% Encoding: UTF-8
@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}
}
@Book{Abbas16,
title = {Group Theory in Particle, Nuclear, and Hadron Physics},
publisher = {CRC Press},
year = {2016},
author = {Abbas, S. A.},
isbn = {9781498704663},
address = {Boca Raton, FL}
}
@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,
author = {H. D. I. Abarbanel and R. Brown and M. B. Kennel},
title = {Variation of {Lyapunov} exponents on a strange attractor},
journal = {J. Nonlin. Sci.},
year = {1991},
volume = {1},
pages = {175--199}
}
@Article{Abdusalam,
author = {A. Abdusalam},
title = {Renormalization group method and {Julia} sets},
journal = {Chaos Solit. Fract.},
volume = {12},
pages = {2001}
}
@Article{ABLM98,
author = {Avnir, D. and Biham, O. and Lidar, D. and Malcai, O.},
title = {Is the geometry of nature fractal?},
journal = {Science},
year = {1998},
volume = {279},
pages = {39--40},
doi = {10.1126/science.279.5347.39}
}
@Book{ablowbook,
title = {Solitions, Nonlinear Evolution Equations and Inverse Scattering},
publisher = {Cambridge Univ. Press},
year = {1992},
author = {M. J. Ablowitz and P. A. Clarkson},
isbn = {9780521387309},
address = {Cambridge}
}
@Article{Abraham95,
author = {N. B. Abraham and U. A. Allen and E. Peterson and A. Vicens and R. Vilaseca and V. Espinosa and G. L. Lippi},
title = {Structural similarities and differences among attractors and their intensity maps in the laser-{Lorenz} model},
journal = {Optics Comm.},
year = {1995},
volume = {117},
pages = {367--384},
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?. }
}
@Book{AbrMars78,
title = {Foundations of Mechanics},
publisher = {Benjamin-Cummings},
year = {1978},
author = {R. Abraham and J. E. Marsden},
address = {Reading, Mass.}
}
@Book{AbrSte64,
title = {Handbook of Mathematical Functions},
publisher = {Dover},
year = {1965},
author = {Abramowitz, M. and Stegun, I. A.},
edition = {third}
}
@Book{AbSh92,
title = {Dynamics - The Geometry of Behavior},
publisher = {Wesley},
year = {1992},
author = {R. H. Abraham and C. D. Shaw},
address = {Reading, MA}
}
@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},
addendum = {\arXiv{1203.3701}},
doi = {10.1017/jfm.2013.75}
}
@Article{ACL93,
author = {R. Artuso and G. Casati and R. Lombardi},
title = {Periodic orbit theory of anomalous diffusion},
journal = {Phys. Rev. Lett.},
year = {1993},
volume = {71},
pages = {62},
doi = {10.1103/PhysRevLett.71.62}
}
@Article{ACL94,
author = {R. Artuso and G. Casati and R. Lombardi},
title = {Periodic orbit theory of deterministic diffusion},
journal = {Physica A},
year = {1994},
volume = {205},
pages = {412--419},
doi = {10.1016/0378-4371(94)90520-7}
}
@Book{Addison97,
title = {Fractals and Chaos: An illustrated course},
publisher = {CRC Press},
year = {1997},
author = {Addison, P. S.},
isbn = {9780750304009},
address = {Boca Raton FL}
}
@Book{Adler95,
title = {{Quaternionic Quantum Mechanics and Quantum Fields}},
publisher = {Oxford Univ. Press, UK},
year = {1995},
author = {S. L. Adler},
isbn = {9780195345063}
}
@Misc{ADSK14,
author = {Avila, A. and De Simoi, J. and Kaloshin, V.},
title = {An integrable deformation of an ellipse of small eccentricity is an ellipse},
year = {2014},
comment = {\arXiv{1412.2853}}
}
@Article{AdWei67,
author = {Adler, R. L. and Weiss, B.},
title = {Entropy, a complete metric invariant for automorphisms of the torus},
journal = {Proc. Natl. Acad. Sci. USA},
year = {1967},
volume = {57},
pages = {1573--1576},
url = {http://www.ncbi.nlm.nih.gov/pmc/articles/PMC224513/}
}
@Article{afind,
author = {O. Biham and W. Wenzel},
title = {Characterization of unstable periodic orbits in chaotic attractors and repellers},
journal = {Phys. Rev. Lett.},
year = {1989},
volume = {63},
pages = {819--822},
doi = {10.1103/PhysRevLett.63.819}
}
@Article{Afraimovich87,
author = {V. S. Afraimovich and B. B. Bykov and L. P. Shilnikov},
title = {On the appearence and the structure of the {Lorenz} attractor},
journal = {Dokl. Akad. Nauk SSSR},
year = {1987},
volume = {234},
pages = {336--339},
note = {In Russian}
}
@Article{AG93,
author = {Aurell, E. and Gilbert, A. D.},
title = {Fast dynamos and determinants of singular integral operators},
journal = {Geophys. Astrophys. Fluid Dynam.},
year = {1993},
volume = {73},
pages = {5--32},
doi = {10.1080/03091929308203617}
}
@Article{AGHks89,
author = {D. Armbruster and J. Guckenheimer and P. Holmes},
title = {{Kuramoto-Sivashinsky} dynamics on the center-unstable manifold},
journal = {SIAM J. Appl. Math.},
year = {1989},
volume = {49},
pages = {676--691}
}
@Article{AGHO288,
author = {D. Armbruster and J. Guckenheimer and P. Holmes},
title = {Heteroclinic cycles and modulated travelling waves in systems with {O}(2) symmetry},
journal = {Physica D},
year = {1988},
volume = {29},
pages = {257--282}
}
@Article{Aguir08,
author = {Aguirregabiria, J. M.},
title = {Robust chaos with variable {Lyapunov} exponent in smooth one-dimensional maps},
journal = {Chaos Solit. Fract.},
year = {2009},
volume = {42},
pages = {2531--2539},
addendum = {\arXiv{0810.3781}}
}
@Article{ahuja_template-based_2007,
author = {S. Ahuja and {I.G.} Kevrekidis and {C.W.} Rowley},
title = {Template-based stabilization of relative equilibria in systems with continuous symmetry},
journal = {J. Nonlin. Sci.},
year = {2007},
volume = {17},
pages = {109--143},
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{Airy1838,
author = {G. B. Airy},
title = {On the intensity of light in the neighbourhood of a caustic},
journal = {Trans. Camb. Phil. Soc.},
year = {1838},
volume = {6},
pages = {379--403},
url = {https://archive.org/stream/transactionsofca06camb#page/n401/mode/2up}
}
@Book{Ait,
title = {Determinants and Matrices},
publisher = {Oliver and Boyd},
year = {1956},
author = {A. C. Aitken},
address = {Edinburgh}
}
@Article{AKcgl02,
author = {I. S. Aranson and L. Kramer},
title = {The world of the complex {Ginzburg-Landau} equation},
journal = {Rev. Mod. Phys.},
year = {2002},
volume = {74},
pages = {99--143}
}
@Article{AlbChe98,
author = {A. Albouy and A. Chenciner},
title = {Le probl{\`e}me des $N$ corps et les distances mutuelles},
journal = {Inv. Math.},
year = {1998},
volume = {31},
pages = {151--184}
}
@Article{AleYak81,
author = {V. M. Alekseev and M. V. Yakobson},
title = {Symbolic dynamics and hyperbolic dynamic systems},
journal = {Phys. Rep.},
year = {1981},
volume = {75},
pages = {290--325},
doi = {10.1016/0370-1573(81)90186-1}
}
@Article{AlIsPo91,
author = {G. D'Alessandro and S. Isola and A. Politi},
title = {Geometric properties of the pruning front},
journal = {Progr. Theor. Phys.},
year = {1991},
volume = {86},
pages = {1149--1157},
doi = {10.1143/ptp/86.6.1149}
}
@Article{Allah08,
author = {Allahverdyan, A. E.},
title = {Entropy of hidden {Markov} processes via cycle expansion},
journal = {J. Stat. Phys.},
year = {2008},
volume = {133},
pages = {535--564},
addendum = {\arXiv{0810.4341}}
}
@Article{AllgGeorg88,
author = {E. L. Allgower and K. Georg},
title = {Numerically stable homotopy methods without an extra dimension},
journal = {Lect. Appl. Math.},
year = {1990},
volume = {26},
pages = {1--13},
url = {http://www.math.colostate.edu/emeriti/georg/homExtra.pdf}
}
@Book{almeida88,
title = {Hamiltonian systems: Chaos and Quantization},
publisher = {Cambridge Univ. Press},
year = {1988},
author = {Ozorio de Almeida, A. M.},
isbn = {9780521386708},
address = {Cambridge}
}
@Article{alvarez_monodromy_2005,
author = {M. Alvarez},
title = {Monodromy and stability for nilpotent critical points},
journal = {Int. J. Bifur. Chaos},
volume = {15},
pages = {1253--1266},
date = {2005},
issn = {0218-1274}
}
@Article{AMGN,
author = {D. Alonso and D. MacKernan, P. Gaspard and G. Nicolis},
title = {Statistical approach to nonhyperbolic chaotic systems},
journal = {Phys. Rev. E},
year = {1996},
volume = {54},
pages = {2474}
}
@Incollection{ampatt,
author = {M. van Hecke and P. C. Hohenberg and van Saarloos, W.},
title = {Amplitude equations for pattern forming systems},
booktitle = {Fundamental Problems in Statistical Mechanics},
publisher = {North-Holland},
year = {1994},
editor = {van Beijeren, H. and M. H. Ernst},
volume = {VIII},
address = {Amsterdam}
}
@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},
number = {1},
pages = {10007},
doi = {10.1209/0295-5075/90/10007}
}
@Book{AnAr88,
title = {Dynamical systems {I}: {Ordinary} Differential Equations and Smooth Dynamical Systems},
publisher = {Springer},
year = {1988},
author = {D. V. Anosov and V. I. Arnol'd},
address = {New York}
}
@Article{AnBoAi07,
author = {Ando, H. and Boccaletti, S. and Aihara, K.},
title = {Automatic control and tracking of periodic orbits in chaotic systems},
journal = {Phys. Rev. E},
year = {2007},
volume = {75},
pages = {066211}
}
@Article{AndCom07,
author = {N. Andersson and G. L. Comer},
title = {Relativistic fluid dynamics: {Physics} for many different scales},
journal = {Living Reviews in Relativity},
year = {2007},
volume = {10},
pages = {1--83},
doi = {10.12942/lrr-2007-1}
}
@Article{AnDiMa08,
author = {Antoneli, F. and Dias, A. P. S. and Matthews, P. C.},
title = {Invariants, equivariants and characters in symmetric bifurcation theory},
journal = {Proc. Royal Soc. Edinburgh A},
year = {2008},
volume = {138},
pages = {477--512},
doi = {10.1017/S0308210506001119}
}
@Article{angen88,
author = {S. B. Angenent},
title = {The periodic orbits of an area preserving twist-map},
journal = {Commun. Math. Phys.},
year = {1988},
volume = {115},
pages = {353--374}
}
@Article{AngMor12,
author = {C. Angstmann and G .P. Morriss},
title = {An approximate formula for the diffusion coefficient for the periodic {Lorentz} gas},
journal = {Phys. Lett. A},
year = {2012},
volume = {376},
pages = {1819--1822},
doi = {10.1016/j.physleta.2012.04.045},
abstract = {An approximate stochastic model for the topological dynamics of the
periodic triangular Lorentz gas is constructed. The model, together with an
extremum principle, is used to find a closed form approximation to the
diffusion coefficient as a function of the lattice spacing. This approximation
is superior to the popular Machta and Zwanzig result and agrees well with a
range of numerical estimates. },
issn = {0375-9601}
}
@Article{AnoSin67,
author = {D. V. Anosov and Ya. G. Sinai},
title = {Some smooth ergodic systems},
journal = {Russ. Math. Surv.},
year = {1967},
volume = {22},
pages = {103--167},
doi = {10.1070/RM1967v022n05ABEH001228},
abstract = {CONTENTS Introduction Lecture 1. The Maupertuis-Lagrange-Jacobi principle and reduction of a dynamical system to a geodesic flow. Some general properties of smooth dynamical systems Lecture 2. Y -systems Lecture 3. Verification of the Y -conditions for a geodesic flow on manifolds of negative curvature Lecture 4. Transversal foliations Lecture 5. Measurability and absolute continuity of transversal foliations for Y -systems Conclusion Appendix. G.{~}A. Margulis, Y -flows on three-dimensional manifolds References}
}
@Article{anosov67,
author = {D. V. Anosov},
title = {Geodesic flows on compact {Riemannian} manifolds of negative curvature},
journal = {Proc. Steklov. Inst. of Math.},
year = {1967},
volume = {90},
annote = {His famous paper about anosov systems?}
}
@Book{AP86,
title = {Inward Bound: of Matter and Forces in the Physical World},
publisher = {Oxford Univ. Press},
year = {1986},
author = {A. Pais},
address = {Oxford UK}
}
@Book{AP91,
title = {Niels Bohr's Times, in Physics, Philosophy and Polity},
publisher = {Oxford Univ. Press},
year = {1991},
author = {A. Pais},
address = {Oxford UK}
}
@Book{ArKoNe88,
title = {Mathematical Aspects of Classical and Celestial Mechanics},
publisher = {Springer},
year = {1988},
author = {V. I. Arnol'd and V. V. Kozlov and A. I. Neishtadt},
address = {New York}
}
@Book{ArnAve,
title = {Ergodic Problems of Classical Mechanics},
publisher = {Addison-Wesley},
year = {1989},
author = {V. I. Arnol'd and A. Avez},
isbn = {0201094061},
address = {Redwood City}
}
@Book{arnold89,
title = {Mathematical Methods for Classical Mechanics},
publisher = {Springer},
year = {1989},
author = {V. I. Arnol'd},
address = {New York}
}
@Article{arnold91k,
author = {V. I. Arnol'd},
title = {Kolmogorov's hydrodynamic attractors},
journal = {Proc. R. Soc. Lond. A},
year = {1991},
volume = {434},
pages = {19--22},
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},
publisher = {Springer},
year = {1992},
author = {V. I. Arnol'd},
address = {New York}
}
@Article{art03,
author = {R. Artuso and P. Cvitanovi{\'c} and G. Tanner},
title = {Cycle expansions for intermittent maps},
journal = {Proc. Theo. Phys. Supp.},
year = {2003},
volume = {150},
pages = {1--21},
doi = {10.1143/PTPS.150.1}
}
@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}
}
@Incollection{ArtBur14,
author = {Artuso, R. and Burioni, R.},
title = {Anomalous Diffusion: Deterministic and Stochastic Perspectives},
booktitle = {Large Deviations in Physics},
publisher = {Springer},
year = {2014},
editor = {Vulpiani, A. and Cecconi, F. and Cencini, M. and Puglisi, A. and Vergni, D.},
isbn = {978-3-642-54250-3},
pages = {263--293},
address = {Berlin},
doi = {10.1007/978-3-642-54251-0_10}
}
@Article{ArtCri03,
author = {R. Artuso and G. Cristadoro},
title = {Weak chaos and anomalous transport: a deterministic approach},
journal = {Commun. Nonlinear Sci. Numer. Simul.},
year = {2003},
volume = {8},
pages = {137--148},
doi = {10.1016/S1007-5704(03)00025-X},
abstract = {We review how transport properties for chaotic dynamical systems may be studied through cycle expansions, and show how anomalies can be quantitatively described by hierarchical sequences of periodic orbits. }
}
@Article{ArtMaz65,
author = {M. Artin and B. Mazur},
title = {On periodic points},
journal = {Ann. Math.},
year = {1965},
volume = {81},
pages = {82}
}
@Article{ArtStr97,
author = {R. Artuso and R. Strepparava},
title = {Reycling diffusion in sawtooth and cat maps},
journal = {Phys. Lett. A},
year = {1997},
volume = {236},
pages = {469--475},
doi = {10.1016/S0375-9601(97)00792-5},
abstract = {We study diffusion in sawtooth and cat maps on the cylinder. The quasi-linear approximation is studied via periodic orbit expansions. }
}
@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},
publisher = {Academic},
year = {2005},
author = {G. B. Arfken and H. J. Weber},
address = {New York},
edition = {6}
}
@Article{AS87,
author = {M. S. Acarlar and C. R. Smith},
title = {A study of hairpin vortices in a laminar boundary layer},
journal = {J. Fluid Mech.},
year = {1987},
volume = {175},
pages = {1--41 and 45--83}
}
@Article{Asa06,
author = {Asamizuya, T.},
title = {Statistical properties of periodic orbits in a 4-disk billiard system: {The} pruning-proof property},
journal = {Progr. Theor. Phys.},
year = {2006},
volume = {116},
pages = {247--271},
doi = {10.1143/PTP.116.247}
}
@Article{Ascher95,
author = {U. Ascher and S. Ruuth and B. Wetton},
title = {Implicit-explicit methods for time-dependent partial differential equations},
journal = {SIAM J. Numer. Anal.},
year = {1995},
volume = {32},
number = {3},
pages = {797--823}
}
@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.}
}
@Book{AshMer,
title = {Solid State Physics},
publisher = {Holt, Rinehart and Winston},
year = {1976},
author = {Ashcroft, N. W. and Mermin, N. D.},
isbn = {0030839939}
}
@Article{AstMelb06,
author = {P. Aston and I. Melbourne},
title = {{Lyapunov} exponents of symmetric attractors},
journal = {Nonlinearity},
year = {2006},
volume = {19},
pages = {2455-2466},
doi = {10.1088/0951-7715/19/10/010}
}
@Book{ASY96,
title = {Chaos, An Introduction to Dynamical Systems},
publisher = {Springer},
year = {1996},
author = {K. T. Alligood and T. D. Sauer and J. A. Yorke},
address = {New York},
doi = {10.1007/b97589}
}
@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,
author = {Cvitanovi{\'c}, P. and Borrero-Echeverry, D. and Carroll, K. and Robbins, B. and Siminos, E.},
title = {Cartography of high-dimensional flows: {A} visual guide to sections and slices},
journal = {Chaos},
year = {2012},
volume = {22},
pages = {047506},
addendum = {\arXiv{1209.4915}},
doi = {10.1063/1.4758309}
}
@Article{aub95ant,
author = {S. Aubry},
title = {Anti-integrability in dynamical and variational problems},
journal = {Physica D},
year = {1995},
volume = {86},
pages = {284--296}
}
@Article{Aubry88,
author = {N. Aubry and P. Holmes and J. L. Lumley and E. Stone},
title = {The dynamics of coherent structures in the wall region of turbulent boundary layer},
journal = {J. Fluid Mech.},
year = {1988},
volume = {192},
pages = {115--173}
}
@Book{ausloos_logistic_2005,
title = {The Logistic Map and the Route to Chaos: From the Beginnings to Modern Applications},
publisher = {Springer},
year = {2005},
author = {M. Ausloos and M. Dirickx},
address = {New York}
}
@Article{AusYor80,
author = {Auslander, J. and Yorke, J. A.},
title = {Interval maps, factors of maps, and chaos},
journal = {Tohoku Math. J.},
year = {1980},
volume = {32},
pages = {177--188},
doi = {10.2748/tmj/1178229634}
}
@Manual{auto,
title = {{AUTO 97}: Continuation and Bifurcation Software for Ordinary Differential Equations (with {H}omCont)},
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abstract = {semiclassical trace formula for a symmetry-reduced
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azimuthal quantum number. For $m \neq 0$, these orbits
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abstract = {In this paper we overview, compare and elaborate on the
invariant representations of periodic systems. Precisely, with reference
to discrete-time systems, we first introduce the concept of periodic
transfer function from which a notion of generalized frequency response
can be worked out. Then we discuss the following four reformulations:
(i) time lifted, (ii) cyclic, (iii) frequency lifted and
(iv) Fourier. A number of interesting links will be established, and
many theoretical aspects somewhat overlooked in the existing literature
will be clarified. All reformulations are first worked out from the
input--output description and then elaborated in a state-space formalism.
