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@article{AACI,
Author = {R. Artuso and E. Aurell and P. Cvitanovi{\'c}},
Journal = {Nonlinearity},
Pages = {325--359},
Title = {Recycling of strange sets: {I}. Cycle expansions},
Volume = {3},
Year = {1990}}
@article{AACII,
Author = {R. Artuso and E. Aurell and P. Cvitanovi{\'c}},
Journal = {Nonlinearity},
Pages = {361},
Title = {Recycling of strange sets: {II}. Applications},
Volume = {3},
Year = {1990}}
@book{ablowbook,
Address = {Cambridge},
Author = {M. J. Ablowitz and P. A. Clarkson},
Publisher = {Cambridge Univ. Press},
Title = {Solitions, Nonlinear Evolution Equations and Inverse Scattering},
Year = {1992}}
@book{AbrMars78,
Address = {Reading, Mass.},
Author = {R. Abraham and J. E. Marsden},
Publisher = {Benjamin-Cummings},
Title = {Foundations of Mechanics},
Year = {1978}}
@article{Absh05,
Author = {J. Abshagen and J. M. Lopez and F. Marques and G. Pfister},
Journal = {J. Fluid Mech.},
Pages = {269--300},
Title = {Mode competition of rotating waves in reflection-symmetric {Taylor-Couette} flow},
Volume = {540},
Year = {2005}}
@article{Absh05a,
Author = {J. Abshagen and J. M. Lopez and F. Marques and G. Pfister},
Journal = {Phys. Rev. Lett.},
Pages = {074501},
Title = {Symmetry breaking via global bifurcations of modulated rotating waves in hydrodynamics},
Volume = {94},
Year = {2005}}
@Book{AbSh92,
title = {Dynamics - The Geometry of Behavior},
publisher = {Addison-Wesley},
year = {1992},
author = {R. H. Abraham and C. D. Shaw},
address = {Reading, MA}
}
@Article{ACHKW11,
author = {Willis, A. P. and Cvitanovi{\'c}, P. and Avila, M.},
title = {Revealing the state space of turbulent pipe flow by symmetry reduction},
journal = {J. Fluid Mech.},
year = {2013},
volume = {721},
pages = {514--540},
note = {\arXiv{1203.3701}},
doi = {10.1017/jfm.2013.75}
}
@book{Adams:2003wi,
title={Sobolev spaces},
author={Adams, R. A. and Fournier, J. J. F.},
year={2003},
publisher={Academic press},
address = {New York},
}
@article{afind,
Author = {O. Biham and W. Wenzel},
Journal = {Phys. Rev. Lett.},
Pages = {819},
Title = {Characterization of unstable periodic orbits in chaotic attractors and repellers},
Volume = {63},
Year = {1989}}
@article{Afraimovich87,
Author = {V. S. Afraimovich and B. B. Bykov and L. P. Shilnikov},
Journal = {Dokl. Akad. Nauk SSSR},
Keywords = {Lorenz, attractor},
Note = {In Russian},
Pages = {336--339},
Title = {On the appearence and the structure of the {Lorenz} attractor},
Volume = {234},
Year = {1987}}
@Article{AGHks88,
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},
abstract = {Consider the normal form on the center manifold for
the KSe and discuss bifurcations with intrinsic O(2) symmetry.}
}
@article{AGHks89,
Author = {D. Armbruster and J. Guckenheimer and P. Holmes},
Journal = {SIAM J. Appl. Math.},
Pages = {676--691},
Title = {{Kuramoto-Sivashinsky} Dynamics on the Center-Unstable manifold},
Volume = {49},
Year = {1989}}
@article{AHMNPTB03,
Author = {M. E. Brachet and M. Abid and C. Huepe and S. Metens
and C. Nore and C. T. Pham and L. S. Tuckerman},
Doi = {10.1016/j.fluiddyn.2003.09.001},
Journal = {Fluid Dyn. Research},
Pages = {509--544},
Volume = {33},
Year = {2003}
}
@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.}
}
@Book{airPollution,
title = {Fundamentals of Air Pollution},
publisher = {Academic Press},
year = {2008},
author = {Vallero, D.},
address = {New York},
edition = {4\textsuperscript{th}}
}
@article{AKcgl02,
Author = {I. S. Aranson and L. Kramer},
Journal = {Rev. Mod. Phys.},
Pages = {99--143},
Title = {The World of the Complex {Ginzburg-Landau} Equation},
Volume = {74},
Year = {2002}}
@article{AllgGeorg88,
Author = {E. L. Allgower and K. Georg},
Journal = {Lect. Appl. Math.},
Pages = {1--13},
Title = {Numerically stable homotopy methods without an extra dimension},
Url = {www.math.colostate.edu/emeriti/georg/homExtra.pdf},
Volume = {26},
Year = {1990}
}
@Article{AltTel08,
author = {Altmann, E. G. and T{\'e}l, T.},
title = {Poincar\'e recurrences and transient chaos in systems with leaks},
journal = {Phys. Rev. E},
year = {2009},
volume = {79},
pages = {016204},
note = {\arXiv{0808.3785}},
abstract = {Simulating observational and experimental
situations, we consider a leak in the phase space of a chaotic
dynamical system. We obtain an expression for the escape rate
of the survival probability applying the theory of transient
chaos. This expression improves previous estimates based on the
properties of the closed system and explains dependencies on
the position and size of the leak and on the initial ensemble.
With a subtle choice of the initial ensemble, we obtain an
equivalence to the classical problem of Poincare recurrences in
closed systems, which is treated in the same framework.
Finally, we show how our results apply to weakly chaotic
systems and justify a split of the invariant saddle in
hyperbolic and non-hyperbolic components, related,
respectively, to the intermediate exponential and asymptotic
power-law decays of the survival probability.},
numpages = {12}
}
@Article{AmbWil08,
author = {Ambrose, D. M. and Wilkening, J.},
title = {Computation of time-periodic solutions of the {Benjamin-Ono} equation},
journal = {J. Nonlinear Science},
year = {2010},
volume = {20},
pages = {277--308},
note = {\arXiv{0804.3623}},
doi = {10.1007/s00332-009-9058-x}
}
@article{AmbWil09,
title={Global paths of time-periodic solutions of the {Benjamin-Ono}
equation connecting pairs of traveling waves},
author={Ambrose, D. M. and Wilkening, J.},
journal={Commun. App. Math. and Comp. Sci.},
volume={4},
pages={177--215},
year={2009},
DOI = {10.2140/camcos.2009.4.177},
Note = {\arXiv{0811.4205}}
}
@Article{AMdABH11,
author = {Avila, K. and Moxey, D. and de Lozar, A. and Avila, M. and Barkley, D. and Hof, B.},
title = {The onset of turbulence in pipe flow},
journal = {Science},
year = {2011},
volume = {333},
pages = {192--196},
doi = {10.1126/science.1203223},
abstract = {Shear flows undergo a sudden transition from laminar to
turbulent motion as the velocity increases, and the onset of turbulence
radically changes transport efficiency and mixing properties. Even for
the well-studied case of pipe flow, it has not been possible to determine
at what Reynolds number the motion will be either persistently turbulent
or ultimately laminar. We show that in pipes, turbulence that is
transient at low Reynolds numbers becomes sustained at a distinct
critical point. Through extensive experiments and computer simulations,
we were able to identify and characterize the processes ultimately
responsible for sustaining turbulence. In contrast to the classical
Landau-Ruelle-Takens view that turbulence arises from an increase in the
temporal complexity of fluid motion, here, spatial proliferation of
chaotic domains is the decisive process and intrinsic to the nature of
fluid turbulence.},
bdsk-url-1 = {http://dx.doi.org/10.1126/science.1203223}
}
@incollection{ampatt,
Address = {Amsterdam},
Author = {M. van Hecke and P. C. Hohenberg and W. van Saarloos},
Booktitle = {Fundamental Problems in Statistical Mechanics},
Editor = {H. van Beijeren and M. H. Ernst},
Publisher = {North-Holland},
Title = {Amplitude equations for pattern forming systems},
Volume = {VIII},
Year = {1994}}
@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,
Address = {New York},
Author = {D. V. Anosov and V. I. Arnol'd},
Publisher = {Springer},
Title = {Dynamical systems {I}: {Ordinary} Differential Equations and Smooth Dynamical Systems},
Year = {1988}}
@article{AndCom07,
Author = {N. Andersson and G. L. Comer},
Doi = {10.12942/lrr-2007-1},
Journal = {Living Rev. Relat.},
Title = {Relativistic fluid dynamics: {Physics} for many different scales},
Volume = {10},
Year = {2007}
}
@article{angen88,
Author = {S. B. Angenent},
Journal = {Commun. Math. Phys.},
Pages = {353--374},
Title = {The periodic orbits of an area preserving twist-map},
Volume = {115},
Year = {1988}}
@Article{AnSo01,
author = {Antoulas, A. C. and Sorensen, D. C.},
title = {{Lyapunov}, {Lanczos}, and inertia},
journal = {Linear Algebra and its Applications},
year = {2001},
volume = {326},
pages = {137--150}
}
@article{arnold91k,
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.},
Author = {V. I. Arnold},
Journal = {Proc. R. Soc. Lond. A},
Pages = {19--22},
Title = {Kolmogorov's hydrodynamic attractors},
Volume = {434},
Year = {1991}}
@book{arnold92,
Address = {New York},
Author = {V. I. Arnol'd},
Publisher = {Springer},
Title = {Ordinary Differential Equations},
Year = {1992}}
@Article{ARS,
author = {U. M. Ascher and S. J. Ruuth and R. J. Spiteri},
title = {Implicit-explicit {R}unge-{K}utta methods for time-dependent partial differential equations},
journal = {Appl. Num. Math.},
year = {1997},
volume = {25},
pages = {151--167}
}
@article{art03int,
Author = {R. Artuso and P. Cvitanovi{\'c} and G. Tanner},
Journal = {Proc. Theo. Phys. Supp.},
Pages = {1--21},
Title = {Cycle expansions for intermittent maps},
Volume = {150},
Year = {2003}}
@Article{ArtMaz65,
author = {Artin, M. and Mazur, B.},
title = {On periodic points},
journal = {Ann. Math.},
year = {1965},
volume = {81},
pages = {82--99},
bdsk-url-1 = {http://www.jstor.org/stable/1970384},
url = {http://www.jstor.org/stable/1970384}
}
@book{ArWe05,
Address = {New York},
Author = {G. B. Arfken and H. J. Weber},
Edition = {6},
Publisher = {Academic Press},
Title = {Mathematical Methods for Physicists: A Comprehensive Guide},
Year = {2005}}
@article{AS87,
Author = {M. S. Acarlar and C. R. Smith},
Journal = {J. Fluid Mech.},
Pages = {1--41, 45--83},
Title = {A study of hairpin vortices in a laminar boundary layer},
Volume = {175},
Year = {1987}}
@article{Ascher95,
Author = {U. Ascher and S. Ruuth and B. Wetton},
Journal = {SIAM J. Numer. Anal.},
Pages = {797--823},
Title = {Implicit-explicit methods for time-dependent partial differential equations},
Volume = {32},
Year = {1995}}
@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.},
Doi = {10.1063/1.4758309},
Journal = {Chaos},
Note = {\arXiv{1209.4915}},
Pages = {047506},
Title = {Cartography of high-dimensional flows: {A} visual guide to sections and slices},
Volume = {22},
Year = {2012}
}
@article{aub95ant,
Author = {S. Aubry},
Journal = {Physica D},
Pages = {284--296},
Title = {Anti-integrability in dynamical and variational problems},
Volume = {86},
Year = {1995}}
@article{Aubry88,
Author = {N. Aubry and P. Holmes and J. L. Lumley and E. Stone},
Journal = {J. Fluid Mech.},
Pages = {115--173},
Title = {The dynamics of coherent structures in the wall region of turbulent boundary layer},
Volume = {192},
Year = {1988}}
@incollection{autod89,
Address = {Amsterdam},
Author = {A. Griewank},
Booktitle = {Mathematical Programming: Recent Developments and Applications},
Editor = {M. Iri and K. Tanabe},
Pages = {83--108},
Publisher = {Kluwer},
Title = {On Automatic Differentiation},
Year = {1989}}
@article{ave99,
Author = {V. M. Egu{\'\i}luz and P. Alstr{\o}m and E. Hern{\'a}ndez-Garc{\'\i}a and O. Piro},
Journal = {Phys. Rev. E},
Pages = {2822},
Title = {Average patterns of spatiotemporal chaos: {A} boundary effect},
Volume = {59},
Year = {1999}}
@article{AvMeRoHo13,
Author = {Avila, M and Mellibovsky, F and Roland, N and Hof, B},
Journal = {Phys. Rev. Lett.},
Pages = {224502},
Title = {Streamwise-localized solutions at the onset of turbulence in pipe flow},
Volume = {110},
Year = {2013},
doi = {10.1103/PhysRevLett.110.224502}
}
@article{AVWIHO10,
Author = {Avila, M. and Willis, A. P. and Hof, B.},
Doi = {10.1017/S0022112009993296},
Journal = {J. Fluid Mech.},
Pages = {127--136},
Title = {On the transient nature of localized pipe flow turbulence},
Volume = {646},
Year = {2010}
}
@UNPUBLISHED{Axen11,
author = {Axenides, M.},
title = {Non {Hamiltonian} chaos from {Nambu} dynamics of surfaces},
note = {\arXiv{1109.0470}},
year = {2011}
}
@Article{AxFl08,
author = {Axenides, M. and Floratos, E.},
title = {{Nambu-Lie} 3-algebras on fuzzy 3\mbox{-}{m}anifolds},
journal = {JHEP},
year = {2009},
volume = {902},
pages = {39},
note = {\arXiv{0809.3493}},
doi = {10.1088/1126-6708/2009/02/039}
}
@ARTICLE{AxFl09,
author = {Axenides, M. and Floratos, E.},
title = {Strange attractors in dissipative {Nambu} mechanics: {Classical}
and quantum aspects},
journal = {JHEP},
year = {2010},
volume = {1004},
pages = {036},
note = {\arXiv{0910.3881}},
doi = {10.1007/JHEP04(2010)036}
}
@ARTICLE{AxFlNi09,
author = {Axenides, M. and Floratos, E.G. and Nicolis, S.},
title = {{Nambu} quantum mechanics on discrete 3-tori},
journal = {J. Phys. A},
year = {2009},
volume = {42},
pages = {275201},
note = {\arXiv{0901.2638}},
doi = {10.1088/1751-8113/42/27/275201}
}
@article{B84,
Author = {D. J. Benney},
Journal = {Stud. Appl. Math.},
Pages = {1--19},
Title = {The evolution of disturbances in shear flows at high {R}eynolds numbers},
Volume = {70},
Year = {1984}}
@article{Bagheri13,
Author = {Bagheri, S.},
title = {Koopman-mode decomposition of the cylinder wake},
journal = {J. Fluid Mech.},
volume = {726},
year = {2013},
pages = {596--623},
doi = {10.1017/jfm.2013.249}
}
@article{Bandelow1998,
Author = {U. Bandelow, L. Recke and B. Sandstede},
Journal = {Opt. Commun.},
Pages = {212--218},
Title = {Frequency regions for forced locking of self-pulsating multi-section {DFB} lasers},
Volume = {147},
Year = {1998}}
@Article{baranger88,
author = {M. Baranger and K. T. R. Davies and J. H. Mahoney},
title = {The calculation of periodic trajectories},
journal = {Ann. Phys.},
year = {1988},
volume = {186},
pages = {95--110},
doi = {10.1016/S0003-4916(88)80018-6}
}
@article{BarChor98,
Author = {G. I. Barenblatt and A. J. Chorin},
Journal = {SIAM Review},
Pages = {265--291},
Title = {New Perspectives in Turbulence: Scaling Laws, Asymptotics, and Intermittency},
Volume = {40},
Year = {1998}}
@article{BARK11,
Author = {Barkley, D.},
Doi = {10.1103/PhysRevE.84.016309},
Journal = {Phys. Rev. E.},
Pages = {016309},
Title = {Simplifying the complexity of pipe flow},
Volume = {84},
Year = {2011}
}
@article{Barkley05,
Author = {Barkley, D. and {Tuckerman}, L.~S.},
Journal = {Phys. Rev. Lett.},
Note = {\arXiv{physics/0403142}},
Pages = {014502-+},
Title = {Computational study of turbulent laminar patterns in {Couette} flow},
Volume = 94,
Year = 2005}
@Article{Barkley99,
author = {Barkley, D. and {Tuckerman}, L.~S.},
title = {Stability analysis of perturbed plane {Couette} flow},
journal = {Phys. Fluids},
year = {1999},
volume = {11},
pages = {1187--1195},
note = {\arXiv{physics/0312048}}
}
@article{BarTuck07,
Author = {D. Barkley and L. S. Tuckerman},
Doi = {10.1017/S002211200600454X},
Journal = {J. Fluid Mech.},
Pages = {109--138},
Title = {Mean flow of turbulent-laminar patterns in plane {Couette} flow},
Volume = {576},
Year = {2007}
}
@article{Batch46,
Author = {Batchelor, G. K.},
Journal = {Nature},
Pages = {883--884},
Title = {Double velocity correlation function in turbulent motion},
Volume = 158,
Year = 1946}
@book{baumgarten2006mixture,
Address = {New York},
Author = {Baumgarten, C.},
Publisher = {Springer},
Title = {Mixture Formation in Internal Combustion Engines},
Year = {2006}}
@inproceedings{BayOrt04,
Author = {E. Bayro-Corrochano and J. Ortegon-Aguilar},
Booktitle = {International Conference on Pattern Recognition},
Doi = {10.1109/ICPR.2004.1334036},
Pages = {56--59},
Title = {Lie algebra template tracking},
Volume = {2},
Year = {2004}
}
@article{BaZeVanH11,
