
Lorenz attractor. We show that sensitivity derivatives of statistical
quantities can be accurately estimated using a single, short trajectory
(over a time interval of 20) on the Lorenz attractor.}
}
@Article{WanZhoE10,
Title = {Study of the noise-induced transition and the exploration of the phase space for the {Kuramoto-Sivashinsky} equation using the minimum action method},
Author = {Wan, X. and Zhou, X. and E, W.},
Journal = {Nonlinearity},
Year = {2010},
Pages = {475},
Volume = {23}
}
@Book{Wat07,
Title = {The Matrix Eigenvalue Problem: {GR} and {Krylov} Subspace Methods},
Author = {Watkins, D. S.},
Publisher = {SIAM},
Year = {2007},
Address = {Philadelphia},
DOI = {10.1137/1.9780898717808},
ISBN = {978-0-89871-641-2}
}
@Book{Wat10,
Title = {Fundamentals of Matrix Computations},
Author = {Watkins, D. S.},
Publisher = {Wiley},
Year = {2010},
Address = {New York},
ISBN = {978-0-470-52833-4}
}
@Article{Wat2005,
title = {{Product Eigenvalue Problems}},
author = {Watkins, D. S.},
booktitle = {SIAM Review},
doi = {10.1137/S0036144504443110},
number = {1},
pages = {3--40},
volume = {47},
year = {2005}
}
@Article{WeBa88,
Title = {Wavenumber selection for single-wave steady states in a nonlinear baroclinic system},
Author = {Weng, {H-Y.} and Barcilon, A.},
Journal = {J. Atmos. Sci.},
Year = {1988},
Pages = {1039--1051},
Volume = {45},
DOI = {10.1175/1520-0469(1988)045<1039:WSFSWS>2.0.CO;2}
}
@Misc{WeBoHaZa13,
Title = {Three-dimensional nonlinear states in the {Blasius} boundary layer},
Author = {Wedin, H. and Bottaro, A. and Hanifi, A. and Zampogna, G.},
Year = {2013}
}
@Article{WeHa89,
Title = {Chaotic behavior and subcritical formation of flow patterns of baroclinic waves for finite dissipation},
Author = {Weimer, W. and Haken, H.},
Journal = {J. Atmos. Sci.},
Year = {1989},
Pages = {1207--1218},
Volume = {46},
DOI = {10.1175/1520-0469(1989)046<1207:CBASFO>2.0.CO;2}
}
@Article{WeiMaWu02,
Title = {Periodic orbit quantization of the closed three-disk billiard as an example of a chaotic system with strong pruning},
Author = {K. Weibert and J. Main and G. Wunner},
Journal = {Nonlin. Phenom. Complex Sys.},
Year = {2002},
Pages = {393--406},
Volume = {5},
Abstract = {Classical chaotic systems with symbolic dynamics but strong pruning
present a particular challenge for the application of semiclassical
quantization methods. In the present study we show that the technique
of periodic orbit quantization by harmonic inversion of trace formulae,
which does not rely on the existence of a complete symbolic dynamics
or other specific properties, lends itself ideally to calculating
semiclassical eigenvalues from periodic orbit data even in strongly
pruned systems. We apply the harmonic inversion technique to cross-correlated
periodic orbit sums, which allows us to reduce the required number
of orbits. The power, and the limitations, of the method in its present
form are demonstrated for the closed three-disk billiard as a prime
example of a classically chaotic bound system with strong pruning.}
}
@Article{weiss82,
Title = {The {P}ainlev\'{e} property for partial differential equations},
Author = {J. Weiss and M. Tabor and G. Carnevale},
Journal = {J. Math. Phys.},
Year = {1983},
Pages = {522},
Volume = {24}
}
@Article{weiss83,
author = {J. Weiss},
title = {The {P}ainlev\'{e} for partial differential equations {II}: {B\"{a}}cklund transformation, {Lax} pairs, and the {Schwarzian} derivative},
journal = {J. Math. Phys.},
year = {1983},
volume = {24},
pages = {1405}
}
@Article{weiss86,
author = {J. Weiss},
title = {{B\"{a}}cklund transformation and the {P}ainlev\'{e} property},
journal = {J. Math. Phys.},
year = {1986},
volume = {27},
pages = {1293}
}
@Book{Weyl31,
Title = {The Theory of Groups and Quantum Mechanics},
Author = {Weyl, H.},
Publisher = {Dover},