},
doi = {10.1016/S0005-1098(00)00087-X}
}
@Inbook{BitCol01,
chapter = {Periodic Control},
pages = {240-- 253},
title = {Periodic Control},
publisher = {Wiley},
year = {1999},
author = {Bittanti, S. and Colaneri, P.},
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isbn = {9780471346081},
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abstract = {We study algebraic structures underlying 't Hooft's
construction relating classical systems with the quantum harmonic
oscillator. The role of group contraction is discussed. We propose the
use of SU(1,1) for two reasons: because of the isomorphism between its
representation Hilbert space and that of the harmonic oscillator and
because zero point energy is implied by the representation structure.
We comment on the relation between dissipation and
quantization. }
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author = {R. Barrio and M. Lefranc and M. A. Mart{\'{i}}nez and S. Serrano},
title = {Symbolic dynamical unfolding of spike-adding bifurcations in chaotic neuron models},
journal = {Europhys. Lett.},
year = {2015},
volume = {109},
pages = {20002},
abstract = {We characterize the systematic changes in the topological
structure of chaotic attractors that occur as spike-adding and
homoclinic bifurcations are encountered in the slow-fast dynamics of
neuron models. This phenomenon is detailed in the simple Hindmarsh-Rose
neuron model, where we show that the unstable periodic orbits appearing
after each spike-adding bifurcation are associated with specific
symbolic sequences in the canonical symbolic encoding of the dynamics
of the system. This allows us to understand how these bifurcations
modify the internal structure of the chaotic attractors.},
doi = {10.1209/0295-5075/109/20002}
}
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author = {Berger, C. and Song, Z. and Li, T. and Li, X. and Ogbazghi, A. Y. and Feng, R. and Dai, Z. and Marchenkov, A. N. and Conrad, E. H. and First, P. N. and De Heer, W. A.},
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author = {G. Bluman},
title = {Connections between symmetries and conservation laws},
journal = {SIGMA},
year = {2005},
volume = {1},
pages = {011},
addendum = {\arXiv{math-ph/0511035}},
doi = {10.3842/SIGMA.2005.011},
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.},
url = {http://www.emis.de/journals/SIGMA/2005/Paper011/}
}
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isbn = {9780521017947},
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@Unpublished{BoPo10,
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title = {Covariant {Lyapunov} vectors for rigid disk systems},
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addendum = {\arXiv{1005.1172}}
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@Article{BoSmWi36,
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@Book{Botti89,
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author = {Ottino, J. M.},
isbn = {9780521368780},
address = {Cambridge}
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@Unpublished{BoWe14,
author = {{Borthwick}, D. and {Weich}, T.},
title = {Symmetry reduction of holomorphic iterated function schemes and factorization of Selberg zeta functions},
year = {2014},
addendum = {\arXiv{1407.6134}}
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@Book{bowen,
title = {Equilibrium States and the Ergodic Theory of {Anosov} Diffeomorphisms},
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abstract = {It is generally argued that the energy dissipation of three-dimensional turbulent flow is concentrated on a set with non-integer Hausdorff dimension. Recently, in order to explain experimental data, it has been proposed that this set does not possess a global dilatation invariance: it can be considered to be a multifractal set. The authors review the concept of multifractal sets in both turbulent flows and dynamical systems using a generalisation of the beta -model.}
}
@Article{BPTV88,
author = {Bessis, D. and Paladin, G. and Turchetti, G. and Vaienti, S.},
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year = {1988},
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pages = {109--134},
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@Article{BrackMurthy:93,
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@Incollection{brand03,
author = {A. Brandenburg},
title = {Computational aspects of astrophysical {MHD} and turbulence},
booktitle = {Advances in Nonlinear Dynamos},
publisher = {Taylor \& Francis},
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editor = {A. Ferriz-Mas and M. N{\'{u}\~{n}ez}},
pages = {269--344},
address = {London},
addendum = {\arXiv{astro-ph/0109497}}
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@Incollection{breakt,
author = {G. Casati and B. V. Chirikov and F. M. Izrailev and J. Ford},
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@Book{Bredon72,
title = {Introduction to Compact Transformation Groups},
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@Article{BrEllGBAHo10,
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addendum = {\arXiv{1012.0076}},
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@Unpublished{Bridges_priv,
author = {T. J. Bridges},
note = {private communication}
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@Article{Bridges08,
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year = {2008},
volume = {244},
number = {7},
pages = {1629--1674},
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{brjuno71,
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@Article{brjuno72,
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@Incollection{BrKevr96,
author = {Brown, H. S. and Kevrekidis, I. G.},
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editor = {D. Benest and C. Froeschl{\'e}},
pages = {45--66},
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@Book{BroDie85,
title = {Representations of Compact {Lie} Groups},
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@Article{broer_quasi-periodic_2007,
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author = {H. W. Broer and H. M. Osinga and G. Vegter},
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@Article{BrOgYuRe05,
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@Article{bronski2005,
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year = {2006},
volume = {19},
pages = {2023--2039},
addendum = {\arXiv{math/0508481}},
doi = {10.1088/0951-7715/19/9/002}
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year = {1975},
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doi = {10.1007/BF01595390},
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.}
}
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@Article{BTHmaw01,
author = {L. Brusch and A. Torcini and M. van Hecke and M. G. Zimmermann and M. B{\"{a}}r},
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year = {2015},
volume = {25},
pages = {073112},
addendum = {\arXiv{1411.3303}},
doi = {10.1063/1.4923742}
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@PhdThesis{BudanurThesis,
author = {Budanur, N. B.},
title = {Exact Coherent Structures in Spatiotemporal Chaos: From Qualitative Description to Quantitative Predictions},
school = {School of Physics, Georgia Inst. of Technology},
year = {2015},
address = {Atlanta},
url = {http://ChaosBook.org/projects/theses.html}
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@Article{budapest,
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@Article{BudCvi14,
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year = {2015},
volume = {114},
pages = {084102},
addendum = {\arXiv{1405.1096}},
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@Unpublished{BudCvi15,
author = {Budanur, N. B. and Cvitanovi{\'c}, P.},
title = {Torus breakdown in the symmetry-reduced state space of the {Kuramoto-Sivashinsky} system},
year = {2015},
addendum = {\arXiv{1509.08133}}
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pages = {295--312},
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@Article{Bunimovich85,
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doi = {10.1063/1.166105}
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pages = {247--280},
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year = {1981},
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year = {1990},
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@Book{ByFu92,
title = {Mathematics of Classical and Quantum Physics},
publisher = {Dover},
year = {1992},
author = {F. W. Byron and R. W. Fuller},
address = {New York}
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@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--89},
doi = {10.1103/PhysRevLett.85.86},
abstract = {MAWs of various periods P occur in phase chaotic
states. The pair of MAWs cease to exist via a
saddle-node bifurcation. For periods beyond this
bifurcation, incoherent near-MAW evolve toward
defects.}
}
@Article{C02,
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pages = {35--97}
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@Misc{C77,
author = {P. Cvitanovi{\'c}},
title = {Classical and exceptional {Lie} algebras as invariance algebras},
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note = {{Oxford} {Univ.} preprint 40/77, unpublished.},
url = {http://birdtracks.eu/refs/OxfordPrepr.pdf}
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@Misc{CaCaKi63,
author = {Cash, J. and Carter Cash, J. and Kilgore, M.},
title = {The Ring of Fire},
year = {1963},
address = {New York},
publisher = {Columbia Records}
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@Article{CagGol03,
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year = {1997},
volume = {10},
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doi = {10.1088/0951-7715/10/5/004},
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. }
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journal = {Nonlinear Anal.},
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year = {2007},
volume = {232},
number = {1},
pages = {314--328},
issn = {0022-0396}
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}
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representation. By computing the power-phase spectral entropy and the
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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.}
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@Misc{CNS,
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addendum = {\arXiv{chao-dyn/9811003}},
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title = {Chemical Applications of Group Theory},
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isbn = {978-0-471-51094-9}
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title = {Handbook for Matrix Computations},
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number = {2},
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abstract = {Developed ETDRK methods, later improved by Kassam and
Trefethen and used to solve Kuramoto-Sivashinsky
system.}
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title = {{Regular Polytopes}},
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title = {{Introduction to Geometry}},
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author = {Coxeter, H. S. M.},
isbn = {978-0-471-50458-0},
address = {New York},
edition = {2},
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year = {1975},
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pages = {329--348},
doi = {10.1016/0022-0396(75)90047-9}
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@Incollection{CPR79,
author = {Churchill, R. C. and Pecelli, G. and Rod, D. L.},
title = {A survey of the {H{\'e}non-Heiles Hamiltonian} with applications to related examples},
booktitle = {Stochastic Behavior in Classical and Quantum {Hamiltonian} Systems},
publisher = {Springer},
year = {1979},
editor = {Casati, G. and Ford, J.},
isbn = {978-3-540-09120-2},
pages = {76--136},
address = {Berlin},
doi = {10.1007/BFb0021739}
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author = {R. C Churchill and G. Pecelli and D. L Rod},
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journal = {Arch. Rational Mech. Anal.},
year = {1980},
volume = {73},
pages = {329--348},
doi = {10.1016/0022-0396(75)90047-9}
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pages = {265--317}
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author = {J. D. Crawford},
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year = {1995},
volume = {2},
pages = {97--128}
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author = {S. C. Creagh and R. G. Littlejohn},
title = {Semiclassical trace formulas in the presence of continuous symmetries},
journal = {Phys. Rev. A},
year = {1991},
volume = {44},
pages = {836--850},
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,
author = {S. C. Creagh and R. G. Littlejohn},
title = {Semiclassical trace formulas for systems with non-Abelian symmetry},
journal = {J. Phys. A},
year = {1992},
volume = {25},
pages = {1643--1669},
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,
author = {S C Creagh},
title = {Semiclassical mechanics of symmetry reduction},
journal = {J. Phys. A},
year = {1993},
volume = {26},
pages = {95--118},
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,
author = {S. C. Creagh},
title = {Quantum zeta function for perturbed cat maps},
journal = {Chaos},
year = {1995},
volume = {5},
pages = {477--493}
}
@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},
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.},
addendum = {\arXiv{1212.0673}},
doi = {10.1088/1751-8113/46/27/272001}
}
@Article{Cristad06,
author = {G. Cristadoro},
title = {Fractal diffusion coefficient from dynamical zeta functions},
journal = {J. Phys. A},
year = {2006},
volume = {39},
pages = {L151},
doi = {10.1088/0305-4470/39/10/L01},
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.}
}
@Article{CRMRF02,
author = {A. {C.-L.} Chian and E. L. Rempel and E. E. Macau and R. R. Rosa and F. Christiansen},
title = {High-dimensional interior crisis in the {Kuramoto-Sivashinsky} equation},
journal = {Phys. Rev. E},
year = {2002},
volume = {65},
pages = {035203},
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.}
}
@Book{CRob94,
title = {Dynamical Systems: Stability, Symbolic Dynamics, and Chaos},
publisher = {CRC Press},
year = {1995},
author = {Robinson, C.},
isbn = {9780849384950},
address = {Boca Raton FL}
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author = {J. J. Crofts and R. L. Davidchack},
title = {Efficient detection of periodic orbits in chaotic systems by stabilizing transformations},
journal = {SIAM J. Sci. Comp.},
year = {2006},
volume = {28},
pages = {1275--1288},
addendum = {\arXiv{nlin.CD/0502013}}
}
@Article{CroDav09,
author = {J. J. Crofts and R. L. Davidchack},
title = {On the use of stabilizing transformations for detecting unstable periodic orbits in high-dimensional flows},
journal = {Chaos},
year = {2009},
volume = {19},
pages = {033138},
doi = {10.1063/1.3222860}
}
@PhdThesis{Crofts07thesis,
author = {J. J. Crofts},
title = {Efficient Method for Detection of Periodic Orbits in Chaotic Maps and Flows},
school = {Univ. of Leicester},
year = {2007},
address = {Leicester, UK},
addendum = {\arXiv{nlin.CD/0706.1940}}
}
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year = {1994},
volume = {263},
pages = {1569--1570},
abstract = {The phenomena of spatiotermporal chaos are introduced
based on both rotating and regular Rayleigh-Bernard
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.}
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author = {B. B. Mandelbrot},
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volume = {123},
pages = {325--351},
doi = {10.1007/s00229-007-0099-x}
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title = {Global Aspects of Classical Integrable Systems},
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author = {Cushman, R. H. and Bates, L. M.},
address = {Boston}
}
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author = {Cvitanovi{\'c}, P. and Vattay, G.},
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year = {1993},
volume = {71},
pages = {4138--4141},
addendum = {\arXiv{chao-dyn/9307012}},
doi = {10.1103/PhysRevLett.71.4138}
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title = {Spectra of Graphs},
publisher = {Academic},
year = {1980},
author = {D. M. Cvektovi{\'c} and M. Doob and H. Sachs},
address = {New York}
}
@Unpublished{Cvi07,
author = {P. Cvitanovi\'{c}},
title = {Continuous symmetry reduced trace formulas},
note = {Unpublished},
year = {2007},
url = {http://ChaosBook.org/~predrag/papers/Cvi07.pdf}
}
@Inproceedings{Cvi92,
author = {Cvitanovi{\'c}, P.},
title = {The power of chaos},
booktitle = {Applications of Chaos},
year = {1992},
editor = {J. H. Kim and J. Stringer},
address = {New York},
organization = {EPRI workshop},
publisher = {John Wiley \& Sons},
url = {http://chaosbook.org/~predrag/papers/preprints.html#epri}
}
@Article{CviGib10,
author = {Cvitanovi{\'c}, P. and Gibson, J. F.},
title = {Geometry of turbulence in wall-bounded shear flows: {Periodic} orbits},
journal = {Phys. Scr. T},
year = {2010},
volume = {142},
pages = {014007},
doi = {10.1088/0031-8949/2010/T142/014007}
}
@Inproceedings{CviLip12,
author = {P. Cvitanovi{\'c} and D. Lippolis},
title = {Knowing when to stop: {How} noise frees us from determinism},
booktitle = {Let's Face Chaos through Nonlinear Dynamics},
year = {2012},
editor = {M. Robnik and V. G. Romanovski},
pages = {82--126},
address = {Melville, New York},
publisher = {Am. Inst. of Phys.},
addendum = {\arXiv{1206.5506}},
doi = {10.1063/1.4745574}
}
@Article{CviMyr89,
author = {Cvitanovi{\'c}, P. and Myrheim, J.},
title = {Complex universality},
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year = {1989},
volume = {121},
number = {2},
pages = {225--254},
doi = {10.1007/BF01217804}
}
@Proceedings{CviPerWirz92,
title = {Quantum Chaos - Quantum Measurement},
year = {1992},
editor = {Cvitanovic, P. and Percival, I. and Wirzba, A.},
volume = {358},
series = {NATO Sci. Ser. C},
doi = {10.1007/978-94-015-7979-7}
}
@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,
author = {Cvitanovi{\'c}, P. and Eckhardt, B.},
title = {Symmetry decomposition of chaotic dynamics},
journal = {Nonlinearity},
year = {1993},
volume = {6},
pages = {277--311},
addendum = {\arXiv{chao-dyn/9303016}},
doi = {10.1088/0951-7715/6/2/008}
}
@Article{Cvitanovic1995109,
author = {Cvitanovi{\'c}, P.},
title = {Dynamical averaging in terms of periodic orbits},
journal = {Physica D},
year = {1995},
volume = {83},
pages = {109--123},
doi = {10.1016/0167-2789(94)00256-P}
}
@Inproceedings{CvitLanCrete02,
author = {P. Cvitanovi{\'c} and Y. Lan},
title = {Turbulent fields and their recurrences},
booktitle = {Correlations and Fluctuations in QCD : Proceedings of 10th International Workshop on Multiparticle Production},
year = {2003},
editor = {N. Antoniou},
isbn = {978-981-238-455-3},
pages = {313--325},
address = {Singapore},
publisher = {World Scientific},
addendum = {\arXiv{nlin.CD/0308006}},
doi = {10.1142/9789812704641_0032}
}
@Book{cvt89b,
title = {Universality in Chaos},
publisher = {Adam Hilger},
year = {1989},
author = {P. Cvitanovi{\'c}},
isbn = {9780852742600},
address = {Bristol}
}
@Incollection{CVW96,
author = {Cvitanovi{\'c}, P. and Vattay, G. and Wirzba, A.},
title = {Quantum fluids and classical determinants},
booktitle = {Classical, Semiclassical and Quantum Dynamics in Atoms},
publisher = {Springer},
year = {1997},
editor = {H. Friedrich and B. Eckhardt},
pages = {29--62},
address = {New York},
addendum = {\arXiv{chao-dyn/9608012}},
doi = {10.1007/BFb0105968}
}
@Article{D_q,
author = {P. Grassberger},
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year = {1983},
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pages = {227}
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@Article{Dahlqv94,
author = {P. Dahlqvist},
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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,
author = {P. Dahlqvist},
title = {Approximate zeta functions for the {Sinai} billiard and related systems},
journal = {Nonlinearity},
year = {1995},
volume = {8},
pages = {11},
doi = {10.1088/0951-7715/8/1/002},
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.}
}
@Article{dana,
author = {I. Dana},
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year = {1990},
volume = {64},
pages = {2339}
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@Article{Dana_1,
author = {Dana},
journal = {Physica D},
year = {1984},
volume = {13},
pages = {55}
}
@Article{dana89,
author = {Dana},
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journal = {Physica D},
year = {1989},
volume = {39},
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@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 = {2016},
chapter = {{World} in a mirror},
address = {Copenhagen},
url = {http://ChaosBook.org/paper.shtml#discrete}
}
@Unpublished{Davidchack_priv,
author = {R. L. Davidchack},
note = {private communication},
year = {2007}
}
@Article{DavisSIAM08,
author = {Davis, P. J.},
title = {Spanning multiple worlds},
journal = {SIAM News},
year = {2008},
volume = {41},
url = {http://www.siam.org/news/news.php?id=1476}
}
@Article{dawson_collections_1997,
author = {S. P. Dawson and A. M. Mancho},
title = {Collections of heteroclinic cycles in the {Kuramoto-Sivashinsky} equation},