title = {Time-periodic solitons in a damped-driven nonlinear Schr\"odinger equation},
author = {Barashenkov, I. V. and Zemlyanaya, E. V. and van Heerden, T. C.},
journal = {Phys. Rev. E},
volume = {83},
pages = {056609},
year = {2011},
doi = {10.1103/PhysRevE.83.056609},
}
@article{BBJ85,
Author = {Bak, P. and Bohr, T. and Jensen, M. H.},
Journal = {Phys. Scr.},
Pages = {50--58},
Title = {Mode-locking and the transition to chaos in dissipative systems},
Volume = {T9},
Year = {1985}}
@article{BC89,
Author = {D. J. Benney and K. A. Chow},
Journal = {Stud. Appl. Math.},
Pages = {37--73},
Title = {A mean flow first harmonic theory for hydrodynamic instabilities},
Volume = {80},
Year = {1989}}
@Inbook{BC96,
chapter = {Bifurcation sequences in problems of thermal convection and of plane {Couette} flow},
pages = {209--226},
title = {Waves and Nonlinear Processes in Hydrodynamics},
publisher = {Springer},
year = {1996},
author = {F. H. Busse and R. M. Clever},
address = {New York}
}
@Inproceedings{BC96a,
author = {Busse, F. H. and Clever, R. M.},
title = {Bifurcation sequences in fluid flows and coherent structures in the turbulent state},
booktitle = {Advances in Turbulence VI},
year = {1996},
editor = {Gavrilakis, S. and Machiels, L. and Monkewitz, P. A.},
series = {Proceedings of the Sixth European Turbulence Conference},
pages = {309--312}
}
@Article{BC98,
author = {Bottin, S. and Chat{\'e}, H.},
title = {Statistical analysis of the transition to turbulence in plane {Couette} flow},
journal = {Eur. Phys. J. B},
year = {1998},
volume = {6},
pages = {143--155},
doi = {10.1007/s100510050536}
}
@article{BCFLR08,
Author = {R. Bachelard and C. Chandre and D. Fanelli and X. Leoncini and S. Ruffo},
Journal = {Phys. Rev. Lett.},
Pages = {260603},
Title = {Abundance of regular orbits and out-of-equilibrium phase transitions in the thermodynamic limit for long-range systems},
Volume = {101},
Year = {2008}}
@Article{BDMD98,
author = {S. Bottin and F. Daviaud and P. Manneville and O. Dauchot},
title = {Discontinuous transition to spatiotemporal intermittency in plane {Couette} flow},
journal = {Europhys. Lett.},
year = {1998},
volume = {43},
pages = {171--176},
doi = {10.1140/epje/i2014-14034-1}
}
@Incollection{BDMD98a,
author = {Bottin, S. and Daviaud, F. and Manneville, P. and Dauchot, O.},
title = {Discontinuous Transition to Spatio-Temporal Intermittency in the Plane {Couette} Flow},
booktitle = {Advances in Turbulence VII},
publisher = {Springer},
year = {1998},
editor = {Frisch, U.},
isbn = {978-94-010-6151-3},
pages = {117--120},
doi = {10.1007/978-94-011-5118-4_28}
}
@article{BDS98,
Author = {Briggs, K. M. and Dixon, T. W. and Szekeres, G.},
Journal = {Int. J. Bif. Chaos},
Pages = {347--357},
Title = {Analytic solutions of the {Cvitanovic-Feigenbaum} and {Feigenbaum-Kadanoff-Shenker} equations},
Volume = {8},
Year = {1998}}
@Article{bene80a,
author = {Benettin, G. and Galgani, L. and Giorgilli, A. and Strelcyn, J. M.},
title = {{Lyapunov} characteristic exponents for smooth dynamical systems; a method for computing all of them. {P}art 1: theory},
journal = {Meccanica},
year = {1980},
volume = {15},
pages = {9--20}
}
@Article{bene80b,
author = {Benettin, G. and Galgani, L. and Giogilli, A. and Strelcyn, J. M.},
title = {{Lyapunov} characteristic exponents for smooth dynamical systems; a method for computing all of them. {P}art 2: numerical application},
journal = {Meccanica},
year = {1980},
volume = {15},
pages = {21--30}
}
@article{benks66,
Author = {D. J. Benney},
Journal = {J. Math. Phys},
Pages = {150--155},
Title = {Long waves in liquid films},
Volume = {45},
Year = {1966}}
@article{Berkooz94,
Author = {G. Berkooz and P. Holmes and J. L. Lumley and N. Aubry and E. Stone},
Journal = {Phys. Fluids},
Pages = {1574--1578},
Title = {Observations regarding `{C}oherence and chaos in a model of turbulent boundary layer' by {X}. {Z}hou and {L}. {S}irovich},
Volume = {6},
Year = {1994}}
@Article{BerlinPF94,
author = {S. Berlin and A. Lundbladh and D. Henningson},
title = {Spatial simulations of oblique transition in a boundary layer},
journal = {Phys. Fluids},
year = {1994},
volume = {6},
number = {6},
pages = {1949--1951},
month = jun
}
@Article{Berry84,
author = {Berry, M. V.},
title = {Quantal phase factors accompanying adiabatic changes},
journal = {Proc. R. Soc. Lond. A},
year = {1984},
volume = {392},
pages = {45--57}
}
@article{BeTh04,
Author = {W.-J. Beyn and V. Th\"ummler},
Journal = {SIAM J. Appl. Dyn. Syst.},
Pages = {85--116},
Title = {Freezing solutions of equivariant evolution equations},
Volume = {3},
Year = {2004}}
@article{BG81,
Author = {D. J. Benney and L. H. Gustavsson},
Journal = {Stud. Appl. Math.},
Pages = {185--209},
Title = {A new mechanism for linear and nonlinear hydrodynamic instability},
Volume = {64},
Year = {1981}}
@article{BGMWW92,
Author = {A. Back and J. Guckenheimer and M. R. Myers and F. J. Wicklin and P. A. Worfolk},
Journal = {Notices of the AMS},
Pages = {303--309},
Title = {DsTool: {Computer} assisted exploration of dynamical systems},
Volume = {39},
Year = {1992}}
@book{BH86,
Address = {New York},
Author = {N. Bleistein and R. A. Handelsman},
Publisher = {Dover},
Title = {Asymptotic Expansions of Integrals},
Year = {1986}}
@Article{biham90c,
author = {O. Biham and W. Wenzel},
title = {Unstable periodic orbits and the symbolic dynamics of the complex {H\'e}non map},
journal = {Phys. Rev. E},
year = {1990},
volume = {42},
pages = {4639},
abstract = {Previous work is extended to the complex plane so that
2^n complex orbits are found for symbol sequences of length n.}
}
@article{Birkhoff,
Author = {G. D. Birkhoff},
Journal = {Science},
Pages = {51--55},
Title = {Recent advances in dynamics},
Volume = {51},
Year = {1920}}
@Article{BJNRZ12,
author = {B. Barker and M. A. Johnson and P. Noble and L. M. Rodrigues and K. Zumbrun},
title = {Stability of periodic { Kuramoto-Sivashinsky} waves},
journal = {Appl. Math. Lett.},
year = {2012},
volume = {25},
pages = {824--829},
doi = {10.1016/j.aml.2011.10.026},
issn = {0893-9659}
}
@article{BK90,
Author = {M. V. Berry and J. P. Keating},
Journal = {J. Phys. A},
Pages = {4839--4849},
Title = {A rule for quantizing chaos},
Volume = {23},
Year = {1990}}
@article{bknk,
Author = {N. Bekki and K. Nozaki},
Journal = {Phys. Lett.},
Pages = {133--135},
Title = {Formations of spatial patterns and holes in the generalized {Ginzburg-Landau} equation},
Volume = {110A},
Year = {1985}}
@article{bknk83,
Author = {K. Nozaki and N. Bekki},
Journal = {Phys. Rev. Lett.},
Pages = {2171},
Title = {Pattern selection and spatiotemporal transition to chaos in the {Ginzburg-Landau} equation},
Volume = {51},
Year = {1983}}
@article{bknk84,
Author = {K. Nozaki and N. Bekki},
Journal = {J. Phys. Soc. Japan},
Pages = {1581--1582},
Title = {Exact Solutions of the Generalized {Ginzburg-Landau} Equation},
Volume = {53},
Year = {1984}}
@Article{BKRS96,
author = {Baladi, V. and Kitaev a. and Ruelle, D. and Semmes, S.},
title = {Sharp determinants and kneading operators for holomorphic maps},
journal = {Proc. Steklov. Inst. of Math.},
year = {1997},
volume = {216},
pages = {186--228}
}
@book{bl,
Address = {New York},
Author = {J. Stoer and R. Bulirsch},
Publisher = {Springer},
Title = {Introduction to Numerical Analysis},
Year = {1983}}
@article{blackwelder76,
Author = {Blackwelder, R. F. and Kaplan, R.},
Journal = {J. Fluid Mech.},
Pages = {89},
Title = {On the wall structure of the turbulent boundary layer},
Volume = {76},
Year = {1976}}
@article{blackwelder87,
Author = {Swearingen, J. D. and Blackwelder, R. F.},
Journal = {J. Fluid Mech.},
Pages = {255},
Title = {The growth and breakdown of streamwise vortices in the presence of a wall},
Volume = {182},
Year = {1987}}
@Article{BlMaLo05,
author = {H. M. Blackburn, F. Marques and Lopez,J. M.},
title = {Symmetry breaking of two-dimensional time-periodic wakes},
journal = {J. Fluid Mech.},
year = {2005},
volume = {522},
pages = {395--411},
doi = {10.1017/S0022112004002095},
issn = {1469-7645},
numpages = {17}
}
@article{BLZ08,
Author = {P. Bates, K. Lu, and C. Zeng},
Journal = {Inventiones mathematicae},
Pages = {355--433},
Title = {Approximately invariant manifolds and global dynamics of spike states},
Volume = {55},
Year = {2008}}
@article{BMS71,
Author = {N. H. Baker and D. W. Moore and E. A. Spiegel},
Journal = {Quatr. J. Mech. and Appl. Math.},
Pages = {391},
Title = {Aperiodic behavior of a non-linear oscillator},
Volume = {24},
Year = {1971}}
@inproceedings{BodEck11,
Addr = {Berlin},
Author = {E. Bodenschatz and M. Eckert},
Booktitle = {Highlights in the History of Turbulence},
Editor = {P. A. Davidson and Y. Kaneda and H. K. Moffatt and K. R. Sreenivasan},
Publisher = {Springer},
Title = {Prandtl and the {G\"ottingen School}},
Year = {2011}}
@book{bohr98tur,
Address = {Cambridge},
Author = {T. Bohr and M. H. Jensen and G. Paladin and A. Vulpiani},
Publisher = {Cambridge Univ. Press},
Title = {Dynamical Systems Approach to Turbulence},
Year = {1998}}
@article{bollt07,
Author = {Bollt, E.},
Journal = {Int. J. Bifur. Chaos},
Pages = {1199--1219},
Title = {Attractor modeling and empirical nonlinear model reduction of dissipative dynamical systems},
Volume = {17},
Year = {2007}}
@article{borrero10,
Author = {D. Borrero-Echeverry and M. F. Schatz and R. Tagg},
Journal = {Phys. Rev. E},
Pages = {025301},
Title = {Transient turbulence in {Taylor-Couette} flow},
Volume = {81},
Year = {2010}}
@Article{Bottin98,
author = {Bottin, S. and {Dauchot}, O. and {Daviaud}, F. and {Manneville}, P.},
title = {Experimental evidence of streamwise vortices as finite amplitude solutions in transitional plane {Couette} flow},
journal = {Phys. Fluids},
year = {1998},
volume = {10},
pages = {2597--2607},
doi = {10.1063/1.869773}
}
@unpublished{Bouch03,
Author = {E. Bouchbinder},
Title = {The nonlinear {Schr\"odinger} equation},
url = {http://ChaosBook.org/library/Bouch03.pdf},
Note = {Unpublished notes},
Year = {2003}}
@book{bowen,
Address = {Berlin},
Author = {R. Bowen},
Publisher = {Springer},
Title = {Equilibrium States and the Ergodic Theory of {Anosov} Diffeomorphisms},
Year = 1975}
@Article{Bowen72,
author = {R. Bowen},
title = {Periodic orbits for hyperbolic flows},
journal = {Amer. J. Math.},
year = {1972},
volume = {94},
pages = {1--30}
}
@Article{Bowen73,
author = {R. Bowen},
title = {Symbolic dynamics for hyperbolic flows},
journal = {Amer. J. Math.},
year = {1973},
volume = {95},
pages = {429--460}
}
@book{BoydiP06,
Address = {New York},
Author = {W. E. Boyce and R. C. DiPrima},
Publisher = {Wiley-Interscience},
Title = {Elementary Differential Equations and Boundary Value Problems},
Year = {2006}}
@article{BQT91,
Author = {K. M. Briggs and G. R. W. Quispel and C. J. Thompson},
Journal = {J. Phys. A},
Keywords = {Feigenbaum constant, Mandelbrot set},
Pages = {3363--3368},
Title = {Feigenvalues for {Mandelsets}},
Volume = {24},
Year = {1991}}
@Article{BR96,
author = {Baladi, V. and Ruelle, D.},
title = {Sharp determinants},
journal = {Invent. Math.},
year = {1996},
volume = {123},
pages = {553--574},
doi = {10.1007/s002220050040},
bdsk-url-1 = {http://dx.doi.org/10.1007/s002220050040}
}
@incollection{brand03,
Address = {London},
Author = {A. Brandenburg},
Booktitle = {Advances in Nonlinear Dynamos},
Editor = {A. Ferriz-Mas and M. N\'u\~nez},
Note = {\arXiv{astro-ph/0109497}},
Pages = {269--344},
Publisher = {Taylor \& Francis},
Title = {Computational aspects of astrophysical {MHD} and turbulence},
Year = {2003}}
@misc{Bridges_priv,
Author = {T. J. Bridges},
Note = {private communication}}
@article{Bridges08,
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.},
Author = {T. J. Bridges},
Journal = {J. Diff. Eqn.},
Pages = {1629--1674},
Title = {Degenerate relative equilibria, curvature of the momentum map,and homoclinic bifurcation},
Volume = {244},
Year = {2008}}
@article{Briggs91,
Author = {Briggs, K.},
Journal = {Math. Computation},
Pages = {435--439},
Title = {A precise calculation of the {Feigenbaum} constants},
Volume = {57},
Year = {1991}}
@inproceedings{BrKevr96,
Address = {Providence, RI},
Author = {Brown, H. S. and Kevrekidis, I. G.},
Booktitle = {Pattern Formation: Symmetry Methods and Applications},
Editor = {D. Benest and C. Froeschl\'{e}},
Pages = {45--66},
Publisher = {AMS},
Series = {Fields Inst. Commun.},
Title = {Modulated traveling waves for the {Kuramoto-Sivashinsky} equation},
Volume = {5},
Year = {1996}}
@article{bro94sur,
Author = {G. Brodin and J. Lundberg},
Journal = {Phys. Plasmas},
Pages = {96},
Title = {Nonlinear surface waves in a plasma with a diffuse boundary},
Volume = {1},
Year = {1994}}
@article{broer97,
Author = {H. W. Broer and H. M. Osinga and G. Vegter},
Journal = {Z. Angew. Math. Phys.},
Pages = {480},
Title = {Algorithms for computing {N}ormally {H}yperbolic {I}nvariant {M}anifolds},
Volume = {48},
Year = {1997}}
@article{bronski2005,
Author = {J. C. Bronski and T. N. Gambill},
Journal = {Nonlinearity},
Note = {\arXiv{math/0508481}},
Pages = {2023--2039},
Title = {Uncertainty estimates and {$L_2$} bounds for the {Kuramoto-Sivashinsky} equation},
Volume = {19},
Year = {2006}}
@article{BTAA95,
Author = {K. H. Bech and N. Tillmark and P. H. Alfredsson and H. I. Andersson},
Journal = {J. Fluid Mech.},
Pages = {291--325},
Title = {An investigation of turbulent plane {Couette} flow at low {R}eynolds numbers},
Volume = {286},
Year = {1995}}
@article{BTBmaw03,
Author = {L. Brusch and A. Torcini and M. B{\"a}r},
Journal = {Physica D},
Pages = {152--167},
Title = {Nonlinear analysis of the {E}ckhaus instability: modulated amplitude waves and phase chaos with nonzero average phase gradient},
Volume = {174},
Year = {2003}}
@article{BTHmaw01,
Author = {L. Brusch and A. Torcini and M. van Hecke and M. G. Zimmermann and M. B{\"a}r},
Journal = {Physica D},
Pages = {127--148},
Title = {Modulated amplitude waves and defect formation in the one-dimensional complex {Ginzburg-Landau} equation},
Volume = {160},
Year = {2001}}
@article{BuBoCvSi14,
Author = {Budanur, N. B. and Borrero-Echeverry, D. and Cvitanovi\'c, P.},
Title = {Periodic orbit analysis of a system with continuous symmetry - A tutorial},
journal = {Chaos},
volume = {25},
year = {2015},
doi = {10.1063/1.4923742},
pages = {073112},
Note = {\arXiv{1411.3303}},
}
@unpublished{Bud15,
Author = {Budanur, N. B.},
Note = {In preparation},
Title = {Invariant measurement of torsion within eigenframes of
periodic orbits},
Year = {2015}}
@article{BudCvi14,
Author = {Budanur, N. B. and Cvitanovi\'c, P. and Davidchack, R. L. and Siminos, E.},
Title = {Reduction of the {SO(2)} symmetry for spatially extended dynamical systems},
journal = {Phys. Rev. Lett.},
volume = {114},
pages = {084102},
Note = {\arXiv{1405.1096}},
Year = {2015},
doi = {10.1103/PhysRevLett.114.084102},
}
@unpublished{BudCvi15,
Author = {Budanur, N. B. and Cvitanovi\'c, P.},
Note = {\arXiv{1509.08133}},
Title = {Torus breakdown in the symmetry-reduced state space of
the {Kuramoto-Sivashinsky} system},
Year = {2015}}
@article{BudMez12,
Abstract = {The
method we present in this paper uses trajectory averages of scalar
functions on the state space to: (a) identify invariant sets in the state
space, and (b) to form coherent structures by aggregating invariant sets
that are similar across multiple spatial scales. First, we construct the
ergodic quotient, the object obtained by mapping trajectories to the
space of the trajectory averages of a function basis on the state space.