Year = {1931},
Address = {New York},
ISBN = {9780486602691}
}
@Book{Weyl39,
Title = {The Classical Groups: Their Invariants and Representations},
Author = {Weyl, H.},
Publisher = {Princeton Univ. Press},
Year = {1939},
Address = {Princeton, NJ},
ISBN = {9780691057569}
}
@Unpublished{WFSBC15,
Title = {Relative periodic orbits form the backbone of turbulent pipe flow},
Author = {Willis, A. P. and Farazmand, M. and Short, K. Y. and Budanur, N. B. and Cvitanovi{\'c}, P.},
Note = {In preparation.},
Year = {2015}
}
@Article{wheeler12,
Title = {Attosecond lighthouses from plasma mirrors},
Author = {Wheeler, J. A. and Borot, A. and Monchoc\'e, S. and Vincenti, H. and Ricci, A. and Malvache, A. and Lopez-Martens, R. and Qu\'er\'e, F.},
Journal = {Nat. Photon},
Year = {2012},
Pages = {829--833},
Volume = {6},
Abstract = {The nonlinear interaction of an intense femtosecond laser pulse with
matter can lead to the emission of a train of sub-laser-cycle?attosecond?bursts
of short-wavelength radiation. Much effort has been devoted to producing
isolated attosecond pulses, as these are better suited to real-time
imaging of fundamental electronic processes. Successful methods developed
so far rely on confining the nonlinear interaction to a single sub-cycle
event. Here, we demonstrate for the first time a simpler and more
universal approach to this problem, applied to nonlinear laser?plasma
interactions. By rotating the instantaneous wavefront direction of
an intense few-cycle laser field as it interacts with a solid-density
plasma, we separate the nonlinearly generated attosecond pulse train
into multiple beams of isolated attosecond pulses propagating in
different and controlled directions away from the plasma surface.
This unique method produces a manifold of isolated attosecond pulses,
ideally synchronized for initiating and probing ultrafast electron
motion in matter.},
DOI = {10.1038/nphoton.2012.284}
}
@Article{WHKW07,
author = {Wipf, A. and Heinzl, T. and Kaestner, T. and Wozar, C.},
title = {Generalized {Potts-models} and their relevance for gauge theories},
journal = {SIGMA},
year = {2007},
volume = {3},
pages = {6--14},
doi = {10.3842/SIGMA.2007.006},
url = {http://www.emis.de/journals/SIGMA/2007/006/}
}
@Unpublished{WiAvCv12,
Title = {Symmetries of turbulent flows: {How} to slice them},
Author = {Willis, A. P. and Avila, M. and Cvitanovi{\'c}, P.},
Year = {2012}
}
@Article{Wieczorek2005,
Title = {The dynamical complexity of optically injected semiconductor lasers},
Author = {S. Wieczorek and B. Krauskopf and T. B. Simpson and D. Lenstra},
Journal = {Phys. Rep.},
Year = {2005},
Pages = {1--128},
Volume = {416}
}
@Book{wiggb,
Title = {Global Bifurcations and Chaos},
Author = {S. Wiggins},
Publisher = {Springer},
Year = {1988},
Address = {New York}
}
@Book{Wiggins90,
Title = {Introduction to Applied Dynamical Systems and Chaos},
Author = {S. Wiggins},
Publisher = {Springer},
Year = {1990},
Address = {New York}
}
@Book{Wigner59,
Title = {Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra},
Author = {Wigner, E. P.},
Publisher = {Academic},
Year = {1959},
Address = {New York}
}
@Article{Wigner60,
Title = {The unreasonable effectiveness of mathematics in the natural sciences},
Author = {Wigner, E. P.},
Journal = {Commun. Pure Appl. Math.},
Year = {1960},
Pages = {1--14},
Volume = {13}
}
@Unpublished{wikiHamMat,
Title = {Hamiltonian matrix},
Author = {Wikipedia},
URL = {http://en.wikipedia.org/wiki/Hamiltonian\_matrix}
}
@Article{Wilczak03,
author = {D. Wilczak},
title = {Chaos in the {Kuramoto-Sivashinsky} equations -- a computer-assisted proof},
journal = {J. Diff. Eqn.},
year = {2003},
volume = {194},
pages = {433--459},
doi = {10.1016/S0022-0396(03)00104-9}
}
@Incollection{Wilczak09,
Author = {R. Wilczak},
Booktitle = {{ChaosBook.org/projects}},