journal = {Physica D},
year = {1997},
volume = {100},
pages = {231--256},
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.}
}
@Article{dCH02,
author = {de Carvalho, A. and Hall, T.},
title = {How to prune a horseshoe},
journal = {Nonlinearity},
year = {2002},
volume = {15},
pages = {R19–R68},
doi = {10.1088/0951-7715/15/3/201}
}
@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{DDeSItz83,
author = {B. Derrida and L. De Seze and C. Itzykson and {J Stat Phys 3}, 559},
year = {1983}
}
@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,
author = {D. D. Stretch},
title = {Automated pattern eduction from turbulent flow diagnostics},
booktitle = {Annual Research Briefs},
year = {1990},
pages = {145--157},
publisher = {Center for Turbulence Research, Stanford University}
}
@Unpublished{Dehaye05,
author = {Dehaye, P.-O.},
title = {Averages over classical compact {Lie} groups and {Weyl} characters},
year = {2005}
}
@Article{DeLeo96,
author = {De Leo, S.},
title = {Quaternionic electroweak theory},
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year = {1996},
volume = {22},
number = {8},
pages = {1137},
doi = {10.1088/0954-3899/22/8/004},
url = {http://arxiv.org/abs/hep-th/9605019}
}
@Article{DemMat01,
author = {Demidenko, G.V. and Matveeva, I.I.},
title = {On stability of solutions to linear systems with periodic coefficients},
journal = {Siberian Math. J.},
year = {2001},
volume = {42},
pages = {282--296},
doi = {10.1023/A:1004837029765}
}
@Article{DemMset,
author = {Demidov, E.},
title = {Anatomy of {Mandelbrot} and {Julia} sets},
url = {http://www.ibiblio.org/e-notes/MSet/Contents.htm}
}
@Book{Dennis96,
title = {Numerical Methods for Unconstrained Optimization and Nonlinear Equations},
publisher = {SIAM},
year = {1996},
author = {Dennis, J. E. and Schnabel, R. B.},
addr = {Philadelphia}
}
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author = {S. Denisov and A. V. Ponomarev},
title = {Oscillons: an encounter with dynamical chaos},
journal = {Chaos},
year = {2011}
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year = {1982},
volume = {48},
pages = {7}
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year = {1982},
volume = {26},
pages = {504}
}
@Article{detect,
author = {Champneys, A. R. and Kuznetsov, Y. A.},
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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},
addendum = {\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{deva87,
title = {An Introduction to Chaotic Dynamical systems},
publisher = {Wesley},
year = {1989},
author = {R. L. Devaney},
address = {Redwood City}
}
@Article{Devaney79,
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year = {1979},
volume = {67},
pages = {137--146}
}
@Inbook{devog58,
chapter = {Contribution to the Theory of Nonlinear Oscillations},
pages = {54--84},
publisher = {Princeton Univ. Press},
year = {1958},
author = {De Vog{\'e}laere, R.},
editor = {S. Lefschetz},
address = {Princeton NJ}
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@Article{DFGHMS86,
author = {Dombre, T. and Frisch, U. and Greene, J. M. and H{\'e}non, M. and Mehr, A. and Soward, A. M.},
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journal = {J. Fluid Mech.},
year = {1986},
volume = {167},
pages = {353--391},
doi = {10.1017/S0022112086002859}
}
@Article{DGOY90,
author = {Ding, M. and C. Grebogi and E. Ott and J. A. Yorke},
title = {Transition to chaotic scattering},
journal = {Phys. Rev. A},
year = {1990},
volume = {42},
pages = {7025--7040},
doi = {10.1103/PhysRevA.42.7025}
}
@Book{DGS76,
title = {Ergodic theory on Compact Spaces},
publisher = {Springer},
year = {1975},
author = {M. Denker and C. Grillenberger and K. Sigmund},
volume = {470},
series = {Lecture Notes in Math.}
}
@Article{DHBEks96,
author = {H. Dankowicz and P. Holmes and G. Berkooz and J. Elezgaray},
title = {Local models of spatio-temporally complex fields},
journal = {Physica D},
year = {1996},
volume = {90},
pages = {387--407}
}
@Article{DhGoKu03,
author = {Dhooge, A. and Govaerts, W. and Kuznetsov, Yu. A.},
title = {{MATCONT: A MATLAB} package for numerical bifurcation analysis of {ODEs}},
journal = {ACM Trans. Math. Softw.},
year = {2003},
volume = {29},
pages = {141--164},
doi = {10.1145/779359.779362}
}
@Article{DhoPatZhi15,
author = {G. Dhont and F. Patra and B. I. Zhilinski\'is},
title = {The action of the special orthogonal group on planar vectors: integrity bases via a generalization of the symbolic interpretation of Molien functions},
journal = {J. Phys. A},
year = {2015},
volume = {48},
pages = {035201},
doi = {10.1088/1751-8113/48/3/035201}
}
@Article{DhoZhi13,
author = {G. Dhont and B. I. Zhilinski\'i},
title = {The action of the orthogonal group on planar vectors: invariants, covariants and syzygies},
journal = {J. Phys. A},
year = {2013},
volume = {46},
pages = {455202},
doi = {10.1088/1751-8113/46/45/455202}
}
@Book{Diacu96,
title = {Celestial Encounters: The Origins of Chaos and Stability},
publisher = {Princeton Univ. Press},
year = {1996},
author = {Diacu, F. and Holmes, P.},
address = {Princeton, NJ}
}
@Book{DiacuHolmes96,
title = {Celestial Encounters, The Origins of Chaos and Stability},
year = {1996},
author = {F. Diacu and P. Holmes, (Princeton Univ. Press, Princeton NJ}
}
@Article{diag_Fred,
author = {Cvitanovi{\'c}, P. and S{\o}ndergaard, N. and Palla, G. and Vattay, G. and Dettmann, C. P.},
title = {Spectrum of stochastic evolution operators: {Local} matrix representation approach},
journal = {Phys. Rev. E},
year = {1999},
volume = {60},
pages = {3936--3941},
addendum = {\arXiv{chao-dyn/9904027}},
doi = {10.1103/PhysRevE.60.3936}
}
@Article{DiEl08,
author = {Dieci, L. and Elia, C.},
title = {{SVD} algorithms to approximate spectra of dynamical systems},
journal = {Math. Comput. Simul.},
year = {2008},
volume = {79},
pages = {1235--1254}
}
@Article{diepenbeek_continuation_????,
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},
title = {Continuation and bifurcation of periodic orbits in symmetric {Hamiltonian} systems.}
}
@Article{diepenbeek_persistence_????,
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},
title = {Persistence of {Hamiltonian} relative periodic orbits}
}
@Article{diepenbeek_relative_????,
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},
title = {Relative periodic orbits in symmetric Lagrangian systems}
}
@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,
author = {Dieci, L. and Russell, R. D. and Van Vleck, E. S.},
title = {On the compuation of {Lyapunov} exponents for continuous dynamical systems},
journal = {SIAM J. Numer. Anal.},
year = {1997},
volume = {34},
pages = {402--423}
}
@Book{ditt01cq,
title = {Classical and Quantum Dynamics: From Classical Paths to Path Integrals},
publisher = {Springer},
year = {2001},
author = {W. Dittrich and M. Reuter},
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{DJKLMPPRT05,
author = {M. Dellnitz and O. Junge and W. S. Koon and F. Lekien and M. W. Lo and J. E. Marsden and K. Padberg and R. Preis and S. D. Ross and B. Thiere},
title = {Transport in dynamical astronomy and multibody problems},
journal = {Int. J. Bifur. Chaos},
year = {2005},
volume = {15},
pages = {699--727}
}
@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,
author = {F. Daviaud and J. Lega and P. Berg{\'e} and P. Coullet and M. Dubois},
title = {Spatio-temporal intermittency in a 1{D} convective pattern: theoretical model and experiments},
journal = {Physica D},
year = {1992},
volume = {55},
pages = {287--308},
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,
author = {R. L. Davidchack and Y-C Lai and E. M. Bollt and M. Dhamala},
title = {Estimating generating partitions of chaotic systems by unstable periodic orbits},
journal = {Phys. Rev. E},
year = {2000},
volume = {61},
pages = {1353}
}
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author = {C. P. Dettmann and G. P. Morriss},
title = {Stability ordering; Strong field {L}orentz gas},
journal = {Phys. Rev. Lett.},
year = {1997},
volume = {78},
pages = {4201}
}
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author = {I. Dana and N. W. Murray and I. Percival},
title = {Transport by turnstiles},
journal = {Phys. Rev. Lett.},
year = {1989},
volume = {62},
pages = {233}
}
@Article{doeb94nls,
author = {H.-D. Doebner and G. A. Goldin},
title = {Properties of nonlinear Schr{\"{o}}dinger equations associated with diffeomorphism group representations},
journal = {J. Phys. A},
year = {1994},
volume = {27},
pages = {1771--1780}
}
@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}
}
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author = {A. Doelman},
title = {Finite-dimensional models of the {Ginzburg-Landau} equation},
journal = {Nonlinearity},
year = {1991},
volume = {4},
pages = {231--250}
}
@Article{doercgl87,
author = {C. R. Doering and J. D. Gibbon and D. D. Holm and B. Nikolaenko},
title = {Exact {Lyapunov} dimension of the universal attractor for the complex {Ginzburg-Landau} equation},
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year = {1987},
volume = {59},
number = {26},
pages = {2911--2914}
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@Article{doercgl88,
author = {C. R. Doering and J. D. Gibbon and D. D. Holm and B. Nikolaenko},
title = {Low-dimensional behaviour in the complex {Ginzburg-Landau} equation},
journal = {Nonlinearity},
year = {1988},
volume = {1},
pages = {279--309},
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},
issn = {1007-5704}
}
@Article{DoLa14a,
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address = {Singapore},
booktitle = {From Phase Transitions to Chaos: Topics in Modern Statistical Physics},
doi = {10.1142/9789814355872_0026}
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author = {A. Gasull and H. Giacomini and M. Grau},
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year = {2008},
volume = {8},
pages = {495--509},
addendum = {\arXiv{math.DS/0610151}}
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title = {Computer Algebra Methods for Equivariant Dynamical Systems},
publisher = {Springer},
year = {2000},
author = {Gatermann, K.},
isbn = {9783540671619},
address = {New York}
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isbn = {978-981-02-2079-2},
address = {Singapore}
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publisher = {Plenum},
chapter = {Microscopic simulations of complex hydrodynamic phenomena},
doi = {10.1007/978-1-4899-2314-1_22}
}
@Unpublished{GBHMRV10,
author = {{Gay-Balmaz}, F. and {Holm}, D. D. and {Meier}, D. M. and {Ratiu}, T. S. and {Vialard}, {F.-X.}},
title = {Invariant higher-order variational problems},
year = {2010},
addendum = {\arXiv{1012.5060}}
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note = {\arXiv{nlin/0610042}},
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which are compatible with an exponential mixing hypothesis, first put
forward by [FM]: they do not seem compatible with the stretched
exponentials believed, in spite of [FM], to describe the mixing.}
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@Unpublished{GGJLF10,
author = {R. Gilmore and J.-M. Ginoux and T. Jones and C. Letellier and U. S. Freitas},
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addendum = {\arXiv{0705.3957}},
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@Book{GiBo,
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address = {Cambridge}
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@TechReport{GibsonMovies,
author = {J. F. Gibson and P. Cvitanovi{\'c}},
title = {Movies of plane {Couette}},
institution = {Georgia Inst. of Technology},
year = {2011},
publisher = {Center for Nonlinear Science},
url = {http://ChaosBook.org/tutorials}
}
@PhdThesis{GibsonPhD,
author = {J. F. Gibson},
title = {Dynamical-systems Models of Wall-bounded, Shear-flow Turbulence},
school = {Cornell Univ.},
year = {2002}
}
@Article{GiChLiPo12,
author = {Ginelli, F. and Chat\'{e}, H. and Livi, R. and Politi, A.},
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note = {\arXiv{1212.3961}},
doi = {10.1088/1751-8113/46/25/254005}
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@Article{GiKaChPoAl11,
author = {Ginelli, F. and Takeuchi, K. A. and Chat{\'e}, H. and Politi, A. and Torcini A.},
title = {Chaos in the {Hamiltonian} mean-field model},
journal = {Phys. Rev. E},
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pages = {066211},
addendum = {\arXiv{1109.6452}},
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chapter = {Dynamo theory},
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doi = {10.1103/PhysRevLett.101.200601},
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.
}
}
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author = {Glendinning, P.},
isbn = {9780521425667},
address = {Cambridge UK}
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@Book{GL-Gil02,
title = {The Topology of Chaos},
publisher = {Wiley},
year = {2003},
author = {Gilmore and M. Lefranc},
editor = {2},
isbn = {978-3-527-41067-5},
address = {New York}
}
@Article{GL-Gil07a,
author = {R. Gilmore},
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journal = {Chaos},
year = {2007},
volume = {17},
pages = {013104}
}
@Book{GL-Gil07b,
title = {The Symmetry of Chaos},
publisher = {Oxford Univ. Press},
year = {2007},
author = {R. Gilmore and C. Letellier},
address = {Oxford}
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year = {1987},
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isbn = {978-3-642-05157-9},
pages = {1--17},
address = {Berlin}
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@Article{GL-Let05,
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title = {Large-scale structural reorganization of strange attractor},
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@Article{GL-Let07,
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@Article{GL-Tsa04,
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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.}
}
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@Inproceedings{Golse06,
author = {Golse, F.},
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volume = {3},
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journal = {Proc. ICM (Madrid, 2006)}
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@Article{Golse08,
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pages = {735--749},
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@InCollection{Golse12,
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title = {Recent results on the periodic {Lorentz} Gas},
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pages = {39--99},
doi = {10.1007/978-3-0348-0191-1_2},
isbn = {978-3-0348-0190-4}
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@Book{golubI,
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publisher = {Springer},
year = {1984},
author = {M. Golubitsky and D. G. Schaeffer},
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@Book{golubII,
title = {Singularities and Groups in Bifurcation Theory, vol. II},
publisher = {Springer},
year = {1988},
author = {M. Golubitsky and I. Stewart and D. G. Schaeffer},
address = {New York}
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@Book{golubitsky2002sp,
title = {The Symmetry Perspective},
publisher = {Birkh{\"a}user},
year = {2002},
author = {Golubitsky, M. and Stewart, I.},
address = {Boston}
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abstract = {A brief review of experiments on film flows, surface
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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,
author = {M. Golubitsky and J. W. Swift and Knobloch, J.},
title = {Symmetries and pattern selection in {Rayleigh-Bernard} convection},
journal = {Physica D},
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author = {Goldhirsch, I. and Sulem, P. L. and Orszag, S. A.},
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@Article{Gouy1899,
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@Book{Govaerts00,
title = {Numerical Methods for Bifurcations of Dynamical Equilibria},
publisher = {SIAM},
year = {2000},
author = {W. J. F. Govaerts},
address = {Philadelphia}
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@Book{GoVanLo96,
title = {Matrix Computations},
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year = {1996},
author = {Golub, G. H. and Van Loan, C. F.},
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@Article{GoWen00,
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volume = {34},
pages = {1151--1163},
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cycle sometimes. 2. Two symbol sequence converges to
the same cycle.}
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note = {Russ. Math. Surv., vol.38, No 1},
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author = {{Gurau}, R. and {Rivasseau}, V. and {Sfondrini}, A.},
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addendum = {{\arXiv{1401.5003}}}
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pages = {71--72},
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.}
}
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school = {School of Physics, Georgia Inst. of Technology},
year = {2008},
address = {Atlanta},
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title = {Dynamics and Bifurcations},
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author = {K. T. Hansen},
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school = {Univ. of Oslo},
year = {1993},
address = {Box 1048 Blindern N-0316, Norway},
url = {http://ChaosBook.org/projects/KTHansen/thesis}
}
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@Book{hartmanb,
title = {Ordinary Differential Equations},
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title = {Handbook of Dynamical Systems},
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title = {Emergence of the theory of {Lie} groups: {An} essay in the history of mathematics},
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author = {Hawkins, T.},
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year = {1996},
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@Book{Hecht74,
title = {Optics},
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volume = {18},
pages = {1--34},
doi = {10.1142/S0218348X10004750},
url = {http://www.math.ucla.edu/~heilman/papers/EigHomotopyV9Submit.pdf}
}
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}
@Inproceedings{Hell77,
author = {R. H. G. Helleman},
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booktitle = {Topics in Nonlinear Dynamics, AIP Conf. Proc. La Jolla 1977},
year = {1978},
editor = {S. Jorna},
volume = {46},
pages = {264--85}
}
@Incollection{Hell79,
author = {R. H. G. Helleman and T. Bountis},
title = {Periodic solutions of arbitrary period, variational methods},
booktitle = {Stochastic Behavior in Classical and Quantum Hamiltonian Systems},
publisher = {Springer},
year = {1979},
editor = {Casati, G. and Ford, J.},
isbn = {978-3-540-09120-2},
pages = {353--375},
address = {Berlin},
doi = {10.1007/BFb0021758}
}
@Article{Helleman,
author = {Helleman and Et All Fourier Series Methods}
}
@Article{HeMuAlBrHa07,
author = {Heusler, S. and M\"uller, S. and Altland, A. and Braun, P. and Haake, F.},
title = {Periodic-orbit theory of level correlations},
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year = {2007},
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pages = {044103}
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@Article{HenLipCvi14,
author = {J. M. Heninger and P. Cvitanovi{\'c} and D. Lippolis},
title = {Neighborhoods of periodic orbits and the stationary distribution of a noisy chaotic system},
journal = {Phys. Rev. E},
year = {2015},
volume = {92},
pages = {062922},
addendum = {\arXiv{1507.00462}},
doi = {10.1103/PhysRevE.92.062922},
issue = {6}
}
@Misc{HenLipCvi15,
author = {J. M. Heninger and D. Lippolis and P. Cvitanovi\'c},
title = {Perturbation theory for the {Fokker-Planck} operator in chaos},
year = {2016},
note = {to be submitted to Comm. Nonlinear Sci. Numer. Simul.},
url = {http://arxiv.org/abs/1602.03044}
}
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author = {M. H{\'e}non},
title = {A two-dimensional mapping with a strange attractor},
journal = {Commun. Math. Phys.},
year = {1976},
volume = {50},
pages = {69},
doi = {10.1007/BF01608556}
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@Article{Henon69,
author = {M. H{\'e}non},
title = {Numerical study of quadratic area-preserving mappings},
journal = {Quart. Appl. Math.},
year = {1969},
volume = {27},
pages = {291--312}
}
@Article{henon82,
author = {M. H{\'e}non},
title = {On the numerical computation of {Poincar{\'e}} maps},
journal = {Physica D},
year = {1982},
volume = {5},
pages = {412--414},
doi = {10.1016/0167-2789(82)90034-3}
}
@Book{henonrtb2,
title = {Generating Families in the Restricted Three-Body Problem {II}. Quantitative Study of Bifurcations},
publisher = {Springer},
year = {2001},
author = {M. H{\'e}non},
address = {New York}
}
@Article{HerGot10,
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year = {2010},
volume = {9},
pages = {536--567},
addendum = {\arXiv{1003.5830}}
}
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year = {2008},
volume = {9},
pages = {1--9},
doi = {10.1186/1471-2105-9-167},