Second, we endow the ergodic quotient with a metric structure that
successfully captures how similar the invariant sets are in the state
space. Finally, we parametrize the ergodic quotient using intrinsic
diffusion modes on it. By segmenting the ergodic quotient based on the
diffusion modes, we extract coherent features in the state space of the
dynamical system. The algorithm is validated by analyzing the
Arnold-Beltrami-Childress flow, which was the test-bed for
alternative approaches: the Ulam's approximation of the transfer
operator and the computation of Lagrangian Coherent Structures.
Furthermore, we explain how the method extends the Poincar\'e map analysis
for periodic flows. As a demonstration, we apply the method to a
periodically-driven three-dimensional Hill's vortex flow, discovering
unknown coherent structures in its state space.},
Author = {M. Budi\v{s}i\'c and I. Mezi\'c},
Doi = {10.1016/j.physd.2012.04.006},
Journal = {Physica D},
Note = {\arXiv{1204.2050}},
Pages = {1255--1269},
Title = {Geometry of the ergodic quotient reveals coherent structures in flows},
Volume = {241},
Year = {2012}
}
@article{BuMoMe12,
Author = {Budi{\v s}i{\'c}, M. and Mohr, R. M. and Mezi{\'c}, I.},
Doi = {10.1063/1.4772195},
Journal = {Chaos},
Note = {\arXiv{1206.3164}},
Numpages = {33},
Pages = {047510},
Title = {{Applied Koopmanism}},
Volume = {22},
Year = {2012}
}
@article{Busse04,
Author = {F. H. Busse},
Journal = {Science},
Pages = {1574--1575},
Title = {Visualizing the dynamics of the onset of turbulence},
Volume = {305},
Year = {2004}}
@unpublished{ByMaGr14,
Author = {G. Byrne, C. Marcotte and R. O. Grigoriev},
Note = {To be submitted},
Title = {Unstable and marginally stable solutions of reaction diffusion systems with weakly broken continuous symmetries},
Year = {2014}}
@article{BZMmaw00,
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.},
Author = {L. Brusch and M. G. Zimmermann and M. van Hecke and M. B{\"a}r and A. Torcini},
Journal = {Phys. Rev. Lett.},
Pages = {86},
Title = {Modulated amplitude waves and the transition from phase to defect chaos},
Volume = {85},
Year = {2000}}
@article{C02,
Author = {S. J. Chapman},
Journal = {J. Fluid Mech.},
Pages = {35--97},
Title = {Subcritical transition in channel flows},
Volume = {451},
Year = {2002}}
@misc{C77,
Author = {P. Cvitanovi\'c},
Note = {({Oxford University} preprint 40/77, unpublished)},
Title = {Classical and exceptional {Lie} algebras as invariance algebras},
Url = {www.nbi.dk/GroupTheory},
Year = {1977},
Bdsk-Url-1 = {www.nbi.dk/GroupTheory}}
@ARTICLE{Cantwell92,
author = {B. J. Cantwell},
title = {Exact solution of a restricted {Euler} equation for the velocity
gradient tensor},
journal = {Phys. Fluids A},
year = {1992},
volume = {4},
pages = {782--793},
doi = {10.1063/1.858295},
keywords = {VELOCITY; TENSORS; ANALYTICAL SOLUTION; PRESSURE GRADIENTS; STRAIN
RATE; TURBULENT FLOW; REYNOLDS NUMBER; NAVIERSTOKES EQUATIONS; KINETIC
ENERGY; INCOMPRESSIBLE FLOW; NUMERICAL SOLUTION; ISOTROPY; INITIAL
CONDITIONS}
}
@book{Canuto2006,
Address = {New York},
Author = {C. Canuto and M. Y. Hussaini and A. Quarteroni and T. A. Zang},
Publisher = {Springer},
Title = {Spectral Methods: Fundamentals in Single Domains},
Year = {2006}}
@book{Canuto2007,
Address = {New York},
Author = {C. Canuto and M. Y. Hussaini and A. Quarteroni and T. A. Zang},
Publisher = {Springer},
Title = {Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics},
Year = {2007}}
@book{Canuto88,
Address = {New York},
Author = {C. Canuto and M. Y. Hussaini and A. Quarteroni and T. A. Zang},
Publisher = {Springer},
Title = {Spectral Methods in Fluid Dynamics},
Year = {1988}}
@article{car88,
Author = {F. Cariello and M. Tabor},
Journal = {Physica D},
Pages = {77},
Title = {{P}ainlev{\'e} expansions for nonintegrable evolution equations},
Volume = {39},
Year = {1989}}
@article{CarCha04,
Author = {Carter, B. and Chamel, N.},
Doi = {10.1142/S0218271804004542},
Journal = {Int. J. Mod. Phys. D},
Note = {\arXiv{astro-ph/0305186}},
Pages = {291--325},
Title = {Covariant analysis of {Newtonian} multi-fluid models for
neutron stars {I: Milne-Cartan} structure and variational
formulation},
Volume = {13},
Year = {2004}
}
@article{carl97int,
Author = {C. P. Dettmann and P. Cvitanovi{\'c}},
Journal = {Phys. Rev. E},
Pages = {6687},
Title = {Cycle expansions for intermittent diffusion},
Volume = {56},
Year = {1997}}
@article{CarlSchn62,
Author = {D. Carlson and H. Schneider},
Journal = {Bull. Amer. Math. Soc.},
Pages = {479--484},
Title = {Inertia theorems for matrices: the semi-definite case},
Volume = {68},
Year = {1962}}
@book{CartanMF,
Address = {Paris},
Author = {Cartan, \'E.},
Publisher = {Hermann},
Series = {{Expos\'es} de {G\'eom\'etrie}},
Title = {La m\'ethode du rep\`ere mobile, la th\'eorie des groupes continus, et les espaces g\'en\'eralis\'es},
Volume = {5},
Year = {1935}}
@article{CB92,
Author = {Clever, R. M. and {Busse}, F. H.},
Journal = {J. Fluid Mech.},
Pages = {511--527},
Title = {Three-dimensional convection in a horizontal layer subjected to constant shear},
Volume = {234},
Year = {1992}}
@Article{CB97,
author = {Clever, R. M. and Busse, F. H.},
title = {Tertiary and quaternary solutions for plane {Couette} flow},
journal = {J. Fluid Mech.},
year = {1997},
volume = {344},
pages = {137--153},
abstract = {The manifold of Nagata steady solutions is
explored in the parameter space of the problem and their
instabilities are investigated. These instabilities usually
lead to time-periodic solutions whose properties do not
differ much from those of the steady solutions except that
the amplitude varies in time. In some cases travelling wave
solutions which are asymmetric with respect to the midplane
of the layer are found as quaternary states of flow.
Similarities with longitudinal vortices recently observed in
experiments are discussed.}
}
@incollection{CBconverg,
Address = {Copenhagen},
Author = {R. Artuso and R. Hugh and P. Cvitanovi{\'c}},
Booktitle = {Chaos: Classical and Quantum},
Editor = {P. Cvitanovi{\'c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
Note = {ChaosBook.org/version14},
Publisher = {Niels Bohr Inst.},
Title = {Chapter "{Why} does it work?"},
Year = {2013}}
@MISC{CBook:appendApplic,
author = {P. Cvitanovi\'c and G. Vattay},
title = {Appendix ``{Applications}''},
year = {2011},
note = {in \refref{DasBuch}}
}
@incollection{CBtrace,
Address = {Copenhagen},
Author = {P. Cvitanovi{\'c}},
Booktitle = {Chaos: Classical and Quantum},
Editor = {P. Cvitanovi{\'c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
Note = {ChaosBook.org/version14},
Publisher = {Niels Bohr Inst.},
Title = {Chapter ``{Trace} Formulas''},
Year = {2013}}
@article{CCKFNRS12,
Author = {Churchland, M. M. and Cunningham, J. P. and Kaufman, M. T. and Foster, J. D. and Nuyujukian, P. and Ryu, S. I. and Shenoy, K. V.},
Doi = {10.1038/nature11129},
Journal = {Nature},
Pages = {51--56},
Title = {Neural population dynamics during reaching},
Volume = {487},
Year = {2012}
}
@Article{CCW05,
author = {G. Csordas and M. Charalambides and F. Waleffe},
title = {A new property of a class of {J}acobi polynomials},
journal = {Proc. Amer. Math. Soc.},
year = {2005},
volume = {133},
pages = {3551--3560}
}
@article{CE89,
Author = {Cvitanovi\'c, P. and Eckhardt, B.},
Journal = {Phys. Rev. Lett.},
Pages = {823--826},
Title = {Periodic orbit quantization of chaotic systems},
Volume = {63},
Year = {1989}}
@article{CE91,
Author = {P. Cvitanovi{\'c} and B. Eckhardt},
Journal = {J. Phys. A},
Pages = {L237},
Title = {Periodic orbit expansions for classical smooth flows},
Volume = {24},
Year = {1991}}
@Article{CE95,
author = {A. Cherhabili and U. Ehrenstein},
title = {Spatially localized two-dimensional finite-amplitude states in plane {Couette} flow},
journal = {Eur. J. Mech. B Fluids},
year = {1995},
volume = {14},
pages = {677--696},
url = {http://cat.inist.fr/?aModele=afficheN&cpsidt=2903887}
}
@article{CEEks93,
Abstract = {Proved the analyticity of the solution of the KSe on a
periodic interval near the real axis and the width of the region is
estimated. Numerical calculations show that this width should not
depend on the size of the interval.},
Author = {P. Collet and J.-P. Eckmann and H. Epstein and J. Stubbe},
Journal = {Physica D},
Pages = {321--326},
Title = {Analyticity for the {Kuramoto-Sivashinsky} equation},
Volume = {67},
Year = {1993}}
@Article{CEEksgl93,
author = {P. Collet and J.-P. Eckmann and H. Epstein and J. Stubbe},
title = {A global attracting set for the {Kuramoto-Sivashinsky} equation},
journal = {Commun. Math. Phys.},
year = {1993},
volume = {152},
pages = {203--214},
abstract = {New bounds are given for the L^2 norm of solutions of the KSe (not necessarily antisymmetric).}
}
@book{cellwolf,
Address = {Reading, MA},
Author = {S. Wolfram},
Publisher = {Addison-Wesley},
Title = {Cellular Automata and Complexity},
Year = {1994}}
@incollection{CERRS,
Address = {Cambridge},
Author = {B. Eckhardt and G. Russberg and P. Cvitanovi\'c and P. E. Rosenqvist and P. Scherer},
Booktitle = {Quantum Chaos: Between Order and Disorder},
Editor = {G. Casati and B. Chirikov},
Pages = {483},
Publisher = {Cambridge Univ. Press},
Title = {Pinball scattering},
Year = {1995}}
@Article{CFGT15,
author = {Carmona, V. and Fern{\'a}ndez-S{\'a}nchez, F. and Garc{\'i}a-Medina, E. and Teruel, A. E.},
title = {Noose structure and bifurcations of periodic orbits in reversible
three-dimensional piecewise linear differential systems},
journal = {J. Nonlin. Sci},
year = {2015},
volume = {25},
pages = {1209--1224},
doi = {10.1007/s00332-015-9251-z}
}
@Article{cgole92,
author = {C. Gol{\'e}},
title = {A new proof of the {Aubry-Mather}'s theorem},
journal = {Math. Z.},
year = {1992},
volume = {210},
pages = {441--448},
abstract = {A auxiliary infinite gradient-like flow is constructed
for the area-preserving map such that orbits of the original system
are rest points of the new system. In this way, existence of orbits
of any rotation number is proved.}
}
@Article{CGS92,
author = {Cvitanovi\'c, P. and Gaspard, P. and Schreiber, T.},
title = {Investigation of the {Lorentz} gas in terms of periodic orbits},
journal = {Chaos},
year = {1992},
volume = {2},
pages = {85--90},
doi = {10.1063/1.165902}
}
@article{ChaKer12,
Author = {Chandler, G. J. and Kerswell, R. R.},
Doi = {10.1017/jfm.2013.122},
Journal = {J. Fluid M.},
Note = {\arXiv{1207.4682}},
Pages = {554--595},
Title = {Invariant recurrent solutions embedded in a turbulent two-dimensional {Kolmogorov} flow},
Volume = {722},
Year = {2013},
Bdsk-Url-1 = {http://dx.doi.org/10.1017/jfm.2013.122}}
@article{chand03tf,
Abstract = {Use wavelets to extract local frequencies and use the
major frequencies to identify the characteristics of phase space
motion.},
Author = {C. Chandre and S. Wiggins and T. Uzer},
Journal = {Physica D},
Pages = {171},
Title = {Time-frequency analysis of chaotic systems},
Volume = {181},
Year = {2003}}
@article{chang94,
Author = {H.-C. Chang},
Journal = {Ann. Rev. Fluid Mech.},
Pages = {103--136},
Title = {Wave evolution on a falling film},
Volume = {26},
Year = {1994}}
@techreport{channelflow,
Author = {J. F. Gibson},
Institution = {U. New Hampshire},
Note = {{\tt {Channelflow.org}}},
Publisher = {J. F. Gibson},
Title = {{Channelflow}: {A} spectral {Navier-Stokes} simulator in {C}++},
Year = {2013}}
@misc{channelflowDat,
Author = {J. F. Gibson},
Institution = {U. New Hampshire},
Note = {{\tt {Channelflow.org/database}}},
Publisher = {J. F. Gibson},
Title = {Database of invariant solutions of plane {Couette} flow},
Year = {2013}}
@Article{CHASCH14,
author = {Chantry, M. and Schneider, T. M.},
title = {Studying edge geometry in transiently turbulent shear flows},
journal = {J. Fluid Mech.},
year = {2014},
volume = {747},
pages = {506--517}
}
@article{chate94,
Abstract = {In the Benjamin-Feir stable region, spatiotemporal
intermittency regimes are identified which consists of patches of
linearly stale stable plane waves separated by localized objects with
a well defined dynamics. In the transition reginon, asymptotic states
with an irregular, frozen spatial structure are shown to occur. The
bistable region beyond the Benjamin-Feir stable region is of the same
spatiotemporal intermittency type with phase turbulence as the },
Author = {H. Chat{\'e}},
Journal = {Nonlinearity},
Pages = {185--204},
Title = {Spatiotemporal intermittency regimes of the one-dimensional complex {Ginzburg-Landau} equation},
Volume = {7},
Year = {1994}}
@article{ChaWri81,
Author = {Chang, S.-J. and Wright, J.},
Doi = {10.1103/PhysRevA.23.1419},
Journal = {Phys. Rev. A},
Pages = {1419--1433},
Title = {Transitions and distribution functions for chaotic systems},
Volume = {23},
Year = {1981},
Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevA.23.1419}}
@ARTICLE{ChePuShr99,
author = {Chertkov, M. and Pumir, A. and Shraiman, B. I.},
title = {Lagrangian tetrad dynamics and the phenomenology of turbulence},
journal = {Phys. Fluids},
year = {1999},
volume = { 11},
pages = {2394--2410},
note = {\arXiv{chao-dyn/9905027}}
}
@article{cherh00,
Author = {A. Cherhabili},
Issn = {0179-9614},
Journal = {Notes in Numerical Fluid Mechanics},
Pages = {69--80},
Title = {Nonlinear equilibrium states in plane {Couette} flow: Continuation from {P}oiseuille flow to {Couette} flow},
Volume = {74},
Year = {2000}}
@Article{cherh96,
author = {A. Cherhabili and U. Ehrenstein},
title = {Existence and stability of finite-amplitude states in plane {Couette} flow},
journal = {Fluid Mechanics and its Applications},
year = {1996},
volume = {36},
pages = {317--320},
doi = {10.1007/978-94-009-0297-8_89}
}
@Article{Cherh97,
author = {Cherhabili, A. and {Ehrenstein}, U.},