Publisher = {U. of Chicago},
Year = {2009},
Chapter = {Reducing the state-space of the complex {Lorenz} flow},
Note = {{NSF REU} summer project},
URL = {http://ChaosBook.org/projects/index.shtml#Wilczak}
}
@Article{Williams79,
author = {Williams, R. F.},
title = {The structure of {Lorenz} attractors},
journal = {Publ. Math. IHES},
year = {1979},
volume = {50},
pages = {73--99},
doi = {10.1007/BF02684770}
}
@Article{WilZgl09,
author = {D. Wilczak and P. Zgliczy\'nski},
title = {Computer assisted proof of the existence of homoclinic tangency for the {H\'enon} map and for the forced damped pendulum},
journal = {SIAM J. Appl. Dyn. Syst.},
year = {2009},
volume = {8},
pages = {1632--1663},
doi = {10.1137/090759975}
}
@Book{Winfree1980,
Title = {The Geometry of Biological Time},
Author = {A. T. Winfree},
Publisher = {Springer},
Year = {1980},
Address = {New York}
}
@Article{Winfree73,
author = {A. T. Winfree},
title = {Scroll-shaped Waves of Chemical Activity in 3 Dimensions},
journal = {Science},
year = {1973},
volume = {181},
pages = {937--939}
}
@Article{Winfree91,
author = {A. T. Winfree},
title = {Varieties of spiral wave behavior: {An} experimentalist's approach to the theory of excitable media},
journal = {Chaos},
year = {1991},
volume = {1},
pages = {303--334}
}
@Article{Wirzba99,
author = {A. Wirzba},
title = {Quantum mechanics and semiclassics of hyperbolic n-disk scattering systems},
journal = {Phys. Rep.},
year = {1999},
volume = {309},
pages = {1--116},
doi = {10.1016/S0370-1573(98)00036-2}
}
@Article{WiShCv15,
author = {Willis, A. P. and Short, K. Y. and Cvitanovi{\'c}, P.},
title = {Symmetry reduction in high dimensions, illustrated in a turbulent pipe},
journal = {Phys. Rev. E},
year = {2016},
volume = {93},
pages = {022204},
note = {\arXiv{1504.05825}},
doi = {10.1103/PhysRevE.93.022204}
}
@Article{witt99hol,
author = {R. W. Wittenberg and P. Holmes},
title = {Scale and space localization in the {Kuramoto-Sivashinsky} equation},
journal = {Chaos},
year = {1999},
volume = {9},
pages = {452},
doi = {10.1063/1.166419}
}
@Article{Witte02,
Title = {Dissipativity, analyticity and viscous shocks in the (de)stabilized {Kuramoto-Sivashinsky} equation},
Author = {R. W. Wittenberg},
Journal = {Physics Lett. A},
Year = {2002},
Pages = {407--416},
Volume = {300},
DOI = {10.1016/S0375-9601(02)00861-7}
}
@Article{WiVeBeBo04,
Title = {Classical invariants and the quantization of chaotic systems},
Author = {D. A. Wisniacki and E. Vergini and R. M. Benito and F. Borondo},
Journal = {Phys. Rev. E},
Year = {2004},
Pages = {035202},
Volume = {70}
}
@Article{WK04,
Title = {Exact coherent structures in pipe flow},
Author = {H. Wedin and R. R. Kerswell},
Journal = {J. Fluid Mech.},
Year = {2004},
Pages = {333--371},
Volume = {508}
}
@Article{WK09,
author = {A. P. Willis and R. R. Kerswell},
title = {Turbulent dynamics of pipe flow captured in a reduced model: puff relaminarisation and localised `edge' states},
journal = {J. Fluid Mech.},
year = {2009},
volume = {619},
pages = {213--233}
}
@InProceedings{WKH93,
Title = {On the origin of streaks in turbulent shear flows},
Author = {F. Waleffe and J. Kim and J. Hamilton},
Booktitle = {Turbulent Shear Flows 8: selected papers from the Eighth International Symposium on Turbulent Shear Flows, Munich, Germany, Sept. 9-11, 1991},
Year = {1993},
Address = {New York},
Editor = {F. Durst and R. Friedrich and B. E. Launder and F. W. Schmidt and U. Schumann and J. H. Whitelaw},
Pages = {37--49},
Publisher = {Springer}
}
@Article{WKWTP06,
author = {Wozar, C. and Kaestner, T. and Wipf, A. and Heinzl, T. and Pozsgay, B.},
title = {Phase structure of {$\mathbb{Z}(3)$}-{Polyakov}-loop models},
journal = {Phys. Rev. D},
year = {2006},
volume = {74},
pages = {114501},
doi = {10.1103/PhysRevD.74.114501}
}
@Article{Wold1987,