abstract = {Analysis of sequence composition is a routine task in genome research. Organisms are characterized by their base composition, dinucleotide relative abundance, codon usage, and so on. Unique subsequences are markers of special interest in genome comparison, expression profiling, and genetic engineering. Relative to a random sequence of the same length, unique subsequences are overrepresented in real genomes. Shortest words absent from a genome have been addressed in two recent studies.}
}
@Article{heron87,
author = {J. M. Ghidaglia and B. H{\'e}ron},
title = {Dimension of the attractors associated to the {Ginzburg-Landau} partial differential equation},
journal = {Physica D},
year = {1987},
volume = {28},
pages = {282--304}
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year = {2007},
volume = {48},
pages = {023514},
doi = {10.1063/1.2426416},
url = {http://geocalc.clas.asu.edu/pdf/CrystalGA.pdf}
}
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author = {Hestenes, D.},
title = {Point Groups and Space Groups in Geometric Algebra},
booktitle = {{Applications of Geometric Algebra in Computer Science and Engineering}},
publisher = {Birkh{\"a}user},
year = {2002},
editor = {Dorst, L. and Doran, C. and Lasenby, J.},
isbn = {978-1-4612-0089-5},
pages = {3--34},
address = {Boston, MA},
doi = {10.1007/978-1-4612-0089-5_1},
url = {http://geocalc.clas.asu.edu/pdf/crystalsymmetry.pdf}
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pages = {691--714},
doi = {10.1119/1.1571836},
url = {http://geocalc.clas.asu.edu/pdf/SpacetimePhysics.pdf}
}
@Misc{Hestenes15,
author = {Hestenes, D.},
title = {The Genesis of {Geometric Algebra}---a personal retrospective},
year = {2015},
url = {http://www-ma2.upc.edu/sxd/--GAT-IMUVA/2015-Hetenes--Genesis%20of%20GA%20(Proceedings%20AGACSE%202015).pdf}
}
@Book{Hestenes66,
title = {{Space-time Algebra}},
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year = {1966},
author = {Hestenes, D.},
isbn = {978-3-319-18413-5},
edition = {2},
doi = {10.1007/978-3-319-18413-5}
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@Incollection{Hestenes86,
author = {Hestenes, D.},
title = {Clifford algebra and the interpretation of quantum mechanics},
booktitle = {{Clifford Algebras and their Applications in Mathematical Physics}},
publisher = {Springer},
year = {1986},
editor = {J. S. R. Chisholm and A. K. Common},
isbn = {978-94-009-4728-3},
pages = {321--346},
address = {Berlin},
doi = {10.1007/978-94-009-4728-3_27},
url = {http://www2.montgomerycollege.edu/departments/planet/planet/Numerical_Relativity/caiqm.pdf}
}
@Book{Hestenes99,
title = {{New Foundations for Classical Mechanics}},
publisher = {Springer},
year = {2006},
author = {Hestenes, D.},
isbn = {9780306471223},
address = {New York},
edition = {2}
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journal = {Nonlinearity},
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volume = {7},
pages = {1055},
doi = {10.1088/0951-7715/7/3/015},
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.}
}
@Inproceedings{hhrugh95,
author = {H. H. Rugh},
title = {Fredholm determinants for real-analytic hyperbolic diffeomorphisms of surfaces},
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pages = {297--303},
address = {Cambridge UK},
publisher = {Cambridge Univ. Press}
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abstract = {The properties of the KSe is discussed in even larger
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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.}
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abstract = {We compare the Gram--Schmidt and covariant phase-space-basis-vector descriptions for three time-reversible harmonic oscillator problems, in two, three, and four phase-space dimensions respectively. The two-dimensional problem can be solved analytically. The three-dimensional and four-dimensional problems studied here are simultaneously chaotic, time-reversible, and dissipative. Our treatment is intended to be pedagogical, for use in an updated version of our book on Time Reversibility, Computer Simulation, and Chaos.}
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author = {O. B. Isaeva and S.P. Kuznetsov},
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addendum = {\arXiv{nlin.CD/0509012}}
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addendum = {\arXiv{nlin.CD/0504063}}
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author = {Ishii, Y.},
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title = {Modern Differential Geometry for Physicists},
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author = {Ishimura, N. and Nakamura, M.},
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author = {S. Isola},
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year = {2002},
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doi = {10.1088/0951-7715/15/5/310},
abstract = {In this paper we introduce Hilbert spaces of holomorphic functions given by generalized Borel and Laplace transforms which are left invariant by the transfer operators of the Farey map and its induced transformation, the Gauss map, respectively. By means of a suitable operator-valued power series we are able to study simultaneously the spectrum of both these operators along with the analytic properties of associated dynamical $\zeta$-functions. This construction establishes an explicit connection between previously unrelated results of Mayer and Rugh.}
}
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author = {S. Isola},
title = {Dynamical zeta functions for non--uniformly hyperbolic transformations},
year = {1995},
note = {published as J. Stat. Phys. 97, 263 (1999)},
url = {http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=95-142}
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author = {Jacobi, C. G. J.},
title = {{{\"U}ber ein leichtes Verfahren die in der Theorie der S{\"a}cul{\"a}rst{\"o}rungen vorkommenden Gleichungen numerisch aufzul{\"o}sen}},
journal = {J. Reine Angew. Math. ({Crelle})},
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title = {Perspectives of Nonlinear Dynamics},
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address = {Cambridge}
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title = {Perspectives of Nonlinear Dynamics},
publisher = {Cambridge Univ. Press},
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isbn = {9780521426336},
address = {Cambridge}
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author = {Jacobi, C. G. J.},
title = {De functionibus alternantibus earumque divisione per productum e differentiis elementorum conflatum},
journal = {J. Reine Angew. Math. ({Crelle})},
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@Article{JePo02,
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journal = {Amer. J. Math.},
year = {2002},
pages = {495--545},
url = {http://www.jstor.org/stable/25099125}
}
@Article{JGB86,
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@Book{jhos,
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author = {J. K. Hale},
address = {New York}
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@Article{JKSN05,
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journal = {Phys. Fluids},
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pages = {015105}
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author = {Jim{\'e}nez, J. and Moin, P.},
title = {The minimal flow unit in near-wall turbulence},
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year = {1991},
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pages = {213--240},
doi = {10.1017/S0022112091002033}
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author = {H. Johnston and J.-G. Liu},
title = {Accurate, stable and efficient {N}avier-{S}tokes solvers based on explicit treatment of the pressure term},
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year = {2004},
volume = {199},
number = {1},
pages = {221--259}
}
@Article{jolly_evaluating_2000,
author = {M. Jolly and R. Rosa and R. Temam},
title = {Evaluating the dimension of an inertial manifold for the {Kuramoto-Sivashinsky} equation},
journal = {Advances in Differential Equations},
year = {2000},
volume = {5},
pages = {31--66}
}
@Article{JoMueSi12,
author = {C. H. Joyner and S. M\"uller and M. Sieber},
title = {Semiclassical approach to discrete symmetries in quantum chaos},
journal = {J. Phys. A},
year = {2012},
volume = {45},
pages = {205102},
addendum = {\arXiv{1202.4998}}
}
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title = {Dynamics Reported: Expositions in Dynamical Systems},
publisher = {Springer},
year = {1992},
author = {Jones, C. K. R. T. and Kirchgraber, U. and Walther, H.-O. and Bielawski, R.},
address = {New York}
}
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author = {C. A. Jones and M. R. E. Proctor},
title = {Strong spatial resonance and travelling waves in {Benard} convection},
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year = {1987},
volume = {121},
pages = {224--228},
doi = {10.1016/0375-9601(87)90008-9}
}
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journal = {Experiment. Math.},
year = {2002},
volume = {11},
pages = {339}
}
@Misc{JordanJava,
author = {A. Jordan},
title = {Iterations of the complex {H{\'e}non} map},
url = {http://www.shef.ac.uk/~daj/henon/chm.html}
}
@Book{JoSa,
title = {Classical dynamics - A contemporary approach},
publisher = {Cambridge Univ. Press},
year = {1998},
author = {J. V. Jos{\'e} and E. J. Salatan},
isbn = {9780521636360},
address = {Cambridge}
}
@Book{Joshi77,
title = {{Elements of Group Theory for Physicists}},
publisher = {New Age International},
year = {1997},
author = {Joshi, A.W.},
isbn = {9788122409758},
address = {New Delhi, India},
}
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author = {P. Gaspard and J. R. Dorfman},
title = {Chaotic scattering theory, ...},
journal = {Phys. Rev. E},
year = {1995},
volume = {52},
pages = {3525},
note = {+Lyapunovs --> transport coefficients for thermostats}
}
@Article{JRT01,
author = {M. Jolly and R. Rosa and R. Temam},
title = {Accurate computations on inertial manifolds},
journal = {SIAM J. Sci. Comput.},
year = {2001},
volume = {22},
pages = {2216}
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pages = {147--148}
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author = {G. Julia},
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year = {1918},
volume = {4},
pages = {47}
}
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author = {H.W.E. Jung},
title = {Uber ganze birationale Transformationen der Ebene},
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year = {1942},
volume = {184},
pages = {161--174},
note = {MR 0008915 (5:74f)}
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@Unpublished{Jung97,
author = {W. Jung},
title = {Some Explicit Formulas for the Iteration of Rational Functions},
year = {1997}
}
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author = {C. Jung and P. Richter},
journal = {J. Phys. A},
year = {1990},
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pages = {2847}
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@Article{JuScho87,
author = {C. Jung and H. J. Scholz},
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year = {1987},
volume = {20},
pages = {3607}
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year = {1984},
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pages = {181--193}
}
@Inproceedings{KaCoCa02,
author = {{Kalnay}, E. and {Corazza}, M. and {Cai}, M.},
title = {Are bred vectors the same as {Lyapunov} vectors?},
booktitle = {EGS XXVII General Assembly, Nice, 21-26 April 2002},
year = {2002}
}
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journal = {Physica D},
year = {1985},
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pages = {75}
}
@Article{KaSa05,
author = {Kawasaki, M. and Sasa, S.},
title = {Statistics of unstable periodic orbits of a chaotic dynamical system with a large number of degrees of freedom},
journal = {Phys. Rev. E},
year = {2005},
volume = {72},
pages = {037202},
addendum = {\arXiv{9801020}}
}
@Article{Kato03,
author = {S. Kato and M. Yamada},
title = {Unstable periodic solutions embedded in a shell model turbulence},
journal = {Phys. Rev. E},
year = {2003},
volume = {68},
pages = {025302},
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.}
}
@Article{Katok80,
author = {A. Katok},
title = {{Lyapunov} exponents, entropy and periodic orbits for diffeomorphisms},
journal = {Publ. Math. IHES},
year = {1980},
volume = {51},
pages = {137--173},
url = {http://www.numdam.org/item?id=PMIHES_1980__51__137_0}
}
@Book{Katok95,
title = {Introduction to the Modern Theory of Dynamical Systems},
publisher = {Cambridge Univ. Press},
year = {1995},
author = {A. Katok and B. Hasselblatt},
isbn = {9780521575577},
address = {Cambridge}
}
@Article{kavousanakis_projective_2007,
author = {{M.E.} Kavousanakis and R. Erban and {A.G.} Boudouvis and {C.W.} Gear and {I.G.} Kevrekidis},
title = {Projective and coarse projective integration for problems with continuous symmetries},
journal = {J. Comput. Physics},
year = {2007},
volume = {225},
pages = {382--407},
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{KawKida01,
author = {G. Kawahara and S. Kida},
title = {Periodic motion embedded in plane {Couette} turbulence: {Regeneration} cycle and burst},
journal = {J. Fluid Mech.},
year = {2001},
volume = {449},
pages = {291--300},
doi = {10.1017/S0022112001006243},
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{Kazantsev98,
author = {E. Kazantsev},
title = {Unstable periodic orbits and attractor of the barotropic ocean model},
journal = {Nonlin. Proc. Geophys.},
year = {1998},
volume = {5},
pages = {193}
}
@InBook{Keane91,
chapter = {Ergodic theory and subshifts of finite type},
pages = {57--66},
title = {Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces},
publisher = {Oxford Univ. Press},
year = {1991},
author = {Keane, M. S.},
editor = {Bedford, T. and Keane, M. S. and Series, C.},
address = {Oxford U. K.},
isbn = {9780198533900}
}
@InCollection{Keating97,
author = {Keating, J. P.},
title = {Resummation and the turning points of zeta functions},
booktitle = {Classical, Semiclassical and Quantum Dynamics in Atoms},
publisher = {Springer},
year = {1997},
editor = {B. Eckhardt and H. Friedrich},
pages = {83--93},
address = {Berlin},
doi = {10.1007/BFb0105970},
isbn = {978-3-540-63004-3}
}
@Book{kellbv,
title = {Numerical Methods for Two-Point Boundary-Value Problems},
publisher = {Dover},
year = {1992},
author = {H. B. Keller},
address = {New York}
}
@Inproceedings{Keller77,
author = {H. B. Keller},
title = {Numerical solution of bifurcation and nonlinear eigenvalue problems},
booktitle = {Applications of Bifurcation Theory},
year = {1977},
editor = {P. H. Rabinowitz},
pages = {359--384},
address = {New York},
publisher = {Academic}
}
@Inproceedings{Keller79,
author = {H. B. Keller},
title = {Global homotopies and {Newton} methods},
booktitle = {Recent Advances in Numerical Analysis},
year = {1979},
editor = {C. de Boor and G. H. Golub},
pages = {73--94},
address = {New York},
publisher = {Academic}
}
@Article{KellerEBK,
author = {J. B. Keller},
title = {Corrected {Bohr--Sommerfeld} quantum conditions for nonseparable systems},
journal = {Ann. Phys. (N. Y.)},
year = {1958},
volume = {4},
pages = {180--188}
}
@Article{KelRug04,
author = {G. Keller and H. H. Rugh},
title = {Eigenfunctions for smooth expanding circle maps},
journal = {Nonlinearity},
year = {2004},
volume = {17},
pages = {1723--1730},
doi = {10.1088/0951-7715/17/5/009}
}
@Article{Kerswell05,
author = {R. R. Kerswell},
title = {Recent progress in understanding the transition to turbulence in a pipe},
journal = {Nonlinearity},
year = {2005},
volume = {18},
pages = {R17--R44},
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.}
}
@Article{kev01ks,
author = {M. E. Johnson and M. S. Jolly and I. G. Kevrekidis},
title = {The {O}seberg transition: visualization of global bifurcations for the {Kuramoto-Sivashinsky} equation},
journal = {Int. J. Bifur. Chaos},
year = {2001},
volume = {11},
number = {1},
pages = {1--18}
}
@Misc{Khol03,
author = {L. Kholodenko},
title = {Designing new apartment buildings for strings and conformal field theories. {First} steps},
addendum = {\arXiv{hep-th/0312294}}
}
@Article{Kim87,
author = {J. Kim and P. Moin and R. Moser},
title = {Turbulence statistics in fully developed channel flow at low {R}eynolds number},
journal = {J. Fluid Mech.},
year = {1987},
volume = {177},
pages = {133--166}
}
@Article{Kimball01,
author = {Kimball, J. C.},
title = {Chaotic properties of the soft-disk {Lorentz} gas},
journal = {Phys. Rev. E},
year = {2001},
volume = {63},
pages = {066216},
doi = {10.1103/PhysRevE.63.066216}
}
@Article{kirby_reconstructing_1992,
author = {M. Kirby and D. Armbruster},
title = {Reconstructing phase space from {PDE} simulations},
journal = {Z. Angew. Math. Phys.},
year = {1992},
volume = {43},
pages = {999--1022},
abstract = {We propose the {Karhunen-Lo{\'e}ve} {(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{KirSto89,
author = {Kirchgraber, U. and Stoffer, D.},
title = {On the definition of chaos},
journal = {Z. angew. Math. Mech.},
year = {1989},
volume = {69},
pages = {175--185},
doi = {10.1002/zamm.19890690703}
}
@Article{KirSto90,
author = {Kirchgraber, U. and Stoffer, D.},
title = {Chaotic behaviour in simple dynamical systems},
journal = {SIAM Rev.},
year = {1990},
volume = {32},
pages = {424--452},
doi = {10.1137/1032078}
}
@Article{Kirwan88,
author = {Kirwan, F.},
title = {The topology of reduced phase spaces of the motion of vortices on a sphere},
journal = {Physica D},
year = {1988},
volume = {30},
pages = {99--123}
}
@InBook{Kitchens,
chapter = {Symbolic dynamics, group automorphisms and Markov partition},
title = {Real and complex dynamical systems},
publisher = {Springer Science \& Business Media},
year = {1995},
author = {B. Kitchens},
editor = {Branner, B.l and Hjorth, P.},
volume = {464},
address = {New York, N. Y.},
isbn = {978-90-481-4565-2}
}
@Book{KJ94,
title = {Chaos: a program collection for the {PC}},
publisher = {Springer},
year = {1994},
author = {Korsch, H. J. and Jodl, H.-J.},
address = {New York},
doi = {10.1007/978-3-540-74867-0}
}
@Article{KK03,
author = {K. Karamanos and I. Kotsireas},
title = {On the arithmetic nature of various constants at the accumulation points of unimodal maps and in particular of the logistic map},
year = {2003},
url = {http://eprints.cecm.sfu.ca/archive/00000016/e-print}
}
@Inproceedings{KK05,
author = {G. Kawahara and S. Kida and M. Nagata},
title = {Unstable periodic motion in plane {Couette} system: The skeleton of turbulence},
booktitle = {One Hundred Years of Boundary Layer Research},
year = {2005},
publisher = {Kluwer}
}
@Article{KKCSG07,
author = {Korabel, N. and Klages, R. and Chechkin, A. V. and Sokolov, I. M. and Gonchar, V. Yu.},
title = {Fractal properties of anomalous diffusion in intermittent maps},
journal = {Phys. Rev. E},
year = {2007},
volume = {75},
pages = {036213},
doi = {10.1103/PhysRevE.75.036213}
}
@Article{KKGOY97,
author = {E. J. Kostelich and I. Kan and C. Grebogi and E. Ott and J. A. Yorke},
title = {Unstable dimension variability: {A} source of nonhyperbolicity in chaotic systems},
journal = {Physica D},
year = {1997},
volume = {109},
pages = {81--90},
doi = {10.1016/S0167-2789(97)00161-9}
}
@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 = {Inv. Math.},
year = {1989},
volume = {98},
pages = {581--597},
doi = {10.1007/BF01393838}
}
@Article{Kla02,
author = {Klages, R.},
title = {Comment on ``Analysis of chaotic motion and its shape dependence in a generalized piecewise linear map''},
journal = {Phys. Rev. E},
year = {2002},
volume = {66},
pages = {018201},
doi = {10.1103/PhysRevE.66.018201}
}
@Article{Kla02a,
author = {Klages, R.},
title = {Suppression and enhancement of diffusion in disordered dynamical systems},
journal = {Phys. Rev. E},
year = {2002},
volume = {65},
pages = {055203},
doi = {10.1103/PhysRevE.65.055203}
}
@Article{KlaDor95,
author = {R. Klages and J. R. Dorfman},
title = {Simple maps with fractal diffusion coefficients},
journal = {Phys. Rev. Lett.},
year = {1995},
volume = {74},
pages = {387--390},
doi = {10.1103/PhysRevLett.74.387}
}
@Article{KlaDor97,
author = {Klages, R. and Dorfman, J. R.},
title = {Dynamical crossover in deterministic diffusion},
journal = {Phys. Rev. E},
year = {1997},
volume = {55},
pages = {R1247--R1250},
doi = {10.1103/PhysRevE.55.R1247}
}
@Article{KlaDor99,
author = {Klages, R. and Dorfman, J. R.},
title = {Simple deterministic dynamical systems with fractal diffusion coefficients},
journal = {Phys. Rev. E},
year = {1999},
volume = {59},
pages = {5361--5383},
doi = {10.1103/PhysRevE.59.5361}
}
@Article{KlaKelHow08,
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--1743},
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{Klein,
author = {M. Klein},
journal = {Z. Naturf. A},
year = {1988},
volume = {43},
pages = {819}
}
@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,
author = {G. {Kreiss} and A. {Lundbladh} and D. S. {Henningson}},
title = {Bounds for threshold amplitudes in subcritical shear flows},
journal = {J. Fluid Mech.},
year = {1994},
volume = {270},
pages = {175--198}
}
@Article{KMOSTY10,
author = {Kunihiro, T. and M\"uller, B. and Ohnishi, A. and Sch\"afer, A. and Takahashi, T. T. and Yamamoto, A.},
title = {Chaotic behavior in classical {Yang-Mills} dynamics},
journal = {Phys. Rev. D},
year = {2010},
volume = {82},
pages = {114015},
doi = {10.1103/PhysRevD.82.114015}
}
@Article{Knauf87,