title = {Finite-amplitude equilibrium states in plane {Couette} flow},
journal = {J. Fluid Mech.},
year = {1997},
volume = {342},
pages = {159--177}
}
@article{CheSu90,
Author = {Cheng, C. Q. and Sun, Y. S.},
Journal = {Celestial Mech. Dynam. Astronom.},
Pages = {275--92},
Title = {Existence of invariant tori in three-dimensional measure-preserving mappings},
Volume = {47},
Year = {1990}}
@article{chfield,
Author = {P. Cvitanovi{\'c}},
Journal = {Physica A},
Note = {\arXiv{nlin.CD/0001034}},
Pages = {61},
Title = {Chaotic field theory: {A} sketch},
Volume = {288},
Year = {2000}}
@article{CHHM98,
Author = {Cendra, H. and Holm, D. D. and Hoyle, M. J. W. and Marsden, J. E.},
Journal = {J. of Math. Phys.},
Pages = {3138--3157},
Title = {The {Maxwell-Vlasov} equations in {Euler-Poincar\'e} form},
Volume = {39},
Year = {1998}}
@article{Chillingworth2000,
Author = {D. Chillingworth},
Journal = {Dyn. Stab. Syst.},
Pages = {101--137},
Title = {Generic multiparameter bifurcation from a manifold},
Volume = {15},
Year = {2000}}
@article{choi94,
Author = {H. Choi and P. Moin and J. Kim},
Journal = {J. Fluid Mech.},
Pages = {75},
Title = {Active turbulence control for drag reduction in wall-bounded flows},
Volume = {262},
Year = {1994}}
@book{Chossat94,
Address = {New York},
Author = {P. Chossat and G. Iooss},
Publisher = {Springer},
Title = {The {Taylor-Couette} Problem},
Year = {1994}}
@BOOK{ChossLaut00,
title = {Methods in Equivariant Bifurcations and Dynamical Systems},
publisher = {World Scientific},
year = {2000},
author = {P. Chossat and R. Lauterbach},
address = {Singapore},
ISBN = {978-981-02-3828-5}
}
@Book{ChPiWa89,
title = {Group Representation Theory for Physicists},
publisher = {World Scientific},
year = {1989},
author = {Chen, J.-Q. and Ping, J. and Wang, F.},
address = {Singapore},
doi = {10.1142/0262}
}
@article{ChriPol96,
Author = {F. Christiansen and A. Politi},
Journal = {Nonlinearity},
Pages = {1623--1640},
Title = {Symbolic encoding in symplectic maps},
Volume = {9},
Year = {1996}}
@article{ChriPol97,
Author = {F. Christiansen and A. Politi},
Journal = {Physica D},
Pages = {32--41},
Title = {Guidelines for the construction of a generating partition in the standard map},
Volume = {109},
Year = {1997}}
@Article{Christiansen97,
author = {F. Christiansen and P. Cvitanovi{\'c} and V. Putkaradze},
title = {Hopf's last hope: {Spatiotempora}l chaos in terms of unstable recurrent patterns},
journal = {Nonlinearity},
year = {1997},
volume = {10},
pages = {55--70},
addendum = {\arXiv{chao-dyn/9606016}},
doi = {10.1088/0951-7715/10/1/004}
}
@article{ChrPol95,
Author = {F. Christiansen and A. Politi},
Journal = {Phys. Rev. E},
Note = {\arXiv{chao-dyn/9411005}},
Pages = {3811},
Title = {A generating partition for the standard map},
Volume = {51},
Year = {1995}}
@article{ChWiKe14,
Author = {Chantry, M. and Willis, A. P. and Kerswell, R. R.},
Doi = {10.1103/PhysRevLett.112.164501},
Journal = {Phys. Rev. Lett.},
Pages = {164501},
Title = {Genesis of streamwise-localized solutions from globally
periodic traveling waves in pipe flow},
Volume = {112},
Year = {2014}
}
@Conference{CKBEG15,
author = {C. Crowley and M. Krygier and D. Borrero-Echeverry and R. Grigoriev and M. Schatz},
title = {Experimental observations of direct laminar-turbulent transition in counter-rotating {Taylor}-Couette flow},
booktitle = {68\textsuperscript{th} Annual Meeting of the APS Division of Fluid Dynamics},
year = {2015},
volume = {60},
series = {Bull. Am. Phys. Soc},
url = {http://meetings.aps.org/link/BAPS.2015.DFD.D29.2}
}
@article{Clever74,
Author = {R. Clever and F. Busse},
Journal = {J. Fluid Mech.},
Keywords = {fluid dynamics},
Pages = {665},
Title = {Large wavelength convection rolls in low Prandt number fluids},
Volume = {65},
Year = {1974}}
@Incollection{CleverBusse00,
author = {Clever, R. M. and Busse, F. H.},
title = {Tertiary and quaternary solutions for plane {Couette} flow with thermal stratification},
booktitle = {Physics of Rotating Fluids},
publisher = {Springer},
year = {2000},
editor = {Egbers, C. and Pfister, G.},
isbn = {978-3-540-67514-3},
pages = {399--416},
doi = {10.1007/3-540-45549-3_22}
}
@article{CM87tran,
Author = {H. Chat{\'e} and P. Manneville},
Journal = {Phys. Rev. Lett.},
Pages = {112},
Title = {Transition to Turbulence via spatiotemporal Intermittency},
Volume = {58},
Year = {1987}}
@misc{CNS,
Note = {Center for Nonlinear Science,\weblink{www.cns.gatech.edu}}}
@phdthesis{Coff76,
Author = {Coffee, T. P.},
School = {City Univ.~of New York, NY.},
Title = {{The stability of plane {Couette} flow}},
Year = 1976}
@article{cohen72,
Author = {B. I. Cohen and J. A. Krommes and W. M. Tang and M. N. Rosenbluth},
Journal = {Nucl. Fusion},
Pages = {971--991},
Title = {Non-linear saturation of the dissipative trapped-ion mode by mode coupling},
Volume = {16},
Year = {1972}}
@BOOK{CoLiSh96,
title = {Ideals, Varieties and Algorithms},
publisher = {Springer},
year = {1996},
author = {D. A. Cox and J. B. Little and D. 0'Shea},
address = {New York}
}
@article{collet80,
Author = {P. Collet and J.-P. Eckmann and O. E. Lanford},
Journal = {Commun. Math. Phys.},
Pages = {211},
Title = {Universal Properties of Maps on an interval},
Volume = {76},
Year = {1980}}
@article{collis04,
Author = {S. S. Collis and R. D. Joslin and A. Seifert and V. Theofilis},
Journal = {Prog. Aerosp. Sci.},
Pages = {237},
Title = {Issues in active flow control: {Theory}, control, simulation, and experiment},
Volume = {40},
Year = {2004}}
@article{condisfam1,
Author = {J.-A. Sepulchre and R. Mackay},
Journal = {Nonlinearity},
Pages = {679--713},
Title = {Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators},
Volume = {10},
Year = {1997}}
@article{conjug_Fred,
Author = {P. Cvitanovi\'{c} and C. P. Dettmann and R. Mainieri and G. Vattay},
Journal = {Nonlinearity},
Note = {\arXiv{chao-dyn/9811003}},
Pages = {939},
Title = {Trace formulae for stochastic evolution operators: {Smooth} conjugation method},
Volume = {12},
Year = {1999}}
@book{constantin_integral_1989,
Address = {New York},
Author = {P. Constantin and C. Foias and B. Nicolaenko and R. Temam},
Publisher = {Springer},
Title = {Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations},
Year = {1989}}
@book{contebook,
Address = {New York},
Author = {R. Conte},
Publisher = {Springer},
Title = {The {P}ainlev{\'e} property: one century later},
Year = {1999}}
@book{Cornwell97,
Address = {New York},
Author = {Cornwell, J. F.},
Publisher = {Academic Press},
Title = {Group Theory in Physics: An introduction},
Year = {1997}}
@Article{CouTre78,
author = {P. Coullet and C. Tresser},
title = {It\'erations d'endomorphismes et groupe de renormalisation},
journal = {J. de Physique Colloque C},
year = {1978},
volume = {39},
pages = {5--25}
}
@Article{CouTre78a,
author = {P. Coullet and C. Tresser},
title = {It\'erations d'endomorphismes et groupe de renormalisation},
journal = {C. R. Acad. Sci. Paris A},
year = {1978},
volume = {287},
pages = {577--580}
}
@article{cpmani,
Author = {A. J. Homburg and H. M. Osinga and G. Vegter},
Journal = {Z. Angew. Math. Phys.},
Pages = {171},
Title = {On the computation of invariant manifolds of fixed points},
Volume = {46},
Year = {1995}}
@article{Cra72,
Abstract = {New eigenvalue bounds are derived for the linear
stability of inviscid parallel flows, both for homogeneous and for
stratified fluids. The usefulness of these bounds, as compared with
that of previous results, is assessed for several examples. For
homogeneous fluids the new upper bounds for the imaginary part ci of
the complex phase velocity are sometimes better than previous
criteria. For both homogeneous and stratified flows, the new upper
bounds for the wave-number ? of neutrally stable disturbances
improve on previous results, giving values within 10% of the known
exact solution in several cases.},
Author = {A. D. D. Craik},
Journal = {J. Fluid Mech.},
Pages = {657--670},
Title = {Eigenvalue bounds in linear inviscid stability theory},
Volume = {53},
Year = {1972}}
@article{Cra89,
Abstract = {A formulation, previously employed to find exact
Navier-Stokes solutions for planar disturbances in two- and
three-dimensional flows with spatially uniform rates of strain, is here
adapted to incorporate the contribution of various types of body force.
In the absence of body forces, it is known that unbounded flows with
constant vorticity and elliptical streamlines are unstable to certain
planar disturbances, which are amplified by a Floquet mechanism. The
influence of a Coriolis force upon this instability mechanism is here
described in detail, as an illustration of the general formulation. The
results are likely to be of geophysical interest and may also have
relevance to the breakdown of closed-eddy structures in turbulence. The
final section of the paper reviews other systems for which analogous
exact solutions may be obtained.},
Author = {A. D. D. Craik},
Journal = {J. Fluid Mech.},
Pages = {275--292},
Title = {The stability of unbounded two- and three-dimensional flows subject to body forces: some exact solutions},
Volume = {198},
Year = {1989}}
@misc{cresson-2006,
Author = {J. Cresson and C. Guillet},
Note = {\arXiv{math/0601701}},
Title = {Hyperboliity versus partial-hyperbolicity and the transversality-torsion phenomenon},
Year = {2006}}
@Article{CroDav06,
author = {J. J. Crofts and R. L. Davidchack},
title = {Efficient detection of periodic orbits in chaotic systems by stabilizing transformations},
journal = {SIAM J. Sci. Comp.},
year = {2006},
volume = {28},
pages = {1275--1288}
}
@PHDTHESIS{Crofts07thesis,
author = {J. J. Crofts},
title = {Efficient method for detection of periodic orbits in chaotic maps
and flows},
school = {Univ. of Leicester},
year = {2007},
address = {Leicester, UK},
note = {\arXiv{nlin.CD/0706.1940}}
}
@Article{cross93,
author = {M. C. Cross and P. C. Hohenberg},
title = {Pattern formation outside of equilibrium},
journal = {Rev. Mod. Phys},
year = {1993},
volume = {65}
}
@article{cross94sp,
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.},
Author = {M. C. Cross and P. C. Hohenberg},
Journal = {Science},
Pages = {1569--1570},
Title = {Spatiotemporal Chaos},
Volume = {263},
Year = {1994}}
@article{CSS85,
Author = {P. Cvitanovi\'c and B. Shraiman and B. S\"oderberg},
Doi = {10.1088/0031-8949/32/4/003},
Journal = {Phys. Scr.},
Pages = {263--270},
Title = {Scaling laws for mode lockings in circle maps},
Volume = {32},
Year = {1985},
Bdsk-Url-1 = {http://dx.doi.org/10.1088/0031-8949/32/4/003}}
@Article{CV93,
author = {P. Cvitanovi\'c and G. Vattay },
title = {Entire {Fredholm} determinants for evaluation of semiclassical and thermodynamical spectra},
journal = {Phys. Rev. Lett.},
year = {1993},
volume = {71},
pages = {4138--4141},
note = {\arXiv{chao-dyn/9307012}}
}
@unpublished{Cvi07,
Author = {P. Cvitanovi\'{c}},
Note = {\\ \wwwcb{/$\sim$predrag/papers/trace.pdf}},
Title = {Continuous symmetry reduced trace formulas},
Year = {2007}}
@article{CviGib10,
Author = {Cvitanovi{\'c}, P. and Gibson, J. F.},
Journal = {Phys. Scr. T},
Pages = {014007},
Title = {Geometry of turbulence in wall-bounded shear flows: {Periodic} orbits},
Volume = {142},
Year = {2010},
doi = {10.1088/0031-8949/2010/T142/014007}
}
@inproceedings{CviLip12,
Address = {Melville, New York},
Author = {P. Cvitanovi\'c and D. Lippolis},
Booktitle = {Let's Face Chaos through Nonlinear Dynamics},
Doi = {10.1063/1.4745574},
Editor = {M. Robnik and V. G. Romanovski},
Note = {\arXiv{1206.5506}},
Pages = {82--126},
Publisher = {Am. Inst. of Phys.},
Title = {Knowing when to stop: {How} noise frees us from determinism},
Year = {2012},
doi = {10.1063/1.4745574}}
@article{CvitaEckardt,
Author = {P. Cvitanovi{\'c} and B. Eckhardt},
Journal = {Nonlinearity},
Note = {\arXiv{chao-dyn/9303016}},
Pages = {277},
Title = {Symmetry decomposition of chaotic dynamics},
Volume = {6},
Year = {1993}}
@Inproceedings{CvitLanCrete02,
author = {P. Cvitanovi{\'c} and Y. Lan},
title = {Turbulent fields and their recurrences},
booktitle = {Proceedings of 10\textsuperscript{th} International Workshop on Multiparticle Production: Correlations and Fluctuations in QCD},
year = {2003},
editor = {N. Antoniou},
address = {Singapore},
publisher = {World Scientific},
note = {\arXiv{nlin.CD/0308006}}
}
@book{cvt89b,
Address = {Bristol},
Author = {P. Cvitanovi{\'c}},
Publisher = {Adam Hilger},
Title = {Universality in Chaos},
Year = {1989}}
@inproceedings{CvWiAv12,
Author = {Cvitanovi{\'c}, P. and Willis, A. P. and Avila, M.},
Booktitle = {Proceed. ICTAM 2012 Intern. Congr. Theor. and Appl. Mech.},
Editor = {Jianxiang Wang},
Title = {Revealing the state space of turbulent pipe flow by symmetry reduction},
Year = {2012}}
@article{Da73,
Abstract = {conjectured for many years that plane Couette
flow is stable to infinitesimal disturbances although
this has never been proved. In this paper we use a,
combination of asymptotic analysis and numerical
computation to examine the associated Orr-Sommerfeld
differential problem in a systematic manner. We obtain
new evidence that the conjecture is, in all probability,
correct.},
Author = {A. Davey},
Journal = {J. Fluid Mech.},
Pages = {369--380},
Title = {On the stability of plane {Couette} flow to infinitesimal disturbances},
Volume = {57},
Year = {1973}}
@Article{Dahlqv94,
author = {P. Dahlqvist},
title = {Determination of resonance spectra for bound chaotic systems},
journal = {J. Phys. A},
year = {1994},
volume = {27},
pages = {763--785},
doi = {10.1088/0305-4470/27/3/020},
abstract = {We consider the computation of the eigenvalues of the
evolution operator-the resonance spectrum-by means of the zeros of
a zeta function. In particular we address the problems of applying
this formalism to bound chaotic systems, caused by e.g.