Title = {Principal component analysis},
Author = {Wold, S. and Esbensen, K. and Geladi, P.},
Journal = {Chemometr. Intell. Lab.},
Year = {1987},
Note = {Proceedings of the Multivariate Statistical Workshop for Geologists and Geochemists},
Pages = {37--52},
Volume = {2},
DOI = {10.1016/0169-7439(87)80084-9}
}
@Article{WolfSwift85,
author = {Wolf, A. and Swift, J. B. and Swinney, H. L. and Vastano, J. A.},
title = {Determining {Lyapunov} exponents from a time series},
journal = {Physica D},
year = {1985},
volume = {16},
pages = {285--317}
}
@Article{WoSa07,
Title = {An efficient method for recovering {Lyapunov} vectors from singular vectors},
Author = {Wolfe, C. L. and Samelson, R. M.},
Journal = {Tellus A},
Year = {2007},
Pages = {355--366},
Volume = {59},
Abstract = { Standard techniques for computing Lyapunov vectors produce results
which are norm-dependent and lack invariance under the linearized
flow. An efficient, norm-independent method for constructing the
n most rapidly growing Lyapunov vectors from n-1 leading forward
and n leading backward asymptotic singular vectors is proposed. The
Lyapunov vectors so constructed are invariant under the linearized
flow in the sense that, once computed at one time, they are defined,
in principle, for all time through the tangent linear propagator.
An analogous method allows the construction of the n most rapidly
decaying Lyapunov vectors from n decaying forward and n-1 decaying
backward singular vectors.}
}
@Article{WRT92,
Title = {The semiclassical helium atom},
Author = {Wintgen, D. and Richter, K. and Tanner, G.},
Journal = {Chaos},
Year = {1992},
Pages = {19--33},
Volume = {2},
DOI = {10.1063/1.165920}
}
@Article{Wulff00,
author = {C. Wulff},
title = {Transitions from relative equilibria to relative periodic orbits},
journal = {Doc. Math.},
year = {2000},
volume = {5},
pages = {227--274}
}
@Article{Wulff01,
Title = {Bifurcation from relative periodic solutions},
Author = {Wulff, C. and Lamb, J. S. W. and Melbourne, I.},
Journal = {Ergod. Theor. Dynam. Syst.},
Year = {2001},
Pages = {605--635},
Volume = {21}
}
@Article{Wulff02,
Title = {Spiral waves and {Euclidean} symmetries},
Author = {Wulff, C.},
Journal = {Z. Phys. Chem.},
Year = {2002},
Pages = {535},
Volume = {216},
DOI = {10.1524/zpch.2002.216.4.535}
}
@Article{Wulff03,
Title = {Persistence of {Hamiltonian} relative periodic orbits},
Author = {C. Wulff},
Journal = {J. Geometry and Physics},
Year = {2003},
Pages = {309--338},
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Editor = {T. Mullin and R. R. Kerswell},
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Year = {2008},
Pages = {024101},
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Abstract = {Inspired by recent results on differences in fluctuations of finite-time
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note = {\arXiv{1008.1941}},
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systems and quadratic for dissipative systems, respectively. The
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structure in Lyapunov vectors corresponding to near-zero Lyapunov
exponents is strongly reduced if CLVs are used instead, whereas for
highly hyperbolic dissipative systems the significance of HLMs is
nearly identical for CLVs and OLVs. In contrast the HLM significance
of Hamiltonian systems is always comparable for CLVs and OLVs irrespective
of hyperbolicity. We also find that in Hamiltonian systems different
symmetry relations between conjugate pairs are observed for CLVs
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indistinguishable in consequence of the microreversibility of Hamiltonian
systems. Transformation properties of Lyapunov exponents, CLVs, and
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}
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