author = {Knauf, A.},
title = {Ergodic and topological properties of {Coulombic} periodic potentials},
journal = {Commun. Math. Phys.},
year = {1987},
volume = {110},
pages = {89--112}
}
@Article{KniKla11a,
author = {Knight, G. and Klages, R.},
title = {Capturing correlations in chaotic diffusion by approximation methods},
journal = {Phys. Rev. E},
year = {2011},
volume = {84},
pages = {041135},
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,
author = {Knobloch, J. and Vanderbauwhede, A.},
title = {A general reduction method for periodic solutions in conservative and reversible systems},
journal = {J. Diff. Eqn.},
year = {1996},
volume = {8},
pages = {71}
}
@Inproceedings{knobloch_hopf_1994,
author = {Knobloch, J. and Vanderbauwhede, A.},
title = {Hopf bifurcation at k-fold resonances in equivariant reversible systems},
booktitle = {Dynamics, Bifurcation and Symmetry, New Trends and New Tools},
year = {1994},
editor = {P. Chossat},
pages = {167},
address = {Doredrecht},
publisher = {Kluwer}
}
@Article{knobloch_hopf_1996,
author = {Knobloch, J. and Vanderbauwhede, A.},
title = {Hopf bifurcation at k-fold resonances in conservative systems},
journal = {Prog. Nonlinear Diff. Equations and Their Applications},
year = {1996},
volume = {19},
pages = {155--170}
}
@Article{KNSks90,
author = {I. G. Kevrekidis and B. Nicolaenko and J. C. Scovel},
title = {Back in the saddle again: a computer assisted study of the {Kuramoto-Sivashinsky} equation},
journal = {SIAM J. Appl. Math.},
year = {1990},
volume = {50},
pages = {760--790},
doi = {10.1137/0150045},
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{KoFo,
author = {A. N. Kolmogorov and S. V. Fomin and {Elementy teorii funktsij i funktsional'nogo analiza}, (Nauka, Moskow},
title = {Elements of the Theory of Functions and Functional Analysis Dover,1999},
year = {1980}
}
@Article{kolm91,
author = {A. N. Kolmogorov},
title = {The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers},
journal = {Proc. R. Soc. Lond. A},
year = {1991},
volume = {434},
number = {1890},
pages = {9--13},
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,
author = {H. Kook and J. D. Meiss},
title = {Periodic orbits for reversible, symplectic mappings},
journal = {Physica D},
year = {1989},
volume = {35},
pages = {65--86},
doi = {10.1016/0167-2789(89)90096-1},
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,
author = {H-T Kook and J. D. Meiss},
title = {Application of {Newton}'s method to {Lagrangian} mappings},
journal = {Physica D},
year = {1989},
volume = {36},
pages = {317--326},
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 = {2016},
chapter = {{Implementing} evolution},
address = {Copenhagen},
url = {http://ChaosBook.org/paper.shtml#appendMeasure}
}
@Article{KoRe96,
author = {Kowalski, K. and Rembielinski, J.},
title = {Groups and nonlinear dynamical systems. {Dynamics} on the {SU(2)} group},
journal = {Physica D},
year = {1996},
volume = {99},
pages = {237--251},
addendum = {\arXiv{chao-dyn/9801019}}
}
@Article{KoRe98,
author = {K. Kowalski and J. Rembielinski},
title = {Groups and nonlinear dynamical systems. {Chaotic} dynamics on the {SU(2)xSU(2)} group},
journal = {Chaos Solit. Fract.},
year = {1998},
volume = {9},
pages = {437--448},
addendum = {\arXiv{chao-dyn/9801020}}
}
@Book{Korn,
title = {Ergodic Theory},
publisher = {Springer},
year = {1982},
author = {I. Kornfeld and S. Fomin and Sinai, Y.},
address = {New York},
isbn = {978-1-4615-6929-9}
}
@Article{Kowal87,
author = {Kowalski, K.},
title = {{Hilbert} space description of classical dynamical systems {I}},
journal = {Physica A},
year = {1987},
volume = {145},
pages = {408--424}
}
@Article{Kowal88,
author = {Kowalski, K.},
title = {{Hilbert} space description of classical dynamical systems {II}},
journal = {Physica A},
year = {1988},
volume = {152},
pages = {98--108}
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@Article{Kowal97,
author = {Kowalski, K.},
title = {Nonlinear dynamical systems and classical orthogonal polynomials},
journal = {J. Math. Phys.},
year = {1997},
volume = {38},
pages = {2483--2505},
addendum = {\arXiv{solv-int/9801018}}
}
@Article{KPR12,
author = {Kirk, V. and Postlethwaite, C. and Rucklidge, A.},
title = {Resonance bifurcations of robust heteroclinic networks},
journal = {SIAM J. Appl. Dyn. Sys.},
year = {2012},
volume = {11},
pages = {1360--1401},
addendum = {\arXiv{1206.4328}},
doi = {10.1137/120864684}
}
@Book{Kr85,
title = {Ergodic Theory},
year = {1985},
author = {U. Krengel and (de Gruyter and Berlin}
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@Article{Kras04,
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title = {Growing random sequences},
journal = {J. Phys. A},
year = {2004},
volume = {37},
pages = {2365--2370},
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,
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},
title = {A survey of methods for computing (un)stable manifolds of vector fields},
journal = {Int. J. Bifur. Chaos},
year = {2005},
volume = {15},
pages = {763--791},
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,
author = {T. Kreilos and B. Eckhardt},
title = {Periodic orbits near onset of chaos in plane {Couette} flow},
journal = {Chaos},
year = {2012},
volume = {22},
pages = {047505},
addendum = {\arXiv{1205.0347}},
doi = {10.1063/1.4757227}
}
@Article{KrKrSa15,
author = {Kraemer, A.S. and Kryukov, N. and Sanders, D.P.},
title = {Efficient algorithms for general periodic {Lorentz} gases in two and three dimensions},
journal = {J. Phys. A},
year = {2015},
volume = {49},
pages = {025001},
doi = {10.1088/1751-8113/49/2/025001}
}
@Article{KRSR67,
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title = {The structure of turbulent boundary layers},
journal = {J. Fluid Mech.},
year = {1967},
volume = {30},
pages = {741--773}
}
@Article{Krupa90,
author = {M. Krupa},
title = {Bifurcations of relative equilibria},
journal = {{SIAM} J. Math. Anal.},
year = {1990},
volume = {21},
pages = {1453--1486},
doi = {10.1137/0521081}
}
@Article{KrupaRobHetCyc97,
author = {M. Krupa},
title = {Robust Heteroclinic Cycles},
journal = {J. Nonlin. Sci.},
year = {1997},
volume = {7},
pages = {129--176},
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,
author = {M. Kruskal},
title = {Asymptotic theory of {Hamiltonian} and other systems with all solutons nearly periodic},
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year = {1962},
volume = {3},
number = {4},
pages = {806},
abstract = {Asymptotic expansion to all orders. A systematic way
to construct the adiabatic invariants.}
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year = {1965},
volume = {15},
pages = {240},
doi = {10.1103/PhysRevLett.15.240}
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@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},
number = {4},
pages = {1214--1233},
abstract = {A contour integral method is presented to evaluate
accurately the matrix functions with removable
singularities.}
}
@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.}
}
@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},
doi = {10.1016/S0167-2789(98)90013-6},
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.}
}
@Article{ksgrim91,
author = {R. Grimshaw and A. P. Hooper},
title = {The non-existence of a certain class of travelling wave solutions of the {Kuramoto-Sivashinsky} equation},
journal = {Physica D},
year = {1991},
volume = {50},
pages = {231--238},
doi = {10.1016/0167-2789(91)90177-B}
}
@Article{ksham95,
author = {S. Bouquet},
title = {Hamiltonian structure and integrability of the stationary {Kuramoto-Sivashinsky} equation},
journal = {J. Math. Phys.},
year = {1995},
volume = {36},
number = {3},
pages = {1242},
abstract = {The stationary KSe can be transformed into a
time-depent 1-d {Hamiltonian} 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},
doi = {10.1016/0165-2125(88)90045-5}
}
@Article{kskent92,
author = {P. Kent and J. Elgin},
title = {Travelling-waves of the {Kuramoto-Sivashinsky} equation: {Period-multiplying} bifurcations},
journal = {Nonlinearity},
year = {1992},
volume = {5},
pages = {899--919},
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,
author = {W. C. Troy},
title = {The existence of steady solutions of the {Kuramoto-Sivashinsky} equation},
journal = {J. Diff. Eqn.},
year = {1989},
volume = {82},
pages = {269--313},
doi = {10.1016/0022-0396(89)90134-4},
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,
author = {J. Jones and W. C. Troy and A. D. MacGillivary},
title = {Steady solutions of the {Kuramoto-Sivashinsky} equation for small wave speed},
journal = {J. Diff. Eqn.},
year = {1992},
volume = {96},
pages = {28--55},
doi = {10.1016/0022-0396(92)90143-B},
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,
author = {T.-S. Yang},
title = {On travelling-wave solutions of the {Kuramoto-Sivashinsky} equation},
journal = {Physica D},
year = {1997},
volume = {110},
pages = {25--42},
doi = {10.1016/S0167-2789(97)00121-8}
}
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author = {L. Kadanoff and C. Tang},
title = {Escape rate from strange repellers},
journal = {Proc. Natl. Acad. Sci. USA},
year = {1984},
volume = {81},
pages = {1276}
}
@Article{KT84,
author = {L. Kadanoff and C. Tang and Proc. Natl. Acad. Sci. USA {81}, 1276},
title = {Escape rate from strange repellers},
year = {1984}
}
@Article{KTH92,
author = {Hansen, K. T.},
title = {Pruning of orbits in four-disk and hyperbola billiards},
journal = {Chaos},
year = {1992},
volume = {2},
pages = {71--75},
doi = {10.1063/1.165900}
}
@Article{KuKuKnMo04a,
author = {A. Yu. Kuznetsova and A. P. Kuznetsov, C. Knudsen and E. Mosekilde and {Int. J. Bif. Chaos 12}, 1241-1266},
title = {Catastrophe theoretic classification of nonlinear oscillators},
year = {2004}
}
@Unpublished{KuPa09,
author = {Kuptsov, Pavel V. and Parlitz, Ulrich},
title = {Strict and fussy modes splitting in the tangent space of the {Ginzburg-Landau} equation},
year = {2009},
addendum = {\arXiv{0912.2261}}
}
@Book{kura84tur,
title = {Chemical Oscillations, Waves and Turbulence},
publisher = {Springer},
year = {1984},
author = {Y. Kuramoto},
address = {New York}
}
@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{Kus83,
author = {M. Ku\v{s}},
title = {Integrals of motion for the {Lorenz} system},
journal = {J. Phys. A},
year = {1983},
volume = {16},
pages = {L689}
}
@Article{kuturb78,
author = {Y. Kuramoto},
title = {Diffusion-Induced Chaos in Reaction Systems},
journal = {Progr. Theor. Phys. Suppl.},
year = {1978},
volume = {64},
pages = {346--367},
doi = {10.1143/PTPS.64.346},
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},
publisher = {Springer},
year = {2004},
author = {Y. A. Kuznetsov},
isbn = {978-1-4419-1951-9},
address = {New York},
edition = {3}
}
@Article{KuzOsb02,
author = {P. Kuznetsov and A.H. Osbaldestin},
title = {Generalized dimensions of {Feigenbaum}'s attractor from renormalization-group functional equations},
journal = {Regul. Chaotic. Dyn.},
year = {2002},
volume = {7},
pages = {32--330},
addendum = {\arXiv{nlin.CD/0204059}},
doi = {10.1070/RD2002v007n03ABEH000214}
}
@Article{LahMir99,
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},
doi = {10.1109/83.784431},
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{LaKo89,
author = {D. P. Lathrop and E. J. Kostelich},
title = {Characterization of an experimental strange attractor by periodic orbits},
journal = {Phys. Rev. A},
year = {1989},
volume = {40},
pages = {4028--4031}
}
@Article{lamb_bifurcationperiodic_2003,
author = {Lamb, J. S. W. and Melbourne, I. and Wulff, C.},
title = {Bifurcation from periodic solutions with spatiotemporal symmetry, including resonances and mode interactions},
journal = {J. Diff. Eqn.},
year = {2003},
volume = {191},
pages = {377--407}
}
@Article{Lamba04,
author = {Lamba, H.},
title = {Chaotic, regular and unbounded behaviour in the elastic impact oscillator},
addendum = {\arXiv{chao-dyn/9310004}}
}
@Article{LamRob98,
author = {J. S. W. Lamb and J. A. G. Roberts},
title = {Time reversal symmetry in dynamical systems: {A} survey},
journal = {Physica D},
year = {1998},
volume = {112},
pages = {1}
}
@Article{lanCvit07,
author = {Lan, Y. and Cvitanovi{\'c}, P.},
title = {Unstable recurrent patterns in {Kuramoto-Sivashinsky} dynamics},
journal = {Phys. Rev. E},
year = {2008},
volume = {78},
pages = {026208},
addendum = {\arXiv{0804.2474}},
doi = {10.1103/PhysRevE.78.026208}
}
@Article{landau44,
author = {L. D. Landau},
title = {On the problem of turbulence},
journal = {Akad. Nauk. Doklady},
year = {1944},
volume = {44},
pages = {339}
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@Book{Landau59a,
title = {{Fluid Mechanics}},
publisher = {Pergamon Press},
year = {1959},
author = {L.D. Landau and E.M. Lifshitz},
address = {Oxford}
}
@Book{Landau59b,
title = {{Mechanics}},
publisher = {Pergamon Press},
year = {1959},
author = {L.D. Landau and E.M. Lifshitz},
isbn = {978-0750628969},
address = {Oxford},
edition = {3},
url = {https://archive.org/search.php?query=creator%3A%22L.D.+Landau+%26+E.M.+Lifshitz%22}
}
@Book{Landau59c,
title = {{Quantum Mechanics: Non-Relativistic Theory}},
publisher = {Pergamon Press},
year = {1959},
author = {L.D. Landau and E.M. Lifshitz},
isbn = {978-0750635394},
address = {Oxford},
url = {https://archive.org/search.php?query=creator%3A%22L.D.+Landau+%26+E.M.+Lifshitz%22}
}
@Book{Landau60,
title = {{Electrodynamics of Continuous Media}},
publisher = {Pergamon Press},
year = {1960},
author = {L.D. Landau and E.M. Lifshitz},
isbn = {978-0750626347},
address = {Oxford},
edition = {2},
url = {https://archive.org/search.php?query=creator%3A%22L.D.+Landau+%26+E.M.+Lifshitz%22}
}
@Book{Landau80a,
title = {{Statistical Physics, Part 1}},
publisher = {Pergamon Press},
year = {1980},
author = {L.D. Landau and E.M. Lifshitz},
isbn = {978-0750633727},
address = {Oxford},
url = {https://archive.org/search.php?query=creator%3A%22L.D.+Landau+%26+E.M.+Lifshitz%22}
}
@Book{Landau80b,
title = {{Statistical Physics, Part 2: Theory of the Condensed State}},
publisher = {Pergamon Press},
year = {1980},
author = {L.D. Landau and E.M. Lifshitz},
isbn = {978-0750626361},
address = {Oxford},
url = {https://archive.org/search.php?query=creator%3A%22L.D.+Landau+%26+E.M.+Lifshitz%22}
}
@Article{Landsberg2006143,
author = {J. M. Landsberg and L. Manivel},
title = {The sextonions and {$E_{7 1/2}$}},
journal = {Adv. Math.},
year = {2006},
volume = {201},
pages = {143--179},
doi = {10.1016/j.aim.2005.02.001}
}
@Article{Lang71,
author = {S. Lang and Lang and Serge (Addison-Wesley, Reading, MA},
title = {Linear Algebra},
year = {1971},
note = {QA184 .L38}
}
@Book{langford_normal_1995,
title = {{Normal Forms and Homoclinic Chaos}},
publisher = {AMS},
year = {1995},
author = {W. F. Langford and W. Nagata},
pages = {294}
}
@Article{lanmaw03,
author = {Y. Lan and N. Garnier and P. Cvitanovi{\'c}},
title = {Stationary modulated-amplitude waves in the 1{D} complex {Ginzburg-Landau} equation},
journal = {Physica D},
year = {2004},
volume = {188},
pages = {193--212},
doi = {10.1016/S0167-2789(03)00289-6}
}
@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,
author = {Y. Lan and P. Cvitanovi{\'c}},
title = {Variational method for finding periodic orbits in a general flow},
journal = {Phys. Rev. E},
year = {2004},
volume = {69},
pages = {016217},
addendum = {\arXiv{nlin.CD/0308008}},
doi = {10.1103/PhysRevE.69.016217}
}
@Article{laquey74,
author = {R. E. LaQuey and S. M. Mahajan and P. H. Rutherford and W. M. Tang},
title = {Nonlinear saturation of the trapped-ion mode},
journal = {Phys. Rev. Lett.},
year = {1974},
volume = {34},
pages = {391--394}
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@Article{laskar90,
author = {Laskar, J.},
title = {The chaotic behavior of the solar system: A numerical estimate of the chaotic zones},
journal = {Icarus},
year = {1990},
volume = {88},
pages = {266--291}
}
@Article{LasYor77,
author = {Lasota, A. and Yorke, J. A.},
title = {On the existence of invariant measures for transformations with strictly turbulent trajectories},
journal = {Bull. Polish Acad. Sci. Math.},
year = {1977},
volume = {25},
pages = {233--238}
}
@Book{latt,
title = {{Quantum Fields on the Computer}},
publisher = {World Scientific},
year = {1992},
author = {M. Creutz},
isbn = {978-981-02-0940-7},
address = {Singapore}
}
@Article{lattref,
author = {B. I. Henry and S. D. Watt and S. L. Wearne},
title = {A lattice refinement scheme for finding periodic orbits},
journal = {Anziam J.},
year = {2000},
volume = {42E},
pages = {C735--C751},
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.}
}
@Article{lau92,
author = {Y.-T. Lau},
title = {The cocoon bifurcations in three-dimensional systems with two fixed points},
journal = {Int. J. Bifur. Chaos},
year = {1992},
volume = {2},
pages = {543--558},
doi = {10.1142/S0218127492000690}
}
@Article{LauMa74,
author = {J. B. Laughlin and P. C. Martin},
title = {Transition to turbulence of a statically stressed fluid},
journal = {Phys. Rev. Lett.},
year = {1974},
volume = {33},
pages = {1189}
}
@Article{Laur91,
author = {B. Lauritzen },
title = {Discrete symmetries and the periodic-orbit expansions},
journal = {Phys. Rev. A},
year = {1991},
volume = {43},
pages = {603--606}
}
@Unpublished{Laurent-Polz04,
author = {Frederic Laurent-Polz},
title = {Relative periodic orbits in point vortex systems},
year = {2004},
addendum = {\arXiv{math/0401022}}
}
@Incollection{lauw86,
author = {H. A. Lauwerier},
booktitle = {Chaos},
publisher = {Princeton Univ. Press},
year = {1986},
editor = {V. Holden},
pages = {39},
address = {Princeton}
}
@Article{LCC06,
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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},
language = {English}
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year = {1993},
note = {Laurea thesis}
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author = {Vanessa L{\'o}pez},
note = {private communication}
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doi = {10.1175/1520-0469(1963)020<0448:TMOV>2.0.CO;2}
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abstract = {A 28-variable model of the atmosphere is constructed by expanding
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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
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to observational errors in the atmosphere to grow to intolerable
errors is strongly dependent upon the current circulation pattern,
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complete circulation pattern are no longer small. The feasibility
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is considered.}
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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.}
}
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point, the tangent space is divided into stable and
unstable subspace corresponding the positive and
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url = {http://mi.mathnet.ru/eng/mmo214}
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year = {1980},
volume = {1},
pages = {219-- 226},
doi = {10.1016/0167-2789(80)90013-5},
abstract = {The Lorenz model is studied in details for $\sigma$ = 10, b = 83 and 145 < r < 170. Between r = 145 and r = 148.4 the Lore nz attractor disaggregates itself into a limit cycle through a cascade of bifurcation with successive undoubling of periods. At r = 166.07 this limit cycle looses its stability through {\textquotedblleft}intermittency{\textquotedblright}, giving rise to a second aperiodic attractor. We give a semi-quantitative interpretation of these processes and discuss their relation with the different transitions to turbulence observed experimentally. }
}
@Article{MaRi83,
author = {{Maxey}, M. R. and {Riley}, J. J.},
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journal = {Phys. Fluids},
year = {1983},
volume = {26},
pages = {883--889}
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@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,
author = {Malige, F. and Robutel, P. and Laskar, J.},
title = {Partial reduction in the {N}-body planetary problem using the angular momentum integral},
journal = {Celestial Mech. Dynam. Astronom.},
year = {2002},
volume = {84},
pages = {283--316}
}
@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}},
publisher = {Cambridge Univ. Press},
year = {1992},
author = {Marsden, J. E.},
isbn = {9780521428446},
address = {Cambridge}
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abstract = {The periodic {Lorentz} gas describes an ensemble of non-interacting point particles in a periodic array of spherical scatterers. We have recently shown that, in the limit of small scatterer density (Boltzmann-Grad limit), the macroscopic dynamics converges to a stochastic process, whose kinetic transport equation is not the linear Boltzmann equation---in contrast to the {Lorentz} gas with a disordered scatterer configuration. This paper focuses on the two-dimensional set-up and reports an explicit, elementary formula for the collision kernel of the transport equation.},
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abstract = {The purpose of this paper is to develop
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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},
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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.}
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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. }