intermittency and non-completeness of the symbolic dynamics. For
bound intermittent systems we derive an approximation of the zeta
function. With the aid of this zeta function it is argued that
bound systems with long time tails have branch cuts in the zeta
function and traces (of the evolution operator) approaching unity
as a power law. We also show that the dominant time scale can be
much longer than the period of the shortest periodic orbit, as is,
for example, the case for the hyperbola billiard. Isolated zeros of
the zeta function for the hyperbola billiard are evaluated by means
of a cycle expansion. Crucial for the success of this approach is
the identification of a sequence of periodic orbit responsible for
a logarithmic branch cut in the zeta function. Semiclassical
implications are briefly discussed.},
bdsk-url-1 = {http://dx.doi.org/10.1088/0305-4470/27/3/020}
}
@article{Dang86,
Author = {Dangelmayr, G.},
Doi = {10.1080/02681118608806011},
Journal = {Dyn. Sys.},
Pages = {159--185},
Title = {Steady-state mode interactions in the presence of {0(2)}-symmetry},
Volume = {1},
Year = {1986},
Bdsk-Url-1 = {http://dx.doi.org/10.1080/02681118608806011}}
@book{DasBuch,
Title = {Chaos: Classical and Quantum},
Author = {P. Cvitanovi{\'c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
Address = {Copenhagen},
Note = {{\tt ChaosBook.org}},
Publisher = {Niels Bohr Inst.},
Year = {2015}}
@misc{DasBuchMirror,
Author = {P. Cvitanovi{\'c}},
Note = {chapter {\em ``World in a mirror''}, in Ref.~\cite{DasBuch}},
Year = {2008}}
@article{DauBer12,
Author = {Dauchot, O. and Bertin, E.},
Doi = {10.1103/PhysRevE.86.036312},
Journal = {Phys. Rev. E},
Numpages = {10},
Pages = {036312},
Title = {Subcritical transition to turbulence: {What} we can learn from the physics of glasses},
Volume = {86},
Year = {2012},
Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevE.86.036312}}
@book{DayanAbbott01,
Address = {Cambridge, MA},
Author = {Dayan, P. and Abbott, L. F.},
Publisher = {MIT Press},
Title = {Theoretical Neuroscience},
Year = {2001}}
@Incollection{DBD96,
author = {Dauchot, O. and Bottin, S. and Daviaud, F.},
title = {Pairs of counter-rotating streamwise vortices: {A} finite amplitude solution in plane {Couette} flow},
booktitle = {Advances in Turbulence VI},
publisher = {Springer},
year = {1996},
editor = {Gavrilakis, S. and Machiels, L. and Monkewitz, P. A.},
volume = {36},
isbn = {978-94-010-6618-1},
pages = {329--332},
doi = {10.1007/978-94-009-0297-8_92}
}
@unpublished{DCTSCD14,
Author = {Ding, X. and Cvitanovi\'c, P. and Takeuchi, K. A. and Siminos, E. and Chat\'e, H. and Davidchack, R. L.},
Note = {In preparation},
Title = {Determination of the physical dimension of a dissipative system by periodic orbits},
Year = {2014}}
@article{DD95,
Author = {O. Dauchot and F. Daviaud},
Journal = {Phys. Fluids},
Pages = {335--343},
Title = {Finite-amplitude perturbation and spots growth mechanism in plane {Couette} flow},
Volume = {7},
Year = {1995}}
@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}
}
@article{DDM98,
Author = {O. Dauchot and F. Daviaud and P. Manneville},
Journal = {Phys. Fluids},
Pages = {2597--2607},
Title = {Experimental evidence of streamwise vortices as finite amplitude solutions in transitional plane {Couette} flow},
Volume = {10},
Year = {1998}}
@inproceedings{DDS90,
Author = {D. D. Stretch},
Booktitle = {Annual Research Briefs},
Pages = {145--157},
Publisher = {Center for Turbulence Research, Stanford University},
Title = {Automated pattern eduction from turbulent flow diagnostics},
Year = {1990}}
@article{Dea63,
Abstract = {The problem of the stability of plane Couette
flow to infinitesimal disturbances is carried numerically
to larger Reynolds numbers than heretofore. The flow is
definitely stable up to R = 1430},
Author = {J. W. Deardorff},
Journal = {J. Fluid Mech.},
Pages = {623--630},
Title = {On the stability of viscous plane {Couette} flow},
Volume = {15},
Year = {1963}}
@article{deLMeAvHo12,
title = {Edge state in pipe flow experiments},
author = {de Lozar, A. and Mellibovsky, F. and Avila, M. and Hof, B.},
journal = {Phys. Rev. Lett.},
volume = {108},
pages = {214502},
year = {2012},
doi = {10.1103/PhysRevLett.108.214502},
}
@Article{Dembo1982,
author = {R. S. Dembo and S. C. Eisenstat and T. Steihaug},
title = {Inexact {Newton} methods},
journal = {SIAM J. Numer. Anal.},
year = {1982},
volume = {19},
pages = {400--408}
}
@article{DeMeMe14,
Author = {Deguchi, K. and Meseguer, A. and Mellibovsky, F.},
Doi = {10.1103/PhysRevLett.112.184502},
Journal = {Phys. Rev. Lett.},
Pages = {184502},
Title = {Subcritical equilibria in {Taylor-Couette} flow},
Volume = {112},
Year = {2014}
}
@Incollection{DeNa12,
author = {Deguchi, K. and Nagata, M.},
title = {Exact coherent state with travelling hairpin vortex in plane {Couette} flow},
booktitle = {Progress in Turbulence and Wind Energy IV},
publisher = {Springer},
year = {2012},
editor = {Oberlack, M. and Peinke, J. and Talamelli, A. and Castillo, L. and H\"olling, M.},
volume = {141},
series = {Proceedings in Physics},
pages = {129--134},
addr = {New York},
bdsk-url-1 = {http://dx.doi.org/10.1007/978-3-642-28968-2_27},
doi = {10.1007/978-3-642-28968-2_27}
}
@book{Dennis96,
Addr = {Philadelphia},
Author = {J.E. Dennis and R.B. Schnabel},
Publisher = {SIAM},
Title = {Numerical Methods for Unconstrained Optimization and Nonlinear Equations},
Year = {1996}}
@article{DesBra13,
Author = {Descalzi, O. and Brand, H. R.},
Doi = {10.1103/PhysRevE.87.022915},
Journal = {Phys. Rev. E},
Pages = {022915},
Title = {Quasi-one-dimensional solutions and their interaction with two-dimensional dissipative solitons},
Volume = {87},
Year = {2013},
Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevE.87.022915}}
@article{DeSo14,
Author = {Dennis, D. J. C. and Sogaro, F. M.},
Journal = {Phys. Rev. Lett.},
Pages = {234501},
Title = {Distinct organizational states of fully developed turbulent pipe flow},
Volume = {113},
doi = {10.1103/PhysRevLett.113.234501},
Year = {2014}}
@article{Deufl84,
Author = {P. Deuflhard},
Journal = {BIT},
Pages = {456},
Title = {Computation of periodic solutions of nonlinear {ODE}},
Volume = {34},
Year = {1984}}
@book{devnmap,
Address = {Reading, MA},
Author = {R. L. Devaney},
Publisher = {Addison-Wesley},
Title = {An Introduction to Chaotic Dynamical systems},
Year = {1989}}
@article{DFGHMS86,
Author = {Dombre, T. and Frisch, U. and Greene, J. M. and H\'enon, M. and Mehr, A. and Soward, A. M.},
Doi = {10.1017/S0022112086002859},
Journal = {J. Fluid Mech.},
Pages = {353--391},
Title = {Chaotic streamlines in the {ABC} flows},
Volume = {167},
Year = {1986},
Bdsk-Url-1 = {http://dx.doi.org/10.1017/S0022112086002859}}
@article{DHB92,
Author = {F. Daviaud and J. Hegseth and P. Berge},
Journal = {Phys. Rev. Lett.},
Pages = {2511--2514},
Title = {Subcritical transition to turbulence in plane {Couette} flow},
Volume = {69},
Year = {1992}}
@article{DHBEks96,
Author = {H. Dankowicz and P. Holmes and G. Berkooz and J. Elezgaray},
Journal = {Physica D},
Pages = {387--407},
Title = {Local models of spatio-temporally complex fields},
Volume = {90},
Year = {1996}}
@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},
volume={48},
pages={035201},
doi = {10.1088/1751-8113/48/3/035201},
year={2015},
}
@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},
volume={46},
pages={455202},
doi = {10.1088/1751-8113/46/45/455202},
year={2013},
}
@book{Diacu96,
Author = {Diacu, F. and Holmes, P.},
Publisher = {Princeton Univ. Press},
Address = {Princeton},
Title = {Celestial Encounters: The Origins of Chaos and Stability},
Year = {1996}}
@article{DIAFRE84,
Author = {Diaconis, P. and Freedman, D.},
Journal = {Ann. Statistics},
Pages = {793--815},
Title = {Asymptotics of graphical projection pursuit},
Volume = {12},
Year = {1984}}
@article{diag_Fred,
Author = {P. Cvitanovi\'{c} and N. S{\o}ndergaard and G. Palla and G. Vattay and C. P. Dettmann},
Journal = {Phys. Rev. E},
Note = {\arXiv{chao-dyn/9904027}},
Pages = {3936},
Title = {Spectrum of stochastic evolution operators: {Local} matrix representation approach},
Volume = {60},
Year = {1999}}
@unpublished{DingCvit14,
Author = {Ding, X. and Cvitanovi\'c, P.},
addendum = {\arXiv{1406.4885}},
Title = {Periodic eigendecomposition and its application
in {Kuramoto-Sivashinsky} system},
Year = {2014}}
@book{ditt01cq,
Address = {New York},
Author = {W. Dittrich and M. Reuter},
Publisher = {Springer},
Title = {Classical and Quantum Dynamics: From Classical Paths to Path Integrals},
Year = {2001}}
@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}
}
@unpublished{DKY05,
Abstract = {The energy gradient theory has been proposed
with the aim of better understanding the mechanism of
flow transition from laminar flow to turbulent flow. In
this theory, it is suggested that the transition to
turbulence depends on the relative magnitudes of the
energy gradient amplifying the disturbance and the
viscous friction damping that disturbance. For a given
flow geometry and fluid properties, when the maximum of K
(the ratio of the energy gradient in the transverse
direction to that in the streamwise direction) in the
flow field is larger than a certain critical value, it is
expected that instability would occur for some initial
disturbances. In this paper, using the energy analysis,
the equation for calculating K for plane Couette flow is
derived. It is demonstrated that the critical value of K
at subcritical transition is about 370 for plane Couette
flow. This value is about the same as for plane
Poiseuille flow and pipe Poiseuille flow (385-389).
Therefore, it is concluded that the critical value of K
at subcritical transition is about 370-389 for wall
bounded parallel shear flows which include both pressure
and shear driven flows.},
Author = {Hua-Shu Dou and Boo Cheong Kho and Khoon Seng Yeo},
Note = {\arXiv{nlin.CD/0501048}},
Title = {Flow transition in plane {Couette} flow},
Year = {2005}}
@article{DLBconv92,
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},
Author = {F. Daviaud and J. Lega and P. Berg{\'e} and P. Coullet and M. Dubois},
Journal = {Physica D},
Pages = {287--308},
Title = {Spatio-temporal intermittency in a 1{D} convective pattern: {T}heoretical model and experiments},
Volume = {55},
Year = {1992}}
@Article{DM97,
author = {C. P. Dettmann and G. P. Morriss},
title = {Stability ordering of cycle expansions},
journal = {Phys. Rev. Lett.},
year = {1997},
volume = {78},
pages = {4201--4204},
doi = {10.1103/PhysRevLett.78.4201},
bdsk-url-1 = {http://dx.doi.org/10.1103/PhysRevLett.78.4201}
}
@article{doeb94nls,
Author = {H.-D. Doebner and G. A. Goldin},
Journal = {J. Phys. A},
Pages = {1771--1780},
Title = {Properties of nonlinear {Schr\"odinger} equations associated with diffeomorphism group representations},
Volume = {27},
Year = {1994}}
@article{doelman91n,
Author = {A. Doelman},
Journal = {Nonlinearity},
Pages = {231--250},
Title = {Finite-dimensional models of the {Ginzburg-Landau} equation},
Volume = {4},
Year = {1991}}
@article{doercgl87,
Author = {C. R. Doering and J. D. Gibbon and D. D. Holm and B. Nikolaenko},
Journal = {Phys. Rev. Lett.},
Pages = {2911--2914},
Title = {Exact {Lyapunov} dimension of the universal attractor for the
Complex {Ginzburg-Landau} equation},
Volume = {59},
Year = {1987}}
@article{doercgl88,
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.},
Author = {C. R. Doering and J. D. Gibbon and D. D. Holm and B. Nikolaenko},
Journal = {Nonlinearity},
Pages = {279--309},
Title = {Low-dimensional behaviour in the complex {Ginzburg-Landau} equation},
Volume = {1},
Year = {1988}}
@Article{DoLa14,
author = {C. Dong and Y. Lan},
title = {Organization of spatially periodic solutions of the steady {Kuramoto-Sivashinsky} equation},
journal = {Commun. Nonlinear Sci. Numer. Simul.},
year = {2014},
volume = {19},
pages = {2140--2153},
doi = {10.1016/j.cnsns.2013.09.040}
}
@Article{DoLa14a,
author = {C. Dong and Y. Lan},
title = {A variational approach to connecting orbits in nonlinear dynamical systems},
journal = {Phys. Lett. A},
year = {2014},
volume = {378},
pages = {705--712},
doi = {10.1016/j.physleta.2014.01.001},
issn = {0375-9601}
}
@article {DOP79,
author = {H. Degn and L. F. Olsen and J. W. Perram},
title = {Bistability, oscillation, and chaos in an enzyme reaction},
journal = {Ann. N. Y. Acad. Sci.},
volume = {316},
doi = {10.1111/j.1749-6632.1979.tb29503.x},
pages = {623--637},
year = {1979},
}
@article{DR_prl,
Author = {P. Dahlqvist and G. Russberg},
Journal = {Phys. Rev. Lett.},
Pages = {2837},
Title = {Existence of stable orbits in the $x^{2}y^{2}$ potential},
Volume = {65},
Year = {1990}}
@Book{DR04,
title = {Hydrodynamic Stability},
publisher = {Cambridge Univ. Press},
year = {2004},
author = {P. Drazin and W. H. Reid},
address = {Cambridge},
edition = {Second}
}
@article{DR91,
Author = {P. Dahlqvist and G. Russberg},
Journal = {J. Phys. A},
Pages = {4763},
Title = {Cycle expansions for intermittent diffusion},
Volume = {24},
Year = {1991}}
@Article{DR91a,
author = {P. Dahlqvist and G. Russberg},
title = {Periodic orbit quantization of bound chaotic systems},
journal = {J. Phys. A},
year = {1991},
volume = {24},
pages = {4763--4778},
doi = {10.1088/0305-4470/24/20/012},
bdsk-url-1 = {http://dx.doi.org/10.1088/0305-4470/24/20/012}
}
@Article{Dragt05,
author = {Dragt, A. J.},
title = {The symplectic group and classical mechanics},
journal = {Ann. N. Y. Acad. Sci.},
year = {2005},
volume = {1045},
pages = {291--307}
}
@techreport{Dragt11bad,
Author = {Dragt, A. J.},
Institution = {Physics Department, U. of Maryland},
Note = {{www.physics.umd.edu/dsat/dsatliemethods.html}},
Title = {Lie methods for nonlinear dynamics with applications to accelerator physics},
Year = {2011}}
@unpublished{DraHab08,
Author = {Dragt, A. J. and Habib, S.},
Note = {\arXiv{quant-ph/9806056}},
Title = {How {Wigner} functions transform under symplectic maps},
Year = {1998}}
@book{Drazin02,
Author = {P.G. Drazin},
Publisher = {Cambridge Univ. Press},
Title = {Introduction to Hydrodynamic Stability},
Year = {2002}}
@article{DreLau90,
title={Ruelle's rotation frequency for a symplectic chain of dissipative oscillators},
author={U. Dressler and W. Lauterborn},
journal={Phys. Rev. A},
volume={41},
pages={6702},
year={1990},
}
@article{DruCip00,
Author = {T. Drummond and R. Cipolla},
Journal = {Intern. J. Computer Vision},
Pages = {21--41},
Title = {Application of {Lie} algebras to visual servoing},
Volume = {37},
Year = {2000}}
@book{DS,
Address = {Philadelphia},
Author = {Dennis Jr., J. E. and R. B. Schnabel},
Publisher = {SIAM},
Title = {Numerical Methods for Unconstrained Optimization and Nonlinear Equations},
Year = {1996}}
@book{DS83,
Address = {Englewood Cliffs, New Jersey},
Author = {Dennis Jr., J. E. and R. B. Schnabel},
Publisher = {Prentice-Hall},
Title = {Numerical Methods for Unconstrained Optimization and Nonlinear Equations},
Year = {1983}}
@article{du87,
Abstract = {A quantitative theory has been developed which shows that
each classical closed orbit that begins and ends near the nucleus
contributes an oscillatory term to the average oscillator strength.},
Author = {M. L. Du and J. B. Delos},
Journal = {Phys. Rev. Lett.},
Pages = {1731},
Title = {Effect of closed classical orbits on quantum spectra: {Ionization} of atoms in a magnetic field},
Volume = {58},
Year = {1987}}
@Article{duguet07,
author = {Duguet, Y. and Willis, A. P. and Kerswell, R. R.},
title = {Transition in pipe flow: the saddle structure on the boundary of turbulence},
journal = {J. Fluid Mech.},
year = {2008},
volume = {613},
pages = {255--274},
note = {\arXiv{0711.2175}},
doi = {10.1017/S0022112008003248}
}
@article{duguet08,
Author = {Duguet, Y. and Pringle, C. C. T. and Kerswell, R. R.},
Journal = {Phys. Fluids},
Note = {\arXiv{0807.2580}},
Pages = {114102},
Title = {Relative periodic orbits in transitional pipe flow},
Volume = {20},
Year = {2008}}
@Article{DuIbKo06,
author = {Dumortier, F. and Ibanez, S. and Kokubu, H.},
title = {Cocoon bifurcation in three-dimensional reversible vector fields},
journal = {Nonlinearity},
year = {2006},
volume = {19},
pages = {305--328},
doi = {10.1088/0951-7715/19/2/004}
}
@book{DuiKol00,
Address = {New York},
Author = {J. J. Duistermaat and J. A. C. Kolk},
Publisher = {Springer},
Title = {Lie Groups},
Year = {2000}}
@misc{DuisterDSsymm,
Author = {J. J. Duistermaat},
Title = {Dynamical Systems with Symmetry}}
@book{dummitfoote,
Author = {Dummit, D.S. and Foote, R.M.},
Publisher = {John Wiley \& Sons, Inc},
Title = {Abstract Algebra},
Year = {2004}}
@article{DuWiKe10,
Author = {Duguet, Y. and Willis, A. P. and Kerswell, R. R.},
Journal = {J. Fluid Mech.},
Pages = {180--208},
Title = {Slug genesis in cylindrical pipe flow},
Volume = {663},
Year = {2010}}
@article{DV01,
Author = {D. Viswanath},
Journal = {IMA J. Numer. Anal.},
Pages = {387--406},
Title = {Global errors of numerical {ODE} solvers and {Lyapunov}'s theory of stability},
Volume = {21},
Year = {2001}}
@article{DV02,
Author = {D. Viswanath},
Journal = {SIAM Review},
Pages = {478--496},
Title = {The {Lindstedt-Poincar\'e} technique as an algorithm for finding periodic orbits},
Volume = {43},
Year = {2002}}
@article{DV03,
Author = {D. Viswanath},
Journal = {Nonlinearity},
Pages = {1035--1056},
Title = {Symbolic dynamics and periodic orbits of the {Lorenz} attractor},
Volume = {16},
Year = {2003}}
@article{DV04,
Author = {D. Viswanath},
Journal = {Physica D},
Pages = {115--128},
Title = {The fractal property of the {Lorenz} attractor},
Volume = {190},
Year = {2004}}
@Article{DV2000,
author = {O. Dauchot and N. Vioujard},
title = {Phase space analysis of a dynamical model for the subcritical transition to turbulence in plane {Couette} flow},
journal = {Eur. Phys. J. B},
year = {2000},
volume = {14},
pages = {377--381},
doi = {10.1007/s100510050142}
}
@Unpublished{DVLindstedt,
author = {D. Viswanath},
title = {An extension of the {Lindstedt-Poincar\'e} algorithm for computing periodic orbits},
note = {unpublished}
}
@unpublished{DWCDDEGHLSPSST13,
Author = {Dijkstra, H. A. and Wubs, F. W. and Cliffe, A. K. and Doedel, E. and Dragomirescu, I. F. and Eckhardt, B. and Gelfgat, A. Y. and Hazel, A. L. and Lucarini, V. and Salinger, A. G. and Phipps, E. T. and Sanchez-Umbria, J. and Schuttelaars, H. and Tuckerman, L. S. and Thiele, U.},
Note = {Commun. Comput. Phys., to appear},
Title = {Numerical bifurcation methods and their application to fluid dynamics: {Analysis} beyond simulation},
Year = {2013}}
@article{EaPoSa10,
Author = {J. Early and J. Pohjanpelto and R. Samelson},
Journal = {Discrete Contin. Dynam. Systems},
Pages = {1571--1586},
Title = {Group foliation of equations in geophysical fluid dynamics},
Volume = {27},
Year = {2010}}
@article{EAS87,
Author = {E. A. Spiegel},
Journal = {Proc. Roy. Soc. A},
Pages = {87},
Title = {Chaos: a mixed metaphor for turbulence},
Volume = {413},
Year = {1987}}
@article{Eben02,
Abstract = {A class of stochastic integer sequences are introducd with every element being a sum of two previous elements. The interplay between randomness and memory leads to a wide variety of behaviors rangin from stretched exponential to log-normal to algebraic growth. The set of all possible sequence values has an intricate structure.},
Author = {E. Ben-Naim and P. L. Krapivsky},
Journal = {J. Phys. A},
Pages = {L557--L563},
Title = {Growth and structure of stochastic sequences},
Volume = {35},
Year = {2002}}
@article{eck81,
Author = {J.-P. Eckmann},
Journal = {Rev. Mod. Phys.},
Pages = {643},
Title = {Roads to turbulence in dissipative dynamical systems},
Volume = {53},
Year = {1981}}
@article{eckcgl,
Author = {B. Janiaud and A. Pumir and D. Bensimon and V. Croquette and H. Richter and L. Kramer},
Journal = {Physica D},
Pages = {269--286},
Title = {The {Eckhaus} instability for travelling waves},
Volume = {55},
Year = {1992}}
@article{eckerg,
Author = {J.-P. Eckmann and D. Ruelle},
Journal = {Rev. Mod. Phys.},
Pages = {617},
Title = {Ergodic theory of chaos and strange attractors},
Volume = {57},
Year = {1985}}
@Incollection{EckhardtFaisst05,
author = {Eckhardt, B. and Faisst, H.},
title = {Dynamical Systems and the Transition to Turbulence},
booktitle = {IUTAM Symposium on Laminar-Turbulent Transition and Finite Amplitude Solutions},
publisher = {Springer},
year = {2005},
editor = {Mullin, T. and Kerswell, R.},
volume = {77},
isbn = {978-1-4020-4048-1},
pages = {35--50},
doi = {10.1007/1-4020-4049-0_2}
}
@article{ecklp88,
Author = {J.-P. Eckmann and C. E. Wayne},
Journal = {J. Stat. Phys.},
Pages = {853--878},
Title = {Liapunov spectra for infinite chains of nonlinear oscillators},
Volume = {50},
Year = {1988}}
@book{eckman80map,
Address = {Boston},
Author = {P. Collet and J.-P. Eckmann},
Publisher = {Birkh{\"a}user},
Title = {Iterated Maps on the Interval as Dynamical Systems},
Year = {1980}}
@Article{EDASRU14,
author = {Engl, T. and Dujardin, J. and Arg{\"u}elles, A. and
Schlagheck, P. and Richter, K. and Urbina, J. D.},
title = {Coherent backscattering in {Fock} space: {A} signature
of quantum many-body interference in interacting bosonic systems},
journal = {Phys. Rev. Lett.},
year = {2014},
volume = {112},
pages = {140403},
doi = {10.1103/PhysRevLett.112.140403}
}
@Misc{EDUR15,
author = {Engl, T. and Urbina, J. D. and Richter, K.},
title = {The semiclassical propagator in {Fock} space: dynamical echo and many-body interference},
year = {2015},
addendum = {\arXiv{1511.07234}}
}
@article{Eggels94,
Author = {J. G. M. Eggels and F. Unger and M. H. Weiss and J. Westerweel and R. J. Adrian and R. Freidrich and F. T. M. Nieuwstadt},
Journal = {J. Fluid Mech.},
Pages = {175--209},
Title = {Fully developed turbulent pipe flow: a comparison between direct numberical simulation and experiment},
Volume = {268},
Year = {1994}}
@Article{egolf00,
author = {D. A. Egolf and I. V. Melnikov and W. Pesch and R. E. Ecke},
title = {Mechanisms of extensive spatiotemporal chaos in {R}ayleigh-{B\'e}nard convection},
journal = {Nature},
year = {2000},
volume = {404},
pages = {733},
abstract = {Numerical study on Boussinesq equation(pseudospectral
method with time splitting scheme). Demonstrate the extensivity of
Lyapunov dimension of the extended system and showed that the
generating mechanism is breaking or connecting rolls mediated by
defects at the domain boundary and exsisting for only a short time.}
}
@article{egolf94,
Abstract = {Numerical study on CGLe. Show that even though the
fractal dimension increases with system size the simple scaling
relation is not correct, i.e. the correlation length can not be used
to directly calculate the fractal dimension.},
Author = {D. A. Egolf and H. S. Greenside},
Journal = {Nature},
Pages = {129},
Title = {Relation between fractal dimension and spatial correlation length for extensive chaos},
Volume = {369},
Year = {1994}}
@article{egolf98,
Author = {D. A. Egolf and I. V. Melnikov and E. Bodenschatz},
Journal = {Phys. Rev. Lett.},
Pages = {3228},
Title = {Importance of local pattern properties in spiral defect chaos},
Volume = {80},
Year = {1998}}
@article{EGP70,
Author = {T. Ellingsen and B. Gjevik and E. Palm},
Journal = {J. Fluid Mech.},
Pages = {97--112},
Title = {On the non-linear stability of plane {Couette} flow},
Volume = {40},
Year = {1970}}
@article{EGtran95,
Abtract = {For the CGLe on a large periodic interval, the transition from defect to phase turbulence is continuous. The conclusion is based on the calculation of the defect density, Lyapunov dimension density and correlation lengths.},
Author = {D. A. Egolf and H. S. Greenside},
Journal = {Phys. Rev. Lett.},
Pages = {1751},
Title = {Characterization of the transition from defect to phase turbulence},
Volume = {74},
Year = {1995}}
@Article{Ehren08,
author = {Ehrenstein, U. and Nagata, M. and Rincon, F.},
title = {Two-dimensional nonlinear plane {Poiseuille}-{Couette} flow homotopy revisited},
journal = {Phys. Fluids},
year = {2008},
volume = {20},
pages = {064103},
doi = {10.1063/1.2943675}
}
@Article{Eisenstat1994,
author = {S. C. Eisenstat and H. F. Walker },
title = {Globally convergent inexact {Newton} methods},
journal = {SIAM J. Optim. },
year = {1994},
volume = {4},
pages = {293--442}
}
@article{EKO04,
Author = {J. P. England and B. Krauskopf and H. M. Osinga},
Journal = {SIAM J. Appl. Dyn. Syst.},
Pages = {161--190},
Title = {Computing one-dimensional stable and unstable sets of planar maps without the inverse},
Volume = {3},
Year = {2004}}
@article{EKR87,
Author = {Eckmann, J.-P. and Kamphorst, S. O. and Ruelle, D.},
Journal = {Europhys. Lett.},
Pages = {973},
Title = {Recurrence plots of dynamical systems},
Volume = {4},
Year = {1987}}
@book{ellip,
Address = {New York},
Author = {D. F. Lawden},
Publisher = {Springer},
Title = {Elliptic Functions and Applications},
Year = {1989}}
@techreport{Elton08,
Author = {Elton, J. R.},
Institution = {School of Physics, Georgia Inst. of Technology},
Note = {{\tt ChaosBook.org/projects}},
Title = {An exploration of moderate {Reynolds} number plane {Couette} turbulence},