}
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title = {{Chaotic Dynamics} - {From} One Dimensional Endomorphism to Two Dimen\-sional Diffeo\-morphism},
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volume = {32},
pages = {267--272},
abstract = {A large number of results from linear time-invariant system
theory can be extended to periodic systems provided an equivalent
time-invariant system can be found. This paper presents a simple and
numerically reliable procedure to achieve the same. It is shown that,
using a stacked representation of periodic systems, under system
equivalence, a minimal-order generalized state-space description can
always be obtained. },
doi = {10.1016/0005-1098(96)85558-0}
}
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year = {2000},
volume = {22},
pages = {6--19}
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@Article{MiZg01,
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title = {Topological entropy for multidimensional perturbations of one dimensional maps},
journal = {Int. J. Bifur. Chaos},
year = {2001},
volume = {5},
pages = {1443--1446},
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])}
}
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author = {S. MacKay and J.D. Meiss and I.C. Percival}
}
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year = {2016},
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year = {1986},
volume = {19},
pages = {89--111},
doi = {10.1016/0167-2789(86)90055-2},
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.}
}
@Book{Moon87,
title = {Chaotic Vibrations: An Introduction for Applied Scientists and Engineers},
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year = {1987},
author = {F. C. Moon},
isbn = {978-0-471-67908-0},
address = {New York}
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pages = {467--521}
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@Article{MorrGree80,
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year = {1980},
volume = {45},
pages = {790--794},
addendum = {see also Phys. Rev. Lett. 48, 569 (1982)}
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@Article{Morrison02,
author = {J. F. Morrison and W. Jiang and B. J. McKeon and A. J. Smits},
title = {Reynolds number dependence of streamwise velocity spectra in turbulent pipe flow},
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year = {2002},
volume = {88},
pages = {214501}
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@Article{Morrison04,
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abstract = {We study {Lyapunov} vectors (LVs) corresponding to
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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
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025202(R) (2007)] for a specific coupled-map lattice.}
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author = {Dahlqvist, P.},
title = {Computing the topological pressure for intermittent maps},
journal = {J. Phys. A},
year = {1997},
volume = {30},
pages = {L351--L358},
doi = {10.1088/0305-4470/30/11/002},
abstract = {The topological pressure is obtained as the leading zero of a
dynamical zeta function. We consider the problem of computing this zero
when it is close to a singularity. In particular we study a family of
intermittent maps, which we argue exhibit a branch point singularity in
its zeta functions. The convergence of the cycle expansion close to
this point is extremely slow. To deal with this problem we consider a
resummation of the cycle expansion. The idea is quite similar to that
of Pad{\'e} approximants, but the ansatz is a generalized series expansion
around the branch point rather than a rational function. The
improvement on convergence of the leading zero is considerable. We also
briefly discuss the relation between correlation decay and the nature
of the branch point.}
}
@Article{Peckham2,
author = {B. Peckham},
title = {Real perturbation of complex analitic families: {Points} to regions},
journal = {Int. J. Bifur. Chaos},
year = {1998},
volume = {8},
pages = {391--414}
}
@Article{Peinke,
author = {J. Peinke and J. Parisi, B. Rohricht and O. E. Rossler},
journal = {Z. Naturf. A},
year = {1987},
volume = {42},
pages = {263}
}
@Book{Peitgen,
title = {The Beauty of Fractals. Images of Complex Dynamical Systems},
publisher = {Springer},
year = {1986},
author = {H. O. Peitgen and P. H. Richter},
address = {Berlin}
}
@Book{PeJuSa04,
title = {Chaos and Fractals: New Frontiers of Science},
publisher = {Springer},
year = {2004},
author = {Peitgen, H.-O. and J{\"u}rgens, H. and Saupe, D.},
isbn = {978-1-4684-9396-2},
address = {New York}
}
@Book{Penr04,
title = {{The Road to Reality: A Complete Guide to the Laws of the Universe}},
publisher = {A. A. Knopf},
year = {2005},
author = {R. Penrose},
address = {New York}
}
@Article{percgl,
author = {A. V. Porubov and M. G. Velarde},
title = {Exact periodic solutions of the complex {Ginzburg-Landau} equation},
journal = {J. Math. Phys.},
year = {1999},
volume = {40},
number = {2},
pages = {884}
}
@Article{percol,
author = {M. V. Entin and G. M. $\mathrm{\acute{E}}$ntin},
journal = {Pis'ma Zh. Eksp. Teor. Fiz.},
year = {1996},
volume = {64},
pages = {427}
}
@Book{perko91,
title = {Differential Equations and Dynamical Systems},
publisher = {Springer},
year = {1991},
author = {L. Perko},
address = {New York}
}
@Book{PerRich82,
title = {Introduction to Dynamics},
publisher = {Cambridge Univ. Press},
year = {1983},
author = {I. Percival and D. Richards},
address = {Cambridge}
}
@Article{PerVivL,
author = {I. Percival and F. Vivaldi},
journal = {Physica D},
year = {1987},
volume = {27},
pages = {373}
}
@Article{Pesin74,
author = {Ya. B. Pesin},
journal = {Funct. Anal. Appl.},
year = {1974},
volume = {8},
pages = {263}
}
@Article{Pesin77,
author = {Ya. B. Pesin},
title = {Characteristic {Lyapunov} exponents and smooth ergodic theory},
journal = {Russian Math. Surveys},
year = {1977},
volume = {32},
pages = {55}
}
@Article{Pesin92,
author = {Ya .B. Pesin},
title = {Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties},
journal = {Ergod. Theor. Dynam. Syst.},
year = {1992},
volume = {12},
pages = {123--151}
}
@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}
}
@Book{Peterson,
title = {Ergodic Theory},
publisher = {Cambridge Univ. Press},
year = {1990},
author = {K. Peterson},
isbn = {9780521389976},
address = {Cambridge UK}
}
@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}
}
@Article{pexp,
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},
doi = {10.1088/0305-4470/24/5/005}
}
@Book{Peyret02,
title = {Spectral Methods for Incompressible Flows},
publisher = {Springer},
year = {2002},
author = {R. Peyret},
address = {New York}
}
@Book{PG97,
title = {{Chaos, Scattering and Statistical Mechanics}},
publisher = {Cambridge Univ. Press},
year = {1997},
author = {P. Gaspard},
isbn = {9780521018258},
address = {Cambridge}
}
@Article{PhysToday04,
author = {R. Fitzgerald},
title = {New experiments set the scale for the onset of turbulence in pipe flow},
journal = {Physics Today},
year = {2004},
volume = {57},
number = {2},
pages = {21--23}
}
@Article{PhysWorld04,
author = {Carlo Barenghi},
title = {Turbulent transition for fluids},
journal = {Physics World},
year = {2004},
volume = {17},
number = {12}
}
@Article{pikovsky,
author = {S. Pikovsky and unpublished.}
}
@Article{pimsimp,
author = {P. Moresco and S. P. Dawson},
title = {The {PIM}-simplex method: an extension of the {PIM}-triple method to saddles with an arbitrary number of expanding directions},
journal = {Physica},
year = {1999},
volume = {126D},
pages = {38},
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,
author = {D. Sweet and H. E. Nusse. and J. A. Yorke},
title = {Stagger-and-step method: {Detecting} and computing chaotic saddles in higher dimensions},
journal = {Phys. Rev. Lett.},
year = {2001},
volume = {86},
pages = {2261--2264},
doi = {10.1103/PhysRevLett.86.2261},
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,
author = {H. E. Nusse and J. A. York},
title = {A procedure for finding numerical trajectories on chaotic saddles},
journal = {Physica D},
year = {1989},
volume = {36},
pages = {137},
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.}
}
@Article{Pinho2009,
author = {Pinho, A. J. and Ferreira, P. J. S. G. and Garcia, S. P. and Rodrigues, J. M. O. S},
title = {On finding minimal absent words},
journal = {BMC Bioinformatics},
year = {2009},
volume = {10},
pages = {1--11},
doi = {10.1186/1471-2105-10-137},
abstract = {The problem of finding the shortest absent words in DNA data has been recently addressed, and algorithms for its solution have been described. It has been noted that longer absent words might also be of interest, but the existing algorithms only provide generic absent words by trivially extending the shortest ones.}
}
@Book{PiPo16,
title = {{Lyapunov} Exponents A Tool to Explore Complex Dynamics},
publisher = {Cambridge Univ. Press},
year = {2016},
author = {A. Pikovsky and A. Politi},
address = {Cambridge UK},
isbn = {9781107030428}
}
@Book{PJD,
title = {Circulant Matrices},
publisher = {Amer. Math. Soc.},
year = {1979},
author = {P. J. Davis},
isbn = {978-0821891650},
address = {New York},
edition = {2}
}
@Article{Plemelj1909,
author = {Plemelj, J.},
title = {{Zur Theorie der Fredholmschen Funktionalgleichung}},
journal = {Monat. Math. Phys.},
year = {1904},
volume = {15},
pages = {93--28},
doi = {10.1007/BF01692293}
}
@Article{PlSiFi91,
author = {Platt, N. and Sirovich, L. and Fitzmaurice, N.},
title = {An investigation of chaotic {Kolmogorov} flows},
journal = {Phys. Fluids A},
year = {1991},
volume = {3},
pages = {681--696},
doi = {10.1063/1.858074}
}
@Article{PoGiYaMa06,
author = {Politi, A. and Ginelli, F. and Yanchuk, S. and Maistrenko, Y.},
title = {From synchronization to {Lyapunov} exponents and back},
journal = {Physica D},
year = {2006},
volume = {224},
pages = {90 - 101},
note = {Dynamics on Complex Networks and Applications },
addendum = {\arXiv{nlin/0605012}},
doi = {http://dx.doi.org/10.1016/j.physd.2006.09.032},
issn = {0167-2789}
}
@Article{Poinc1896,
author = {H. Poincar{\'e}},
title = {Sur les solutions p{\'e}riodiques et le principe de moindre action},
journal = {C. R. Acad. Sci. Paris},
year = {1896},
volume = {123},
pages = {915--918}
}
@Book{poincare,
title = {{Les M{\'e}thodes Nouvelles de la M{\'e}chanique C{\'e}leste}},
publisher = {Guthier-Villars},
year = {1899},
author = {H. Poincar{\'e}},
address = {Paris},
note = {For a very readable exposition of {P}oincar{\'e}'s work and the development of the dynamical systems theory up to 1920's see \rf{JBG97}.}
}
@Book{poinper,
title = {{New Methods in Celestial Mechanics}},
publisher = {Springer},
year = {1992},
author = {H. Poincar{\'e}},
address = {New York}
}
@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}
}
@Inproceedings{Poll-1,
author = {M. Pollicott},
title = {Periodic orbits and zeta functions},
booktitle = {1999 AMS Summer Institute on {Smooth ergodic theory and applications}, Seattle},
year = {1999},
series = {Symposia Pure Applied Math.}
}
@Misc{Poll1206,
author = {Kennedy, R.},
title = {The case of {Pollock}'s fractals focuses on physics},
howpublished = {New York Times, Dec. 2, 2006},
url = {http://www.nytimes.com/2006/12/02/books/02frac.html}
}
@Article{polli,
author = {Pollicott, M.},
title = {Meromorphic extensions of generalised zeta functions},
journal = {Inv. Math.},
year = {1986},
volume = {85},
pages = {147--164},
doi = {10.1007/BF01388795}
}
@Incollection{Pollicott02,
author = {M. Pollicott},
title = {Periodic orbits and zeta functions},
booktitle = {Handbook of Dynamical Systems},
publisher = {Elsevier},
year = {2002},
editor = {Hasselblatt, B. and Katok, A.},
volume = {1, Part A},
pages = {409--452},
address = {New York},
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{Pollicott85,
author = {Pollicott, M.},
title = {On the rate of mixing of {Axiom A} flows},
journal = {Inv. Math.},
year = {1985},
volume = {81},
pages = {413--426},
doi = {10.1007/BF01388579}
}
@Article{Pollicott91,
author = {Pollicott, M.},
title = {A note on the {Artuso-Aurell-Cvitanovi{\'c}} 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}},
publisher = {Cambridge Univ. Press},
year = {1993},
author = {Pollicott, M.},
isbn = {9780521435932},
address = {Cambridge}
}
@Article{PollJ-Sm06,
author = {Jones-Smith, K. and Mathur, H.},
title = {Fractal analysis: {Revisiting Pollock's} drip paintings},
journal = {Nature},
year = {2006},
volume = {444},
pages = {E9--E10},
doi = {10.1038/nature05398}
}
@Article{PollTay06,
author = {Taylor, R. P. and Micolich, A. P. and Jonas, D.},
title = {Fractal analysis: {Revisiting Pollock'}s drip paintings {(Reply)}},
journal = {Nature},
year = {2006},
volume = {444},
pages = {E10--E11},
doi = {10.1038/nature05399}
}
@Article{PollTay99,
author = {Taylor, R. P. and Micolich, A. P. and Jonas, D.},
title = {Fractal analysis of {Pollock}'s drip paintings},
journal = {Nature},
year = {1999},
volume = {399},
pages = {422--422},
doi = {10.1038/20833}
}
@Article{pomeau80,
author = {Y. Pomeau and Manneville, P.},
title = {Intermittent transition to turbulence in dissipative dynamical systems},
journal = {Commun. Math. Phys.},
year = {1980},
volume = {74},
pages = {189}
}
@Book{popebook,
title = {Turbulent Flows},
publisher = {Cambridge Univ. Press},
year = {2000},
author = {S. B. Pope},
isbn = {9780521598866},
address = {Cambridge}
}
@Article{postgal89,
author = {B. Garc\'{i}a-Archilla and J. Novo and E. S. Titi},
title = {Postprocessing the {G\"{a}lerkin} method: {A} novel approach to approximate inertial manifolds},
journal = {SIAM J. Numer. Anal.},
year = {1998},
volume = {35},
pages = {941--972},
abstract = {{G}{\"{a}}lerkin method}
}
@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--270},
doi = {10.1051/jp4:1998636}
}
@Book{PoYu,
title = {Dynamical systems and ergodic theory},
publisher = {Cambridge Univ. Press},
year = {1998},
author = {Pollicott, M. and Yuri, M.},
isbn = {9780521575997},
address = {Cambridge}
}
@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},
url = {http://www.jstor.org/stable/2006982}
}
@Book{PP90,
title = {{Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics}},
publisher = {Ast{\'e}risque, Soci{\'e}t{\'e} math{\'e}matique de France},
year = {1990},
author = {Parry, W. and Pollicott, M.},
volume = {187--188},
url = {http://smf4.emath.fr/en/Publications/Asterisque/1990/187-188/html/smf_ast_187-188.html}
}
@Book{PPRS92,
title = {Encounter with Chaos - {Self} Organised Hierarchical Complexity in Semiconductor Experiments},
publisher = {Springer},
year = {1992},
author = {J. Peinke and J. Parisi and O. E. R\"ossler and R. Stoop},
address = {New York}
}
@Article{pre88top,
author = {Cvitanovi{\'c}, P. and Gunaratne, G. H. and Procaccia, I.},
title = {Topological and metric properties of {H{\'e}non}-type strange attractors},
journal = {Phys. Rev. A},
year = {1988},
volume = {38},
pages = {1503--1520},
doi = {10.1103/PhysRevA.38.1503}
}
@Book{Press86,
title = {Numerical Recipes},
publisher = {Cambridge Univ. Press},
year = {2007},
author = {W. H. Press and B. P. Flannery, and S. A. Teukolsky and W. T. Vetterling},
isbn = {9780521880688},
address = {Cambridge UK},
edition = {3}
}
@Book{Press96,
title = {Numerical Recipes in Fortran},
publisher = {Cambridge Univ. Press},
year = {1996},
author = {W. H. Press and B. P. Flannery and S. A. Teukolsky and W. T. Vetterling},
address = {Cambridge}
}
@Article{Proctor88,
author = {Proctor, R. A.},
title = {Odd symplectic groups},
journal = {Inv. Math.},
year = {1988},
volume = {92},
pages = {307--332}
}
@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,
author = {D. Pingel and P. Schmelcher and F. K. Diakonos},
title = {Detecting unstable periodic orbits in chaotic continuous-time dynamical systems},
journal = {Phys. Rev. E},
year = {2001},
volume = {64},
number = {2},
pages = {026214},
abstract = {Change the stability of the fixed points on the
Poincar{\'e} section to find periodic orbits.}
}
@Article{PSDB00,
author = {Detlef Pingel and P. Schmelcher and F. K. Diakonos and O. Biham},
title = {Theory and applications of the systematic detection of unstable periodic orbits in dynamical systems},
journal = {Phys. Rev. E},
year = {2000},
volume = {62},
pages = {2119},
addendum = {\arXiv{nlin.CD/0006011}}
}
@Misc{PSSTV02,
author = {Polymilis and G. Servizi and Ch. Skokos and G. Turchetti and M. N. Vrahatis},
title = {Locating periodic orbits by {Topological Degree} theory},
year = {2002},
addendum = {\arXiv{nlin/0211044}}
}
@Article{pugh67cl,
author = {C. Pugh},
title = {An improved closing lemma and a general density theorem},
journal = {Amer. J. Math.},
year = {1967},
volume = {89}
}
@Article{R1883,
author = {O. Reynolds},
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},
journal = {Proc. Roy. Soc. Lond. Ser A},
year = {1883},
volume = {174},
pages = {935--982}
}
@Article{R1894,
author = {Osborne Reynolds},
title = {On the dynamical theory of incompressible viscous flows and the determination of the criterion},
journal = {Proc. Roy. Soc. Lond. Ser A},
year = {1894},
volume = {186},
pages = {123--161}
}
@Article{RaHaAb96,
author = {Rauh, A. and Hannibal, L. and Abraham, N. B.},
title = {Global stability properties of the complex {Lorenz} model},
journal = {Physica D},
year = {1996},
volume = {99},
pages = {45--58}
}
@Article{RamSri00,
author = {K. Ramasubramanian and M.S. Sriram},
title = {A comparative study of computation of {Lyapunov} spectra with different algorithms},
journal = {Physica D},
year = {2000},
volume = {139},
pages = {72--86},
doi = {10.1016/S0167-2789(99)00234-1},
issn = {0167-2789}
}
@Article{Rand82,
author = {Rand, D.},
title = {Dynamics and symmetry - predictions for modulated waves in rotating fluids},
journal = {Arch. Rational Mech. Anal.},
year = {1982},
volume = {79},
pages = {1--3}
}
@Book{Rayleigh1877I,
title = {{The Theory of Sound}},
publisher = {Macmillan},
year = {1877},
author = {Rayleigh, J. W. S.},
volume = {1},
address = {London},
url = {https://archive.org/details/theorysound06raylgoog}
}
@Book{Rayleigh1896II,
title = {{The Theory of Sound}},
publisher = {Macmillan},
year = {1896},
author = {Rayleigh, J. W. S.},
volume = {2},
address = {London},
url = {https://archive.org/details/theoryofsound02raylrich}
}
@Article{Rayleigh83,
author = {J.S. Rayleigh},
title = {On the crispation of fluid resting upon vibrating support},
journal = {Philos. Mag.},
year = {1883},
volume = {16},
pages = {50--58},
url = {https://archive.org/details/londonedinburg5161883lond}
}
@Article{rcd,
author = {O. B. Isaeva and S. P. Kuznetsov},
title = {On scaling properties of two-dimensional maps near the accumulation point of the period-tripling cascade},
journal = {Regul. Chaotic Dyn.},
year = {2000},
volume = {5},
pages = {459--476}
}
@Article{RCLdisc,
author = {J. M. Robbins and S. C. Creagh and R. G. Littlejohn},
journal = {Phys. Rev. A},
year = {1989},
volume = {39},
pages = {2838}
}
@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},
abstract = {This paper presents a methodology to study the role played
by nonattracting chaotic sets called chaotic saddles in chaotic
transitions of high-dimensional dynamical systems. Our methodology is
applied to the Kuramoto-Sivashinsky equation. The paper describes a novel technique
that uses the stable manifold of a chaotic saddle to characterize the
homoclinic tangency responsible for an interior crisis, a chaotic
transition that results in the enlargement of a chaotic attractor. The
numerical techniques explained here are important to improve the
understanding of the connection between low-dimensional chaotic systems
and spatiotemporal systems which exhibit temporal chaos and spatial
coherence.}
}
@Article{ReChMi07,
author = {Rempel, E. L. and Chian, A. C. and Miranda, R. A.},
title = {Chaotic saddles at the onset of intermittent spatiotemporal chaos},
journal = {Phys. Rev. E},
year = {2007},
volume = {76},
pages = {056217},
doi = {10.1103/PhysRevE.76.056217},
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,
author = {Rempel, E. L. and Chian, A. C.},
title = {Intermittency induced by attractor-merging crisis in the {Kuramoto-Sivashinsky} equation},
journal = {Phys. Rev. E},
year = {2005},
volume = {71},
pages = {016203},
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.}
}
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volume = {98},
pages = {014101},
abstract = {Nonattracting chaotic sets (chaotic saddles) are shown to
be responsible for transient and intermittent dynamics in an extended
system exemplified by a nonlinear regularized long-wave equation,
relevant to plasma and fluid studies. As the driver amplitude is
increased, the system undergoes a transition from quasiperiodicity to
temporal chaos, then to spatiotemporal chaos. The resulting
intermittent time series of spatiotemporal chaos displays random
switching between laminar and bursty phases. We identify temporally and
spatiotemporally chaotic saddles which are responsible for the laminar
and bursty phases, respectively. Prior to the transition to
spatiotemporal chaos, a spatiotemporally chaotic saddle is responsible
for chaotic transients that mimic the dynamics of the post-transition
attractor.}
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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}
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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.