Year = {2007}}
@article{EM99,
Author = {B. Eckhardt and A. Mersmann},
Journal = {Phys. Rev. E},
Pages = {509--517},
Title = {Transition to turbulence in a shear flow},
Volume = {60},
Year = {1999}}
@book{encDicMath,
Address = {Cambridge, MA},
Author = {It{\^o}, K.},
Publisher = {MIT Press},
Title = {Encyclopedic Dictionary of Mathematics},
Year = {1993}}
@Misc{EnUrRi15,
author = {Engl, T. and Urbina, J. D. and Richter, K.},
title = {Boson sampling as canonical transformation: {A} semiclassical approach in {Fock} space},
year = {2015},
addendum = {\arXiv{1502.07483}}
}
@Article{EPUR14,
author = {Engl, T. and Pl{\"o}ss, P. and Urbina, J. D. and Richter, K.},
title = {The semiclassical propagator in fermionic {Fock} space},
journal = {Theor. Chem. Acc.},
year = {2014},
volume = {133},
pages = {1563},
doi = {10.1007/s00214-014-1563-9}
}
@Article{ErshPot98,
author = {S. V. Ershov and A. B. Potapov},
title = {On the concept of stationary {Lyapunov} basis},
journal = {Physica D},
year = {1998},
volume = {118},
pages = {167--198}
}
@article{ERTW91,
Author = {Ezra, G. S. and Richter, K. and Tanner, G. and Wintgen, D.},
Journal = {J. Phys. B},
Pages = {L413--L420},
Title = {Semiclassical cycle expansion for the helium atom},
Volume = {24},
Year = {1991}}
@article{ESHW07,
Author = {B. Eckhardt and T. M. Schneider and B. Hof and J. Westerweel},
Journal = {Ann. Rev. Fluid Mech.},
Pages = {447--468},
Title = {Turbulence transition in pipe flow},
Volume = {29},
Year = {2007}}
@article{ET02,
Author = {M. Embree and L. Trefethen},
Journal = {SIAM J. Sci. Comput.},
Pages = {83--590},
Title = {Generalizing eigenvalue theorems to pseudospectra theorems},
Volume = {23},
Year = {2002}}
@Inproceedings{etc12,
author = {Cvitanovi{\'c}, P. and Gibson, J. F.},
title = {Geometry of state space in plane {Couette} flow},
booktitle = {Advances in Turbulence XII},
year = {2009},
editor = {A. V. Johansson and B. Eckhardt},
series = {Proc. 12{th} EUROMECH Eur. Turb. Conf.},
pages = {41--43},
address = {New York},
publisher = {Springer}
}
@article{ETFS,
Author = {W. S. Edwards and L. S. Tuckerman and R. A. Friesner and D. C. Sorensen},
Journal = {J. Comput. Physics},
Pages = {82--102},
Title = {Krylov methods for the incompressible {Navier-Stokes} equations},
Volume = {110},
Year = {1994}}
@Article{expholm88,
author = {P. Holmes and J. Marsden and J. Scheurle},
title = {Exponentially small splittings of separatrices with applications to {KAM} theory and degenerate bifurcations},
journal = {Cont. Math., AMS},
year = {1988},
volume = {81},
pages = {213--244}
}
@article{ExpsPCtransition,
Author = {N. Tillmark and P. H. Alfredsson},
Journal = {J. Fluid Mech.},
Pages = {89--102},
Title = {Experiments on transition in plane {Couette} flow},
Volume = {235},
Year = {1992}}
@Article{FaHa12,
author = {M. Farazmand and G. Haller},
title = {Computing {Lagrangian} coherent structures from their variational theory},
journal = {Chaos},
year = {2012},
volume = {22},
pages = {013128},
doi = {10.1063/1.3690153}
}
@article{FAIECK04,
Author = {Faisst, H. \& Eckhardt, B.},
Journal = {J. Fluid Mech.},
Pages = {343--352},
Title = {Sensitive dependence on initial conditions in transition to turbulence},
Volume = {504},
Year = {2004}}
@Article{FaisstPRE00,
author = {H. Faisst and B. Eckhardt},
title = {Transition from {Couette-Taylor} system to plane {Couette} system},
journal = {Phys. Rev. E},
year = {2000},
volume = {61},
pages = {7227--7230},
doi = {10.1103/PhysRevE.61.7227}
}
@misc{Faraz15,
Author = {Farazmand, M.},
Note = {\arXiv{1508.06363}},
Title = {An adjoint-based approach for finding invariant solutions of
{Navier-Stokes} equations},
Year = {2015}}
@article{farazmand2014coherent,
Author = {Farazmand, M. and Haller, G.},
Journal = {arXiv preprint arXiv:1402.4835},
Title = {How coherent are the vortices of two-dimensional turbulence?},
Year = {2014}}
@Article{FarHal13,
author = {M. Farazmand and G. Haller},
title = {Attracting and repelling {Lagrangian} coherent structures from a single computation},
journal = {Chaos},
year = {2013},
volume = {23},
pages = {023101},
doi = {10.1063/1.4800210},
bdsk-url-1 = {http://dx.doi.org/10.1063/1.4800210}
}
@article{farmer87,
Author = {J. D. Farmer and J. J. Sidorowich},
Journal = {Phys. Rev. Lett.},
Pages = {845},
Title = {Predicting Chaotic Time Series},
Volume = {59},
Year = {1987}}
@article{FE03,
Author = {H. Faisst and B. Eckhardt},
Journal = {Phys. Rev. Lett.},
Pages = {224502},
Title = {Traveling waves in pipe flow},
Volume = {91},
Year = {2003}}
@article{FeHa82,
Author = {M. J. Feigenbaum and B. Hasslacher},
Journal = {Phys. Rev. Lett.},
Pages = {605--609},
Title = {Irrational decimations and path-integrals for external noise},
Volume = 49,
Year = 1982}
@article{feigen78,
Author = {M. J. Feigenbaum},
Journal = {J. Stat. Phys.},
Pages = {25--52},
Title = {Quantitative universality for a class of nonlinear transformations},
Volume = {19},
Year = {1978}}
@article{feigen79,
Author = {M. J. Feigenbaum},
Journal = {J. Stat. Phys.},
Note = {Reprinted in \refref{cvt89b}.},
Pages = {669--706},
Title = {The universal metric properties of nonlinear transformations},
Volume = {21},
Year = {1979}}
@article{feigen79scal,
Author = {M. J. Feigenbaum},
Journal = {Phys. Lett. A},
Pages = {375--378},
Title = {The onset spectrum of turbulence},
Volume = {74},
Year = {1979}}
@Inproceedings{FeigKen83,
author = {M. J. Feigenbaum and R. D. Kenway},
title = {The onset of chaos},
booktitle = {Statistical and Particle Physics},
year = {1983},
editor = {K. C. Bowler and A. J. McKane},
pages = {1--100},
address = {Edinburgh},
organization = {Proc. 26\textsuperscript{th} Scottish Universities Summer School in Physics},
publisher = {SUSSP Publ.}
}
@article{Feiste1994,
Author = {U. Feiste and D. J. As and A. Erhardt},
Journal = {IEEE Photon. Technol. Lett.},
Pages = {106--108},
Title = {18 {GHZ} all-optical frequency locking and clock recovery using a self-pulsating two-section laser},
Volume = {6},
Year = {1994}}
@article{FeKaSh82,
Author = {M.J. Feigenbaum and L.P. Kadanoff and S.J. Shenker},
Doi = {10.1016/0167-2789(82)90030-6},
Journal = {Physica D},
Pages = {370--386},
Title = {Quasiperiodicity in dissipative systems: {A} renormalization group analysis},
Volume = {5},
Year = {1982},
Bdsk-Url-1 = {http://dx.doi.org/10.1016/0167-2789(82)90030-6}}
@article{FelCou98,
Author = {Feldman, G. J. and Cousins, R. D.},
Doi = {10.1103/PhysRevD.57.3873},
Journal = {Phys. Rev. D},
Pages = {3873--3889},
Title = {Unified approach to the classical statistical analysis of small signals},
Volume = {57},
Year = {1998}
}
@article{FelsOlver98,
Author = {M. Fels and P. J. Olver},
Journal = {Acta Appl. Math.},
Pages = {161--213},
Title = {Moving coframes: {I. A} practical algorithm},
Volume = {51},
Year = {1998}}
@article{FelsOlver99,
Author = {M. Fels and P. J. Olver},
Journal = {Acta Appl. Math.},
Pages = {127--208},
Title = {Moving coframes: {II. Regularization} and theoretical foundations},
Volume = {55},
Year = {1999}}
@TechReport{fftwmanual,
author = {M. Frigo and S.G. Johnson},
title = {{FFT}W 3.0 User's Manual},
institution = {fftw.org},
year = {2006},
bdsk-url-1 = {www.fftw.org/fftw3.pdf},
publisher = {M. Frigo and S.G. Johnson},
url = {www.fftw.org/fftw3.pdf}
}
@article{FJolly95,
Abstract = {An algebraic approximation of the global attractors is
developed for the {Lorenz} and {KS} system.},
Author = {C. Foias and M. S. Jolly},
Journal = {Nonlinearity},
Pages = {295--319},
Title = {On the numerical algebraic approximation of global attractors},
Volume = {8},
Year = {1995}}
@Article{FKYY83,
author = {P. Frederickson and J. L. Kaplan and E. D. Yorke and J. A. Yorke},
title = {The {Liapunov} dimension of strange attractors},
journal = {J. Diff. Eqn.},
year = {1983},
volume = {49},
pages = {185--207},
doi = {10.1016/0022-0396(83)90011-6}
}
@article{fl92,
Abstract = {Study the complex domains of validity in generalized standard mappings and compare Pad{\'e} methods and periodic orbit methods.},
Author = {C. Falcolini and R. de la Llave},
Journal = {J. Stat. Phys.},
Pages = {645--666},
Title = {Numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors},
Volume = {67},
Year = {1992}}
@UNPUBLISHED{Flor11,
author = {Floratos, E.},
title = {Matrix quantization of turbulence},
note = {\arXiv{1109.1234}},
year = {2011}
}
@article{FNSTks85,
Author = {C. Foias and B. Nicolaenko and G. R. Sell and R. Temam},
Journal = {C. R. Acad. Sci. I-Math},
Pages = {285--288},
Title = {Inertial manifold for the {Kuramoto-Sivashinsky} equation},
Volume = {301},
Year = {1985}}
@article{FNSTks88,
Author = {C. Foias and B. Nicolaenko and G. R. Sell and R. Temam},
Journal = {J. Math. Pure Appl.},
Pages = {197--226},
Title = {Inertial manifold for the {Kuramoto-Sivashinsky} equation and an estimate of their lowest dimension},
Volume = {67},
Year = {1988}}
@Article{focusPOT,
author = {Cvitanovi{\'c}, P.},
title = {Recurrent flows: the clockwork behind turbulence},
journal = {J. Fluid Mech. Focus Fluids},
year = {2013},
volume = {726},
pages = {1--4},
doi = {10.1017/jfm.2013.198}
}
@article{foias88,
Author = {C. Foias and M. S. Jolly and I. G. Kevrekidis and G. R. Sell and E. S. Titi},
Journal = {Phys. Lett. A},
Pages = {433},
Title = {On the computation of inertial manifolds},
Volume = {131},
Year = {1988}}
@article{Foiasks88,
Author = {C. Foias and B. Nicolaenko and G. R. Sell and R. T{\'e}mam},
Journal = {J. Math. Pures et Appl.},
Pages = {197},
Title = {{Kuramoto-Sivashinsky} equation},
Volume = {67},
Year = {1988}}
@Article{FowlerCLE82,
author = {Fowler , A. C. and {Gibbon}, J. D. and {McGuinness}, M. J.},
title = {The complex {Lorenz} equations},
journal = {Physica D},
year = {1982},
volume = {4},
pages = {139--163}
}
@article{FowlerCLE83,
Author = {Fowler , A. C. and Gibbon, J. D. and McGuinness, M. J.},
Journal = {Physica D},
Pages = {139--163},
Title = {The real and complex {Lorenz} equations and their relevance to physical systems},
Volume = {7},
Year = {1983}}
@Article{FrankANM03,
author = {J. Frank and W. Hundsdorfer and J.G. Verwer},
title = {On the stability of implicit-explicit linear multistep methods},
journal = {Appl. Num. Math.},
year = {1997},
volume = {25},
pages = {193--205}
}
@Misc{franzosi-2004,
author = {R. Franzosi and P. Poggi and M. Cerruti-Sola},
title = {{Lyapunov} exponents from unstable periodic orbits in the {FPU}-beta model},
year = {2004},
note = {\arXiv{nlin/0408056}}
}
@Article{FrCv11,
author = {Froehlich, S. and Cvitanovi{\'c}, P.},
title = {Reduction of continuous symmetries of chaotic flows by the method of slices},
journal = {Commun. Nonlinear Sci. Numer. Simul.},
year = {2012},
volume = {17},
pages = {2074--2084},
note = {\arXiv{1101.3037}},
doi = {10.1016/j.cnsns.2011.07.007},
bdsk-url-1 = {http://dx.doi.org/10.1016/j.cnsns.2011.07.007}
}
@Inproceedings{Fried87,
author = {Fried, D.},
title = {Lefschetz formulas for flows},
booktitle = {The Lefschetz Centennial Conference},
year = {1987},
editor = {D. Sundararaman and A. Verjovsky and S. Gitler},
volume = {58},
pages = {41--43},
address = {Providence, RI},
publisher = {AMS},
bdsk-url-1 = {http://dx.doi.org/10.1090/conm/058.3},
doi = {10.1090/conm/058.3}
}
@article{Frigo05,
Author = {M. Frigo and S.G. Johnson},
Journal = {Proceedings of the IEEE},
Note = {Special issue on ``Program Generation, Optimization, and Platform Adaptation''},
Pages = {216--231},
Title = {The Design and Implementation of {FFTW3}},
Volume = {93},
Year = {2005}}
@book{frisch,
Address = {Cambridge, UK},
Author = {U. Frisch},
Publisher = {Cambridge Univ. Press},
Title = {Turbulence},
Year = {1996}}
@article{frisch86,
Author = {U. Frisch and B. Hasslacher and Y. Pomeau},
Journal = {Phys. Rev. Lett.},
Pages = {1505--1508},
Title = {Lattice-gas automata for the navier-stokes equation},
Volume = {56},
Year = {1986}}
@article{frisch87,
Author = {U. Frisch and Z. S. She and P.-L. Sulem},
Journal = {Physica D},
Pages = {382--392},
Title = {Large-scale flow driven by the anisotropic kinetic alpha effect},
Volume = {28},
Year = {1987}}
@book{froyland,
Address = {Bristol},
Author = {Fr\o yland, J.},
Publisher = {Inst. of Phys. Publ.},
Title = {Chaos and coherence},
Year = {1992}}
@Article{froyland_alfsen,
author = {Fr\o yland, J. and Alfsen, K. H.},
title = {{Lyapunov}-exponent spectra for the {Lorenz} model},
journal = {Phys. Rev. A},
year = {1984},
volume = {29},
pages = {2928}
}
@article{FSTks86,
Abstract = {The KSe on the periodic interval has stable steady
states of period 2*pi/N. These },
Author = {U. Frisch and Z. S. She and O. Thual},
Journal = {J. Fluid Mech.},
Pages = {221--240},
Title = {Viscoelastic behavior of cellular solutions to the {Kuramoto-Sivashinsky} model},
Volume = {168},
Year = {1986}}
@unpublished{FWweb,
Author = {F. Waleffe},
Title = {Exact coherent structures},
Url = {www.math.wisc.edu/~waleffe/ECS/RRC-data.html},
Year = {2006}
}
@article{G00,
Author = {S. Grossmann},
Journal = {Rev. Mod. Phys.},
Pages = {603--618},
Title = {The onset of shear turbulence},
Volume = {72},
Year = {2000}}
@article{GaChi88,
Author = {Garrison, J. C. and Chiao, R. Y.},
Journal = {Phys. Rev. Lett.},
Pages = {165--168},
Title = {Geometrical phases from global gauge invariance of nonlinear classical field theories},
Volume = {60},
Year = {1988}}
@article{Gall69,
Abstract = {A proof by Petrov \cite{Petr40} that plane {Couette} flow
and plane Poiseuille flow are always stable with respect to infinitesimal
disturbances is examined and found to be based on a set of incomplete
functions, thus rendering the proof invalid.},
Author = {A. P. Gallagher},
Journal = {SIAM J. Appl. Math.},
Pages = {765--768},
Title = {On a proof by {Petrov} of the stability of plane {Couette} flow and plane {Poiseuille} flow},
Volume = {17},
Year = {1969}}
@article{GallMer62,
Author = {Gallagher, A. P. and Mercer, A. McD},
Journal = {J. Fluid Mech.},
Pages = {623--631},
Title = {On the behaviour of small disturbances in plane {Couette} flow},
Volume = {13},
Year = {1962}}
@article{GallMer64,
Author = {Gallagher, A. P. and Mercer, A. McD},
Journal = {J. Fluid Mech.},
Pages = {350--350},
Title = {On the behaviour of small disturbances in plane {Couette} flow. Part 2.The higher eigenvalues},
Volume = {18},
Year = {1964}}
@Book{GaQu95,
title = {{Lyapunov} Matrix Equation in System Stability and Control},
publisher = {Academic Press},
year = {1995},
author = {Z. Gaji\'c and M. Qureshi},
address = {New York}
}
@article{garnier,
Author = {N. Garnier and A. Chiffaudel and F. Daviaud and A. Prigent},
Journal = {Physica D},
Pages = {1--29},
Title = {Nonlinear dynamics of waves and modulated waves in 1{D} thermocapillary flows},
Volume = {174},
Year = {2003}}
@Article{gasp95,
author = {P. Gaspard and G. Nicolis and A. Provata and S. Tasaki},
title = {Spectral signature of the pitchfork bifurcation: {Liouville} equation approach},
journal = {Phys. Rev. E},
year = {1995},
volume = {51},
pages = {74--94},
keywords = {dynamical systems, Koopman}
}
@BOOK{gatermannHab,
title = {Computer Algebra Methods for Equivariant Dynamical Systems},
publisher = {Springer},
year = {2000},
author = {Gatermann, K.},
address = {New York}
}
@book{gauge,
Address = {Singapore},
Author = {T. S. Bir{\'o} and S. G. Matinyan and B. M{\"u}ller},
Publisher = {World Scientific},
Title = {Chaos and Gauge Field Theory},
Year = {1994}}
@UNPUBLISHED{GGJLF10,
author = {R. Gilmore and J.-M. Ginoux and T. Jones and C. Letellier and U.