}
}
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title = {Microscopic Chaos, Fractals and Transport in Nonequilibrium Statistical Mechanics},
publisher = {World Scientific},
year = {2007},
author = {R. Klages},
isbn = {978-981-256-507-5},
address = {Singapore}
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title = {{Deterministic Diffusion in One-dimensional Chaotic Dynamical Systems}},
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year = {1996},
author = {R. Klages},
address = {Berlin}
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author = {R. Mainieri},
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publisher = {Niels Bohr Inst.},
year = {2016},
chapter = {{A} brief history of chaos},
address = {Copenhagen},
url = {http://ChaosBook.org/paper.shtml#appendHist}
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@Article{RoSa00,
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journal = {Science},
year = {2000},
volume = {290},
pages = {2323--2326},
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.}
}
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year = {2014},
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@PhdThesis{RosenqvThesis,
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school = {Copenhagen Univ.},
year = {1995},
address = {Copenhagen},
url = {ChaosBook.org/projects/theses.html}
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journal = {Nature},
year = {1999},
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pages = {770--772},
doi = {10.1038/44529},
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.}
}
@Misc{Rothman06,
author = {D. Rothman},
title = {Nonlinear Dynamics I: Chaos},
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year = {2003},
volume = {16},
pages = {1257--1275},
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}
}
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@Article{rue04ne,
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journal = {Physics Today},
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number = {5},
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abstract = {New ideas of nonequilibrium physics are introduced.
Emphasize the SRB measure approach and energy
fluctuation theorem.}
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issn = {00029327, 10806377},
publisher = {Johns Hopkins University Press}
}
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author = {D. Ruelle},
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year = {1987},
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year = {1983},
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year = {1982},
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@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}}
}
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@Article{Ruelle09,
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year = {1976},
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year = {1986},
pages = {281}
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@Inproceedings{Ruelle91,
author = {D. Ruelle},
title = {Dynamical Zeta Functions: Where Do They Come from and What Are They Good for?},
booktitle = {Mathematical Physics X},
year = {1992},
isbn = {978-3-642-77305-1},
pages = {43--51},
address = {Berlin},
publisher = {Springer},
doi = {10.1007/978-3-642-77303-7_4}
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author = {Ruelle, D.},
title = {Functional determinants related to dynamical systems and the thermodynamic formalism},
institution = {Inst. Hautes Etud. Sci.},
year = {1995},
number = {IHES-P-95-30},
address = {Bures-sur-Yvette},
url = {http://cds.cern.ch/record/281769}
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addendum = {\arXiv{chao-dyn/9610011}},
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author = {Rugh, H. H.},
title = {Time Evoluation and Correlations in Chaotic Dynamical Systems},
school = {Univ. of Copenhagen},
year = {1992},
address = {Copenhagen}
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author = {M. Rumberger and J. Scheurle},
title = {The orbit space method: theory and application},
booktitle = {Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems},
publisher = {Springer},
year = {2001},
editor = {B. Fiedler},
address = {New York},
url = {http://dynamics.mi.fu-berlin.de/danse/}
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address = {Cambridge},
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year = {2011},
volume = {CORFU2011},
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addendum = {\arXiv{1204.5772}}
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pages = {329--341},
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.}
}
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publisher = {Springer},
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pages = {249--264},
address = {New York}
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year = {2010},
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pages = {1--33},
addendum = {\arXiv{0709.2944}},
doi = {10.1137/070705623}
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@Unpublished{SCD09b,
author = {Siminos, E. and Cvitanovi{\'c}, P. and Davidchack, R. L.},
title = {Recurrent spatio-temporal structures of translationally invariant {Kuramoto-Sivashinsky} flow},
note = {{I}n preparation},
year = {2010}
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school = {Univ. K{\"{o}}ln},
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@PhdThesis{Sieber91,
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year = {1991},
comment = {DESY report 91-030},
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pages = {248--266},
doi = {10.1016/0167-2789(90)90058-W},
abstract = {We study the hyperbola billiard, a strongly chaotic
system whose classical dynamics is the free motion of a particle with
elastic reflections on the boundary. The corresponding quantum
mechanical problem is to determine the bound state energies as
eigenvalues of the Dirichlet Laplacian on D. It is shown that the
classical periodic orbits of the hyperbola billiard can be
effectively enumerated by a ternary code. Combining this code with an
extremum principle, we are able to determine with high precision more
than 500 000 primitive periodic orbits together with their lengths,
multiplicities and {Lyapunov} exponents. The statistical properties of
the length spectrum of the periodic orbits are found to be consistent
with a random walk model, which in turn predicts asymptotically an
exponential proliferation of long periodic orbits and leads to a
novel formula for the topological entropy, whose value turns out to
be approximately 0.6. The periodic orbits are used for a quantitative
test of Gutzwiller's periodic-orbit theory, which plays the role of a
semiclassical quantization rule. We find that the predictions of the
periodic-orbit theory for the Gaussian level density agree at low
energies surprisingly well with the tru results obtained from a
numerical solution of the Schr\"odinger equation.}
}
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@Unpublished{siminos11,
author = {Siminos, E. and B{\'e}nisti, D. and Gremillet, L.},
title = {Stability of nonlinear {Vlasov} equilibria through spectral deformation and {Fourier-{Hermit}e} expansion},
note = {Submitted},
year = {2010}
}
@PhdThesis{SiminosThesis,
author = {Siminos, E.},
title = {Recurrent Spatio-temporal Structures in Presence of Continuous Symmetries},
school = {School of Physics, Georgia Inst. of Technology},
year = {2009},
address = {Atlanta},
url = {http://ChaosBook.org/projects/theses.html}
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address = {Gif-sur-Yvette, France},
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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}},
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author = {Ya. G. Sinai},
address = {Princeton}
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@Incollection{Skokos08,
author = {Skokos, Ch.},
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.},
isbn = {978-3-642-04457-1},
pages = {63--135},
address = {New York},
addendum = {\arXiv{0811.0882}},
doi = {10.1007/978-3-642-04458-8_2}
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@Book{Smale00,
title = {The Collected Papers of {Stephen Smale}},
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author = {S. Smale and F. Cucker and R. Wong},
isbn = {978-981-02-4307-4},
address = {Singapore}
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@Article{Smale70II,
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@Book{Smale80,
title = {The Mathematics of Time},
publisher = {Springer},
year = {1980},
author = {S. Smale},
address = {New York},
annote = {Subtitle: Essays on Dynamical Systems, Economic Processes and related topics}
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@Article{SMcgl02,
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title = {Stable and chaotic solutions of the complex {Ginzburg-Landau} equation with periodic boundary conditions},
journal = {Physica D},
year = {2002},
volume = {161},
pages = {102--115},
abstrac = {The behavior of the solutions of the CGLe without diffusion is studied with a periodic boundary condition.}
}
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year = {1941},
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year = {1994},
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@Book{Sou97,
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year = {1997},
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address = {Boston}
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@Article{Spalart91,
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year = {1991},
volume = {434},
pages = {211--216},
abstract = {A brief survey of modelling equations of fully
developed homogeneous turbulence. Kolmogorov's original
paper is translated.}
}
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title = {The {Lorenz} Equations: {Bifurcations}, Chaos and Strange Attractors},
publisher = {Springer},
year = {1982},
author = {C. Sparrow},
address = {New York}
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@Article{spercgl,
author = {W. Sch{\"{o}}pf and L. Kramer},
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year = {1991},
volume = {66},
pages = {2316}
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@Book{spfunc1,
title = {Special Functions: An Introduction to the Classical Functions of Mathematical Physics},
publisher = {Wiley},
year = {1996},
author = {N. M. Temme},
address = {New York}
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@Misc{SpiekerBlog,
author = {W. Spieker},
title = {Gone fishin' - a blog},
howpublished = {School of Physics, Georgia Inst. of Technology, (Atlanta, )},
year = {2011},
url = {http://ChaosBook.org/projects/Spieker}
}
@Book{sprott03,
title = {Chaos and Time-series Analysis},
publisher = {Oxford University Press},
year = {2003},
author = {Sprott, J. C.},
isbn = {9780198508403},
address = {Oxford}
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@Article{Spruill07,
author = {Spruill, M. C.},
title = {Asymptotic distribution of coordinates on high dimensional spheres},
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year = {2007},
volume = {12},
pages = {234--247}
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@Article{SPScgl92,
author = {B. I. Shraiman and A. Pumir and van Saarloos, W. and Hohenberg, P. C. and H. Chat{\'e} and M. Holen},
title = {Spatiotemporal chaos in the one-dimensional complex {Ginzburg-Landau} equation},
journal = {Physica D},
year = {1992},
volume = {57},
pages = {241--248}
}
@Inproceedings{SreeVanDoo93,
author = {Sreedhar, J. and Van Dooren, P.},
title = {Discrete-time periodic systems: a {Floquet} approach},
booktitle = {Proceed. 1993 Conference on Information Sciences \& Systems, Johns Hopkins Univ.},
year = {1993}
}
@Inproceedings{SreeVanDoo93a,
author = {Sreedhar, J. and Van Dooren, P.},
title = {Pole placement via the periodic {Schur} decomposition},
booktitle = {American Control Conference, 1993},
year = {1993},
pages = {1563--1567},
abstract = {We present a new method for eigenvalue assignment in linear
periodic discrete-time systems through the use of linear periodic state
feedback. The proposed method uses reliable numerical techniques based on
unitary transformations. In essence, it computes the Schur form of the
open-loop monodromy matrix via a recent implicit eigen-decomposition
algorithm, and shifts its eigenvalues sequentially. Given complete
reachability of the open-loop system, we show that we can assign an
arbitrary set of eigenvalues to the closed-loop monodromy matrix in this
manner. Under the weaker assumption of complete control-lability, this
method can be used to place all eigenvalues at the origin, thus solving
the so-called deadbeat control problem. The algorithm readily extends to
more general situations, such as when the system equation is given in
descriptor form.}
}
@InProceedings{SreeVanDoo93b,
author = {Sreedhar, J. and Van Dooren, P.},
title = {Solution of periodic discrete-time {Riccati} and {Lyapunov} equations},
booktitle = {Decision and Control, 1993., Proceedings of the 32\textsuperscript{nd} IEEE Conference on},
year = {1993},
volume = {1},
pages = {361--362},
abstract = {We present a Schur method for the solution of periodic discrete-time Riccati and {Lyapunov} equations. The method can be easily extended to solve the corresponding implicit equations which arise from generalized state space systems},
doi = {10.1109/CDC.1993.325128}
}
@Article{SrViWi95,
author = {Srivastava, Y.N. and Vitiello, G. and Widom, A.},
title = {Quantum Dissipation and Quantum Noise},
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volume = {238},
pages = {200--207}
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author = {Stoop, R. and Steeb, W.-H.},
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journal = {Phys. Rev. E},
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volume = {55},
pages = {7763--7766},
doi = {10.1103/PhysRevE.55.7763}
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@Inbook{S-S97,
chapter = {Geometric integration},
pages = {121--143},
title = {The State of the Art in Numerical Analysis},
publisher = {Clarendon Press},
year = {1997},
author = {J. M. Sanz-Serna},
editor = {I. S. Duff and G. A. Watson},
address = {Oxford UK}
}
@Article{SSL10,
author = {G. S{\'a}nchez-Arriaga and E. Siminos and E. Lefebvre},
title = {Solitary waves with nonvanishing boundary conditions in relativistic plasmas},
journal = {submitted},
year = {2010},
url = {cns.gatech.edu/~siminos/papers/SSL10.pdf}
}
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volume = {LI},
number = {2},
pages = {265--281}
}
@Unpublished{stadium95,
author = {Hansen, K. T. and Cvitanovi{\'c}, P.},
title = {Symbolic dynamics and {Markov} partitions for the stadium billiard},
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title = {Topics in applied mathematics},
note = {MIT course 18.325},
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url = {http://math.mit.edu/~stevenj/18.325/representation-theory.pdf}
}
@Book{StGo09,
title = {{Mathematics for Physics: A Guided Tour for Graduate Students}},
publisher = {Cambridge Univ. Press},
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author = {Stone, M. and Goldbart, P.},
isbn = {9780521854030},
address = {Cambridge}
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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}{\'e}non map is
used to demonstrate the idea.}
}
@Book{Stockmann07,
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note = {Laurea thesis}
}
@Book{strogb,
title = {{Nonlinear Dynamics and Chaos}},
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isbn = {9780813349107},
address = {Boulder, CO}
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}
@Article{SumRules,
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journal = {Nonlinearity},
year = {1998},
volume = {11},
pages = {1209--1232},
addendum = {\arXiv{chao-dyn/9712002}},
doi = {10.1088/0951-7715/11/5/003},
abstract = {The global constraints on chaotic dynamics induced by the
analyticity of smooth flows are used to dispense with individual
periodic orbits and derive infinite families of exact sum rules for
several simple dynamical systems. The associated {Fredholm} determinants
are of particularly simple polynomial form. The theory developed
suggests an alternative to the conventional periodic orbit theory
approach for determining eigenspectra of transfer operators. AMS
classification scheme numbers: 58F20}
}
@Inproceedings{sunil_ahuja_template-based_2006,
author = {S. Ahuja and {I.G.} Kevrekidis and {C.W.} Rowley},
title = {Template-based stabilization of relative equilibria},
booktitle = {2006 American Control Conference, 14-16 June 2006},
year = {2006},
pages = {6},
address = {Piscataway, NJ},
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.},
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journal = {Math. Proc. Cambridge Philos. Soc.},
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pages = {385}
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@Incollection{symb_dyn,
author = {P. Cvitanovi{\'c}},
booktitle = {Chaos: Classical and Quantum},
publisher = {Niels Bohr Inst.},
year = {2016},
chapter = {{Charting} the state space},
address = {Copenhagen},
url = {http://ChaosBook.org/paper.shtml#knead}
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@Unpublished{szendro-2007,
author = {I. G. Szendro and D. Pazo and M. A. Rodriguez and J. M. Lopez},
title = {Spatiotemporal structure of {Lyapunov} vectors in chaotic coupled-map lattices},
year = {2007},
abstract = {useful list of references on {Lyapunov} vectors},
addendum = {\arXiv{0706.1706}}
}
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title = {Chaos and Integrability in Nonlinear Dynamics: An Introduction},
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addendum = {\arXiv{1207.5571}},
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year = {2009},
volume = {103},
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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.},
addendum = {\arXiv{0907.4298}},
doi = {10.1103/PhysRevLett.103.154103}
}
@Article{TaGiCh11,
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url = {http://terrytao.wordpress.com/}
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title = {Chaotic Dynamics: An Introduction Based on Classical Mechanics},
publisher = {Cambridge Univ. Press},
year = {2006},
author = {T{\'e}l, T. and Gruiz, M.},
isbn = {9780521547833},
address = {Cambridge UK}
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abstract = {Estimates on the dimension of the attractors and the
decay rates of Fourier mode of NSe are given. By
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between the large and small eddies are derived. Such
laws open the way to numerically efficient algorithms
via multiscale methods.}
}
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}
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author = {D. Terhesiu and G. Froyland},
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number = {9},
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doi = {10.1088/0951-7715/21/9/001}
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year = {1997},
volume = {55},
pages = {5073},
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.}
}
@Book{thi,
title = {Quantum Mechanics of Atoms and Molecules},
publisher = {Springer},
year = {1979},
author = {W. Thirring},
volume = {3},
address = {New York},
note = {Page 48, Sects. 2.3.21 and 2.3.22 .}
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title = {Derivatives and constraints in chaotic flows: asymptotic behaviour and a numerical method},
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volume = {172},
pages = {139--161},
addendum = {\arXiv{nlin/0101012}},
doi = {10.1016/S0167-2789(02)00588-2}
}
@Article{ThiBoo99,
author = {Thiffeault, {J.-L.} and Boozer, A. H.},
title = {Geometrical constraints on finite-time {Lyapunov} exponents in two and three dimensions},
journal = {Chaos},
year = {2001},
volume = {11},
pages = {16--28},
addendum = {\arXiv{physics/0009017}},
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{Thom1,
author = {J. M. T. Thompson and R. Ghaffari},
journal = {Phys. Lett. A},
year = {1982},
volume = {91},
pages = {5}
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pages = {191--198}
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@Unpublished{tHooft00,
author = {'t Hooft, G.},
title = {Determinism and dissipation in quantum gravity},
year = {1999},
addendum = {\arXiv{hep-th/0003005}},
pages = {397--413}
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author = {'t Hooft, G.},
title = {Determinism in free bosons},
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year = {2003},
volume = {42},
pages = {355--361},
addendum = {\arXiv{hep-th/0104080}},
doi = {10.1023/A:1024459703072}
}
@Unpublished{tHooft01b,
author = {'t Hooft, G.},
title = {Quantum mechanics nd determinism},
year = {2001},
addendum = {\arXiv{hep-th/0105105}}
}
@Article{tHooft07,
author = {'t Hooft, G.},
title = {A mathematical theory for deterministic quantum mechanics},
journal = {J. Phys.: Conf. Series},
year = {2007},
volume = {67},
pages = {012015},
doi = {10.1088/1742-6596/67/1/012015}
}
@Article{tHooft99,
author = {'t Hooft, G.},
title = {Quantum gravity as a dissipative deterministic system},
journal = {Classical and Quantum Gravity},
year = {1999},
volume = {16},
pages = {3263--3279},
doi = {10.1088/0264-9381/16/10/316}
}
@Book{tHooftVelt73,
title = {{DIAGRAMMAR}},
publisher = {CERN},
year = {1973},
author = {'t Hooft, G. and Veltman, M.},
address = {Geneva},
url = {www.library.uu.nl/digiarchief/dip/dispute/2002-0718-131954/Diagrammar - 1973.pdf}
}
@Book{ThoSte02,
title = {Nonlinear Dynamics and Chaos: Geometrical Methods for Engineers and Scientists},
publisher = {Wiley},
year = {2002},
author = {J. M. T. Thompson and H. B. Stewart},
address = {New York}
}
@PhdThesis{Thum05,
author = {V. Th\"ummler},
title = {Numerical Analysis of the Method of Freezing Traveling Waves},
school = {Bielefeld Univ.},
year = {2005}
}
@Book{Tinkham,
title = {{Group Theory and Quantum Mechanics}},
publisher = {Dover},
year = {2003},
author = {Tinkham, M.},
isbn = {9780486131665},
address = {New York}
}
@Article{ToDe94,
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title = {Geometric phases in lasers and liquid flows},
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year = {1994},
volume = {49},
pages = {1392--1399},
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,
author = {Vladislav Yu. Toronov and Vladimir L. Derbov},
title = {Geometric-phase effects in laser dynamics},
journal = {Phys. Rev. A},
year = {1994},
volume = {50},
pages = {878--881},
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{ToDe97a,
author = {Toronov, V. Y. and Derbov, V. L.},
title = {Boundedness of attractors in the complex {Lorenz} model},
journal = {Phys. Rev. E},
year = {1997},
volume = {55},
pages = {3689--3692}
}
@Article{ToDe98,
author = {Toronov, V. Y. and Derbov, V. L.},
title = {Topological properties of laser phase},
journal = {J. Optical Soc. of America B},
year = {1998},
volume = {15},
pages = {1282--1290}
}
@Article{Tomlinson29,
author = {G. A. Tomlinson},
journal = {Philos. Mag. Ser. 7},
year = {1929},
volume = {7},
pages = {905--939},
doi = {10.1080/14786440608564819}
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@Article{tompaid96,
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year = {1996},
volume = {5},
pages = {211--230},
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.}
}
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volume = {77},
pages = {1047},
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,
author = {P. Tak\'{a}\v{c}},
title = {Invariant 2-tori in the time-dependent {Ginzburg-Landau} equation},
journal = {Nonlinearity},
year = {1992},
volume = {5},
pages = {289--321}
}
@Inbook{torino,
chapter = {Chaos for cyclists},
publisher = {Cambridge Univ. Press},
year = {1989},
author = {P. Cvitanovi{\'c}},
editor = {E. Moss and L. A. Lugiato and W. Schleich},
isbn = {9780521384179},
address = {Cambridge},
booktitle = {Noise and Chaos in Nonlinear Dynamical Systems}
}
@Book{Townsend76,
title = {The Structure of Turbulent Shear Flows},
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author = {A. A. Townsend},
address = {Cambridge},
edition = {2}
}
@Article{transgol,
author = {R. Ramshankar and J. P. Gollub},
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journal = {Phys. Fluids A},
year = {1991},
volume = {3},
pages = {1344}
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@Book{Trefethen97,
title = {Numerical Linear Algebra},
publisher = {SIAM},
year = {1997},
author = {L. N. Trefethen and D. Bau},
addr = {Philadelphia}
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@Article{TrePan98,
author = {Trevisan, A. and Pancotti, F.},
title = {Periodic orbits, {Lyapunov} vectors, and singular vectors in the {Lorenz} system},
journal = {J. Atmos. Sci.},
year = {1998},
volume = {55},
pages = {390},
doi = {10.1175/1520-0469(1998)055%3C0390:POLVAS%3E2.0.CO;2},
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.}
}
@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},
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.
},
addendum = {\arXiv{0902.2714}}
}
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year = {2002},
volume = {16},
pages = {91--97},
addendum = {\arXiv{physics/0312051}},
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.}
}
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title = {Heteroclinic Cycles and Periodic Orbits in Reversible Systems},
publisher = {Pitman},
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author = {Vanderbauwhede, A.},
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pages = {250}
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title = {Local Bifurcation and Symmetry},
publisher = {Pitman},
year = {1982},
author = {Vanderbauwhede, A.},
address = {Boston}
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@PhdThesis{vanderbauwhede_lokale_1980,
author = {Vanderbauwhede, A.},
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volume = {4},
pages = {685--700}
}
@Unpublished{vanderbauwhede_short_1997,
author = {Vanderbauwhede, A.},
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year = {1997}
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discrete-time counterpart of the classical theory of Floquet transforms
developed by Floquet and {Lyapunov} in the 1800s for continuous-time
systems. We state and prove a necessary and sufficient condition for a
{\textquotedblleft}discrete-time Floquet transform{\textquotedblright} to exist, and give a construction for
the transform when it does exist. Our results also extend to generalized
state-space, or descriptor, systems. },
doi = {10.1016/0024-3795(94)90400-6}
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title = {On balancing and order reduction of unstable periodic systems},
booktitle = {Periodic Control Systems 2001},
year = {2001},
editor = {S. Bittanti and P. Colaneri},
isbn = {978-0-08-043682-1},
pages = {177--182},
address = {New York},
organization = {IFAC Workshop},
publisher = {Elsevier},
url = {http://elib.dlr.de/22687/1/varga_como01p2.pdf}
}
@Inproceedings{Varga05,
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year = {2005},
volume = {16},
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@Inproceedings{Varga11,
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year = {2011}
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year = {2001},
editor = {S. Bittanti and P. Colaneri},
isbn = {978-0-08-043682-1},
pages = {171--176},
address = {New York},
organization = {IFAC Workshop},
publisher = {Elsevier},
url = {http://elib.dlr.de/11743/1/varga_como01p1.pdf}
}
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addendum = {\arXiv{0801.1918}},
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volume = {21}
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volume = {52},
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addendum = {\arXiv{0901.4968}}
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@Incollection{Visw07a,
author = {D. Viswanath},
title = {The dynamics of transition to turbulence in plane {Couette} flow},
booktitle = {Mathematics and Computation, a Contemporary View. The Abel Symposium 2006},
publisher = {Springer},
year = {2008},
volume = {3},
series = {Abel Symposia},
address = {New York},
addendum = {\arXiv{physics/0701337}}
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author = {D. Viswanath},
title = {Recurrent motions within plane {Couette} turbulence},
journal = {J. Fluid Mech.},
year = {2007},
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author = {D. Viswanath},
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year = {2008},
note = {In press.}
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editor = {A. Knauf and Ya. G. Sinai},
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title = {{Stochastic Processes in Physics and Chemistry}},
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year = {1998},
volume = {8},
pages = {723--729},
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,
author = {Vladimirov, A. G. and Toronov, V. Y. and Derbov, V. L.},
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abstract = {Based on the experimentally generated velocity field,
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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
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It can be used as a discretized version of the similar problem in the
continuous regime.}
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author = {F. Waleffe},
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editor = {Shigeo Kida},
pages = {115--128},
publisher = {National Center of Sciences, Tokyo, Japan}
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author = {F. Waleffe},
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author = {F. Waleffe},
title = {Proposal for a {Self-Sustaining Process} in shear flows},
note = {{Center for Turbulence Research, Stanford University/NASA Ames, unpublished preprint (1990)}},
year = {1990},
url = {www.math.wisc.edu/~waleffe/ECS/sspctr90.pdf}
}
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author = {Watkins, D. S.},
isbn = {978-0-470-52833-4},
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pages = {033029},
addendum = {\arXiv{1311.5128}},
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abstract = {In ballistic open quantum systems, one often observes that
the resonances in the complex-energy plane form a clear chain
structure. Taking the open three-disk system as a paradigmatic model
system, we investigate how this chain structure is reflected in the
resonance states and how it is connected to the underlying classical
dynamics. Using an efficient scattering approach, we observe that
resonance states along one chain are clearly correlated, while
resonance states of different chains show an anticorrelation. Studying
the phase-space representations of the resonance states, we find that
their localization in phase space oscillates between different regions
of the classical trapped set as one moves along the chains, and that
these oscillations are connected to a modulation of the resonance
spacing. A single resonance chain is thus not a WKB quantization of a
single periodic orbit, but the structure of several oscillating chains
arises from the interaction of several periodic orbits. We illuminate
the physical mechanism behind these findings by combining the
semiclassical cycle expansion with a quantum graph model.}
}
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@PhdThesis{Weich14,
author = {Weich, T.},
title = {Singular Equivariant Spectral Asymptotics of Schr{\"o}dinger Operators in R\^{} n and Resonances of {Schottky} Surfaces},
school = {Philipps-Universit{\"a}t Marburg},
year = {2014}
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@Unpublished{WFSBC15,
author = {Willis, A. P. and Farazmand, M. and Short, K. Y. and Budanur, N. B. and Cvitanovi{\'c}, P.},
title = {Relative periodic orbits form the backbone of turbulent pipe flow},
note = {In preparation.},
year = {2016}
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@Book{wiggins1992,
title = {Chaotic Transport in Dynamical Systems},
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address = {N.Y.}
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@Book{Wigner59,
title = {{Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra}},
publisher = {Academic},
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isbn = {9780323152785},
address = {New York}
}
@Misc{wikiHamMat,
author = {Wikipedia},
title = {Hamiltonian matrix},
url = {{en.wikipedia.org/wiki/Hamiltonian\_matrix}}
}
@Misc{wikiRossMap,
author = {Wikipedia},
title = {R\"ossler attractor},
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@Unpublished{Wilczak09,
author = {R. Wilczak},
title = {Reducing the state-space of the complex {Lorenz} flow},
note = {{NSF REU} summer 2009 project, {U. of Chicago}},
year = {2009},
url = {http://ChaosBook.org/projects/Wilczak/blog.pdf}
}
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author = {Wilkinson, A.},
title = {Smooth ergodic theory},
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publisher = {Springer},
year = {2009},
editor = {Meyers, R. A.},
isbn = {978-0-387-75888-6},
pages = {8168--8183},
address = {New York},
doi = {10.1007/978-0-387-30440-3_484},
url = {http://www.math.uchicago.edu/~wilkinso/papers/smoothergodictheory.pdf}
}
@Book{Wimberger14,
title = {{Nonlinear Dynamics and Quantum Chaos}},
publisher = {Springer},
year = {2014},
author = {Wimberger, S.},
address = {New York}
}
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addendum = {\arXiv{chao-dyn/9712015}},
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@Unpublished{wirzbaCasTalk,
author = {A. Wirzba},
title = {A force from nothing into nothing: {Casimir} interactions},
year = {2003)},
addendum = {\\ \wwwcb{/projects/Wirzba/openfull.ps.gz} overheads}
}
@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},
addendum = {\arXiv{1504.05825}},
doi = {10.1103/PhysRevE.93.022204}
}
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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},
abstract = {Using a wavelet basis, the spatiotemporally chaotic
regime of the KSe is explored where a good seperation
of scales is observed. In large scales, the dynamics is
Gaussian. In the intermediate scales, the dynamics is
reminiscent of travelling waves and heteroclinic cyces
which is the typical behavior for small system size. In
the small scales, the dynamics is intermittent. Through
investigation of the interaction between different
scales, we see the intermediate structures give the
defining shape of the cell and the large scales trigger
the spatiotemporal chaos. The small scales dissipate
energy and modify the background in a average sense.}
}
@Article{WiVeBeBo04,
author = {D. A. Wisniacki and E. Vergini and R. M. Benito and F. Borondo},
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pages = {035202}
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journal = {J. Fluid Mech.},
year = {2004},
volume = {508},
pages = {333--371},
doi = {10.1017/S0022112004009346}
}
@Inproceedings{WKH93,
author = {F. Waleffe and J. Kim and J. Hamilton},
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year = {1993},
editor = {F. Durst and R. Friedrich and B. E. Launder and F. W. Schmidt and U. Schumann and J. H. Whitelaw},
pages = {37--49},
address = {New York},
publisher = {Springer}
}
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@Article{WolfSwift85,
author = {Wolf, A. and Swift, J. B. and Swinney, H. L. and Vastano, J. A.},
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journal = {Physica D},
year = {1985},
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pages = {285--317},
doi = {10.1016/0167-2789(85)90011-9}
}
@Article{WoSa07,
author = {Wolfe, C. L. and Samelson, R. M.},
title = {An efficient method for recovering {Lyapunov} vectors from singular vectors},
journal = {Tellus A},
year = {2007},
volume = {59},
pages = {355--366},
doi = {10.1111/j.1600-0870.2007.00234.x},
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,
author = {Wintgen, D. and Richter, K. and Tanner, G.},
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journal = {Chaos},
year = {1992},
volume = {2},
pages = {19--33},
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@Article{Wulff00,
author = {C. Wulff},
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@Article{Wulff01,
author = {Wulff, C. and Lamb, J. S. W. and Melbourne, I.},
title = {Bifurcation from relative periodic solutions},
journal = {Ergod. Theor. Dynam. Syst.},
year = {2001},
volume = {21},
pages = {605--635}
}
@Article{Wulff03,
author = {C. Wulff},
title = {Persistence of {Hamiltonian} relative periodic orbits},
journal = {J. Geometry and Physics},
year = {2003},
volume = {48},
pages = {309--338}
}
@Inproceedings{WW05,
author = {F. Waleffe and J. Wang},
title = {Transition threshold and the {Self-Sustaining Process}},
booktitle = {Non-uniqueness of Solutions to the {Navier-Stokes} Equations and their Connection with Laminar-Turbulent Transition},
year = {2005},
editor = {T. Mullin and R. R. Kerswell},
pages = {85--106},
publisher = {Kluwer}
}
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author = {Xia, Z.},
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@Article{YaIsAk14,
author = {Yadykin, I. B. and Iskakov, A. B. and Akhmetzyanov, A. V.},
title = {Stability analysis of large-scale dynamical systems by sub-{Gramian} approach},
journal = {Intl. J. Robust Nonlin. Control},
year = {2014},
volume = {24},
pages = {1361--1379},
doi = {10.1002/rnc.3116},
abstract = {We consider two methods for solving
differential and algebraic {Lyapunov} equations in the time and frequency
domains. Solutions of these equations are finite and infinite Gramians.
In the first approach, we use the Laplace transform to solve the
equations, and we apply the expansion of the matrix resolvent of the
dynamical system. The expansions are bilinear and quadratic forms of the
Faddeev matrices generated by resolvents of the original matrices. The
second method allows computation of an infinite Gramian of a stable
system as a sum of sub-Gramians, which characterize the contribution of
eigenmodes to the asymptotic variation of the total system energy over an
infinite time interval. Because each sub-Gramian is associated with a
particular eigenvector, the potential sources of instability can easily
be localized and tracked in real time. When solutions of {Lyapunov} equations have low-rank structure typical of large-scale applications,
sub-Gramians can be represented in low-rank factored form, which makes
them convenient in the stability analysis of large systems. Our numerical
tests for Kundur's four-machine two-area system confirm the suitability
of using Gramians and sub-Gramians for small-signal stability analyses of
electric power systems.}
}
@Article{YaRa10,
author = {Yang, H.-l. and Radons, G.},
title = {Comparison between covariant and orthogonal {Lyapunov} vectors},
journal = {Phys. Rev. E},
year = {2010},
volume = {82},
pages = {046204},
note = {\arXiv{1008.1941}},
doi = {10.1103/PhysRevE.82.046204},
abstract = {Two sets of vectors, covariant Lyapunov vectors (CLVs) and orthogonal
Lyapunov vectors (OLVs), are currently used to characterize the linear
stability of chaotic systems. A comparison is made to show their
similarity and difference, especially with respect to the influence
on hydrodynamic Lyapunov modes (HLMs). Our numerical simulations
show that in both Hamiltonian and dissipative systems HLMs formerly
detected via OLVs survive if CLVs are used instead. Moreover, the
previous classification of two universality classes works for CLVs
as well, i.e., the dispersion relation is linear for Hamiltonian
systems and quadratic for dissipative systems, respectively. The
significance of HLMs changes in different ways for Hamiltonian and
dissipative systems with the replacement of OLVs with CLVs. For general
dissipative systems with nonhyperbolic dynamics the long-wavelength
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
and OLVs. Especially, CLVs in a conjugate pair are statistically
indistinguishable in consequence of the microreversibility of Hamiltonian
systems. Transformation properties of Lyapunov exponents, CLVs, and
hyperbolicity under changes of coordinate are discussed in appendices.}
}
@Article{YaTaGiChRa08,
author = {Yang, H.-l. and Takeuchi, K. A. and Ginelli, F. and Chat{\'e}, H. and Radons, G.},
title = {Hyperbolicity and the effective dimension of spatially-extended dissipative systems},
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year = {2009},
volume = {102},
pages = {074102},
addendum = {\arXiv{0807.5073}},
doi = {10.1103/PhysRevLett.102.074102},
abstract = {We show, using covariant {Lyapunov} vectors, that
the chaotic solutions of spatially extended dissipative
systems evolve within a manifold spanned by a finite number
of physical modes hyperbolically isolated from a set of
residual degrees of freedom, themselves individually
isolated from each other. In the context of dissipative
partial differential equations, our results imply that a
faithful numerical integration needs to incorporate at
least all physical modes and that increasing the resolution
merely increases the number of isolated modes.}
}
@Book{Yode88,
title = {Unrolling Time: {Christiaan} {Huygens} and the Mathematization of Nature},
publisher = {Cambridge Univ. Press},
year = {1988},
author = {J. G. Yoder},
isbn = {9780521524810},
address = {Cambridge}
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@Article{YoNo93,
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pages = {1531--1543},
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pages = {12--48}
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pages = {2888}
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pages = {4265},
doi = {10.1103/PhysRevLett.85.4265},
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can identify the dynamical equivalence of two signals
from one process.}
}
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@Article{ZaZha70,
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author = {P. Zgliczynski and K. Mischaikow},
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@Unpublished{ZhCvGo15,
author = {Zhang, T. and Cvitanovi\'c, P. and Goldman, D. I.},
title = {Diffuse globally, compute locally: a cyclist tale},
note = {In preparation},
year = {2016}
}
@Article{ZhLuRo01,
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@PhdThesis{Zimmermann97,
author = {Zimmermann, M.G.},
title = {Global Bifurcations and Chaotic Dynamics in Physical Applications},
school = {Uppsala Univ.},
year = {1997},
address = {Uppsala, Sweden},
abstract = {The complex interactions between spatial and temporal
components of the dynamics, known as spatiotemporal
chaos, only recently begun to be studied using ideas
from fundamental dynamical systems theory (Temam, 1988;
Goren et al. 1995; Christiansen et al. 1997). The
present dissertation is primarily interested in
studying the appearance of global bifurcations in two
specific physical examples, a laser with an injected
signal, and a catalytic reaction on a platinum surface.
Characteristic of these bifurcations is the presence of
a highly degenerate .... One of the main motivations to
study these equations is that some of them display
spatiotemporal chaos, which in many ways, resembles
turbulence in fluid dynamics. Recently, different
efforts to uncover the dynamical origin for these
complex states have been developing (Goren et al. 1995;
Christiansen et al. 1997).},
isbn = {9789155439729},
publisher = {Acta Universitatis Upsaliensis}
}
@Book{Zinn-Justin89,
title = {Quantum Field Theory and Critical Phenomena},
publisher = {Oxford University Press},
year = {1989},
author = {J. Zinn-Justin},
address = {Oxford}
}
@Article{ZumKla,
author = {Zumofen, G. and Klafter, J.},
title = {Scale-invariant motion in intermittent chaotic systems},
journal = {Phys. Rev. A},
year = {1993},
volume = {47},
pages = {851}
}
@Book{Zyg,
title = {Trigonometric Series},
publisher = {Cambridge Univ. Press},
year = {1959},
author = {A. Zygmund},
isbn = {9780521890533},
address = {Cambridge UK}
}