S. Freitas},
title = {Connecting curves for dynamical systems},
note = {\arXiv{1003.1703}},
year = {2010}
}
@unpublished{GHC08a,
Author = {Gibson, J. F. and Cvitanovi{\'c}, P.},
Note = {In preparation},
Title = {Periodic orbits of plane {Couette} flow},
Year = {2010}}
@article{GHCV08,
Author = {J. Halcrow and J. F. Gibson and P. Cvitanovi{\'c} and D. Viswanath},
Journal = {J. Fluid Mech.},
Note = {\arXiv{0808.1865}},
Pages = {365--376},
Title = {Heteroclinic connections in plane {Couette} flow},
Volume = {621},
Year = {2009}}
@article{GHCW07,
Author = {J. F. Gibson and J. Halcrow and P. Cvitanovi{\'c}},
Journal = {J. Fluid Mech.},
Note = {\arXiv{0705.3957}},
Pages = {107--130},
Title = {Visualizing the geometry of state space in plane {Couette} flow},
Volume = {611},
Year = {2008},
doi = {10.1017/S002211200800267X}
}
@article{GiacoOtto05,
Author = {L. Giacomelli and F. Otto},
Journal = {Commun. Pure Appl. Math.},
Pages = {297--318},
Title = {New bounds for the {Kuramoto-Sivashinsky} equation},
Volume = {58},
Year = {2005}}
@article{GibMcCLE82,
Author = {Gibbon, J. D. and McGuinness, M. J.},
Journal = {Physica D},
Pages = {108--122},
Title = {The real and complex {Lorenz} equations in rotating fluids and lasers},
Volume = {5},
Year = {1982}}
@article{GibMcCLE83,
Author = {Gibbon, J. D. and McGuinness, M. J.},
Journal = {Proc. R. Irish. Acad., Ser. A},
Pages = {17--22},
Title = {The real and complex {Lorenz} equations},
Volume = {83},
Year = {1983}}
@Article{GiBr13,
author = {Gibson, J. F. and {Brand}, E. W.},
title = {Spanwise-localized solutions of planar shear flows},
journal = {J. Fluid Mech.},
year = {2014},
volume = {745},
pages = {25--61},
note = {\arXiv{1304.6323}},
doi = {10.1017/jfm.2014.89}
}
@misc{GibsonMovies,
Author = {Gibson, J. F. and Cvitanovi{\'c}, P.},
Institution = {Georgia Inst. of Technology},
Note = {{\tt {ChaosBook.org/tutorials}}},
Title = {Movies of plane {Couette}},
Year = {2013}}
@phdthesis{GibsonPhD,
Author = {J. F. Gibson},
School = {Cornell University},
Title = {Dynamical systems models of wall-bounded, shear-flow turbulence},
Year = {2002}}
@article{GiChLiPo12,
Abstract = {... covariant Lyapunov vectors (CLVs) which span local intrinsic directions
in the phase space of chaotic systems ... we review the basic results
of ergodic theory, with a specific reference to the implications
of Oseledets' theorem for the properties of the CLVs. We then present
a detailed description of a `dynamical' algorithm to compute the
CLVs and show that it generically converges exponentially in time.
We also discuss its numerical performance and compare it with other
algorithms presented in the literature. We finally illustrate how
CLVs can be used to quantify deviations from hyperbolicity with reference
to a dissipative system (a chain of H\'enon maps) and a Hamiltonian
model (a Fermi-Pasta-Ulam chain).},
Author = {Ginelli, F. and Chat\'{e}, H. and R. Livi and Politi, A.},
Doi = {10.1088/1751-8113/46/25/254005},
Journal = {J. Phys. A},
Note = {\arXiv{1212.3961}},
Pages = {254005},
Title = {Covariant {Lyapunov} vectors},
Volume = {46},
Year = {2013},
Bdsk-Url-1 = {http://dx.doi.org/10.1088/1751-8113/46/25/254005}}
@unpublished{GiLe04,
Author = {Ginzburg, V. and {Lerman}, E.},
Note = {\arXiv{math/0402264}},
Title = {Existence of relative periodic orbits near relative equilibria},
Year = 2004}
@article{ginelli-2007-99,
Author = {F. Ginelli and P. Poggi and A. Turchi and H. Chate and R. Livi and A. Politi},
Journal = {Phys. Rev. Lett.},
Note = {\arXiv{0706.0510}},
Pages = {130601},
Title = {Characterizing dynamics with covariant {Lyapunov} vectors},
Volume = {99},
Year = {2007}}
@ARTICLE{GLEM07,
author = {Li, Y. and {Chevillard}, L. and {Eyink}, G. and {Meneveau}, C.},
title = {Matrix exponential-based closures forthe turbulent subgrid-scale
stress tensor},
journal = {Phys. Rev. E},
year = {2009},
volume = {79},
pages = {016305},
note = {\arXiv{0704.3781}}
}
@book{glend94,
Address = {Cambridge},
Author = {P. Glendinning},
Key = {QA372 1994},
Publisher = {Cambridge Univ. Press},
Title = {Stability, instability and chaos: an introduction to the theory of nonlinear differential equations},
Year = {1994}}
@book{GL-Gil07b,
Address = {Oxford},
Author = {R. Gilmore and C. Letellier},
Publisher = {Oxford Univ. Press},
Title = {The Symmetry of Chaos},
Year = {2007}}
@ARTICLE{GL-Mir93,
author = {R. Miranda and E. Stone},
title = {The proto-{Lorenz} system},
journal = {Phys. Letters A},
year = {1993},
volume = {178},
pages = {105}
}
@article{GM00aut,
Author = {J. Guckenheimer and B. Meloon},
Journal = {SIAM J. Sci. Comput.},
Pages = {951--985},
Title = {Computing periodic orbits and their bifurcations with automatic differentiation},
Volume = {22},
Year = {2000}}
@book{golubII,
Address = {New York},
Author = {M. Golubitsky and I. Stewart and D. G. Schaeffer},
Publisher = {Springer},
Title = {Singularities and Groups in Bifurcation Theory, vol. II},
Year = {1988}}
@book{golubitsky2002sp,
Address = {Boston},
Author = {Golubitsky, M. and Stewart, I.},
Publisher = {Birkh{\"a}user},
Title = {{The symmetry perspective}},
Year = {2002}}
@article{golubord,
Abstract = {A brief review of experiments on film flows, surface
waves, and thermal convection. In 1-d, the transition from cellular
states to spatiotemporal chaos is described. In 2-d, periodic and
quasiperiodic patterns including defect-mediating ones are discussed.
Statistical emphasis on the transport and mixing phenomena in fluids
is put and some open problems are stated.},
Author = {J. P. Gollub},
Journal = {Proc. Natl. Acad. Sci. USA},
Pages = {6705},
Title = {Order and disorder in fluid motion},
Volume = {92},
Year = {1995}}
@article{golubsym84,
Author = {M. Golubitsky and J. W. Swift and E. Knobloch},
Journal = {Physica D},
Pages = {249},
Title = {Symmetries and pattern selection in {R}ayleigh-{B}ernard convection},
Volume = {10},
Year = {1984}}
@book{gottlieb77,
Address = {Philadelphia},
Author = {D. Gottlieb and S. A. Orszag},
Publisher = {SIAM},
Title = {Numerical analysis of spectral methods: theory and applications},
Year = {1977}}
@book{GoVanLo96,
Address = {Baltimore, MD},
Author = {Golub, G. H. and Van Loan, C. F.},
Publisher = {J. Hopkins Univ. Press},
Title = {Matrix Computations},
Year = 1996}
@Misc{GPLv2,
author = {Free Software Foudnation},
title = {{GNU G}eneral {P}ublic {L}icense},
month = jun,
year = {1991},
bdsk-url-1 = {www.gnu.org/licenses/gpl.html},
url = {www.gnu.org/licenses/gpl.html}
}
@article{GraPro83,
Author = {Grassberger, P. and Procaccia, I.},
Journal = {Phys. Rev. A},
Pages = {2591--2593},
Title = {Estimation of the {Kolmogorov} entropy from a chaotic signal},
Volume = {28},
Year = {1983}}
@article{grass89,
Abstract = {Discuss Biham-Wenzel method in {H}{\'e}non map and point
out its defficiency: 1. It converges to a limit cycle sometimes. 2.
Two symbol sequence converges to the same cycle.},
Author = {P. Grassberger and H. Kantz and U. Moenig},
Journal = {J. Phys. A},
Pages = {5217--5230},
Title = {On the symbolic dynamics of {H\'enon} map},
Volume = {22},
Year = {1989}}
@article{GrBaPo88,
Author = {P. Grassberger and R. Badii and A. Politi},
Journal = {J. Stat. Phys.},
Pages = {135},
Title = {Scaling laws for invariant measures on hyperbolic and nonhyperbolic attractors},
Volume = {51},
Year = {1988}}
@article{greb90shad,
Author = {C. Grebogi and S. M. Hammel and J. A. Yorke and T. Sauer},
Journal = {Phys. Rev. Lett.},
Pages = {1527--1530},
Title = {Shadowing of physical trajectories in chaotic dynamics: {Containment} and refinement},
Volume = {65},
Year = {1990}}
@book{greshosani,
Address = {New York},
Author = {P. M. Gresho and R. L. Sani},
Publisher = {John Wiley and Sons},
Title = {Incompressible Flow and the Finite Element Method},
Year = {2000}}
@article{Groh64,
Author = {Grohne, D.},
Journal = {Z. Angew. Math. Mech.},
Pages = {350--352},
Title = {Uber das {S}pektrum bei {E}igenschwingungen ebener {L}aminarstromungen},
Volume = {13},
Year = {1964}}
@article{GroTho77,
Author = {S. Grossmann and S. Thomae},
Journal = {Z. Naturforsch. A},
Pages = {1353--1363},
Title = {Invariant distributions and stationary correlation functions of one-dimensional discrete processes},
Volume = {32},
Year = {1977}}
@book{grue_waves_1996,
Address = {New York},
Author = {Grue, J. and Gjevik, B. and Weber, J. E.},
Publisher = {Springer},
Title = {Waves and Nonlinear Processes in Hydrodynamics},
Year = {1996}}
@Article{Guck92,
author = {Guckenheimer, J.},
title = {Symmetry properties of confined convective states},
journal = {Z. Angew. Math. Phys.},
year = {1992},
volume = {43},
pages = {422--429}
}
@book{guckb,
Address = {New York},
Author = {J. Guckenheimer and P. Holmes},
Publisher = {Springer},
Title = {Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields},
Year = {1983}}
@article{Guckenheimer79b,
Author = {J. Guckenheimer and R. Williams},
Journal = {Publ. Math. IHES},
Keywords = {attractor, Lorenz},
Pages = {55--72},
Title = {Structural stability of the {L}orenz attractor},
Volume = {50},
Year = {1979}}
@article{GuHu98,
Annote = {resonant response leading to algebraic growth for small times.
It occurs when the eigenvalues of the Orr-Sommerfeld equation coincide
with the eigenvalues of the homogeneous operator for the streamwise
velocity component. The resonance has been investigated numerically for
plane Couette flow. The aspect ratio of the most rapidly growing wave increases
with the Reynolds number, with its spanwise wavelength approaching
a constant value of about 3 channel heights.},
Author = {L. H. Gustavsson and L. S. Hultgren},
Journal = {J. Fluid Mech.},
Pages = {149--159},
Title = {A resonance mechanism in plane {Couette} flow},
Volume = {98},
Year = {1980}}
@book{GuiSte90,
Address = {Cambridge},
Author = {V. Guillemin and S. Sternberg},
Publisher = {Cambridge Univ. Press},
Title = {Symplectic Techniques in Physics},
Year = {1990}}
@unpublished{GupThy08,
Author = {N. Gupte and N. N. Thyagu},
Note = {\arXiv{0811.3685}; {(Published in the proceedings of the conference ``Recent Developments in Nonlinear Dynamics'', Bharathidasan University, Thiruchirapalli, India, February 2008)}},
Title = {Chaotic advection of inertial particles in two dimensional flows},
Year = {2008}}
@book{gutbook,
Address = {New York},
Author = {M. C. Gutzwiller},
Publisher = {Springer},
Title = {Chaos in Classical and Quantum Mechanics},
Year = {1990}}
@Misc{GutOsi15,
author = {Gutkin, B. and {Osipov}, V.},
title = {Classical foundations of many-particle quantum chaos},
year = {2015},
addendum = {\arXiv{1503.02676}}
}
@article{gutzwiller67,
Author = {M. C. Gutzwiller},
Journal = {J. Math. Phys.},
Pages = {1979--2000},
Title = {Phase-integral approximation in momentum space and the bound states of an atom},
Volume = 8,
Year = 1967}
@article{gutzwiller69,
Author = {M. C. Gutzwiller},
Journal = {J. Math. Phys.},
Pages = {1004--1021},
Title = {Phase-integral approximation in momentum space and the bound states of an atom. II},
Volume = 10,
Year = 1969}
@article{gutzwiller70,
Author = {M. C. Gutzwiller},
Journal = {J. Math. Phys.},
Pages = {1791--1806},
Title = {Energy spectrum according to classical mechanics},
Volume = 11,
Year = 1970}
@article{gutzwiller71,
Author = {M. C. Gutzwiller},
Journal = {J. Math. Phys.},
Pages = {343--358},
Title = {Periodic orbits and classical quantization conditions},
Volume = {12},
Year = {1971}}
@misc{HaFaHa13,
Author = {Hadjighasem, A. and {Farazmand}, M. and {Haller}, G.},
Note = {\arXiv{1302.1732}},
Title = {Detecting invariant manifolds, attractors, and generalized
{KAM} tori in aperiodically forced mechanical systems},
Year = {2013}}
@article {Hagen1839,
author = {G. Hagen},
title = {{\"U}ber die Bewegung des Wassers in engen cylindrischen R{\"o}hren},
journal = {Ann. Phys.},
volume = {122},
doi = {10.1002/andp.18391220304},
pages = {423--442},
year = {1839},
}
@article{HaKiWa95,
Author = {J. M. Hamilton and J. Kim and F. Waleffe},
Journal = {J. Fluid Mech.},
Pages = {317--348},
Title = {Regeneration mechanisms of near-wall turbulence structures},
Volume = {287},
Year = {1995},
doi = {10.1017/S0022112095000978}
}
@techreport{Halc04,
Author = {Halcrow, J.},
Institution = {School of Physics, Georgia Inst. of Technology},
Note = {{\tt ChaosBook.org/projects}},
Title = {Implementation of a {Pressure Poisson Equation} method for plane {Couette} flow},
Year = {2004}}
@phdthesis{HalcrowThesis,
Address = {Atlanta},
Author = {Halcrow, J.},
Note = {{\tt ChaosBook.org/projects/theses.html}},
School = {School of Physics, Georgia Inst. of Technology},
Title = {Geometry of turbulence: An exploration of the state-space of plane {Couette} flow},
Year = {2008}}
@book{haledb,
Address = {New York},
Author = {J. Hale and H. Ko{\c{c}}ak},
Publisher = {Springer},
Title = {Dynamics and Bifurcations},
Year = {1991}}
@book{haleinf84,
Address = {New York},
Author = {J. K. Hale and L. T. Magalhaes},
Publisher = {Springer},
Title = {An Introduction to Infinite Dimensional Systems-Geometric Theory},
Year = {1984}}
@book{haleob,
Address = {New York},
Author = {J. Hale},
Publisher = {Wiley-Interscience},
Title = {Ordinary Differential Equations},
Year = {1969}}
@book{Hall03,
Address = {New York},
Author = {Hall, B. C},
Publisher = {Springer},
Title = {Lie Groups, {Lie} Algebras, and Representations},
Year = {2003}}
@article{Haller11,
Author = {Haller, G.},
Doi = {10.1016/j.physd.2010.11.010},
Journal = {Physica D},
Pages = {574--598},
Title = {A variational theory of hyperbolic {Lagrangian Coherent Structures}},
Volume = {240},
Year = {2011},
Bdsk-Url-1 = {http://dx.doi.org/10.1016/j.physd.2010.11.010}}
@article{Haller11err,
Author = {Farazmand, M. and Haller, G.},
Journal = {Physica D},
Pages = {439--441},
Title = {Erratum and addendum to ``A variational theory of hyperbolic Lagrangian coherent structures''},
Year = {2011}}
@Article{hallerAnnRevFluid,
author = {G. Haller},
title = {Lagrangian coherent structures},
journal = {Annu. Rev. Fluid Mech.},
year = {2015},
volume = {47},
pages = {137--161}
}
@book{halmos:fdvs,
Address = {Princeton},
Author = {Paul R. Halmos},
Edition = {Second},
Publisher = {D. Van Nostrand},
Title = {Finite-Dimensional Vector Spaces},
Year = 1958}
@article{HAMANE05,
Author = {Hall, P. and Marron, J.S. and Neeman, A.},
Journal = {J. R. Statist. Soc. B},
Pages = {427--444},
Title = {Geometric representation of high dimension, low sample size data},
Volume = {67},
Year = {2005}}
@article{HaMe98,
Author = {G. Haller and I. Mezi\'c},
Journal = {Nonlinearity},
Pages = {319--339},
Title = {Reduction of three-dimensional, volume-preserving flows with symmetry},
Volume = {11},
Year = {1998}}
@book{hamer,
Address = {New York},
Author = {Morton Hamermesh},
Publisher = {Dover},
Title = {Group Theory and Its Application to Physical Problems},
Year = {1962}}
@phdthesis{hansen,
Address = {Box 1048, Blindern, N-0316, Oslo, Norway},
Author = {K. T. Hansen},
Note = {{\tt ChaosBook.org/projects/KTHansen/thesis}},
School = {University of Oslo},
Title = {Symbolic Dynamics in Chaotic systems},
Year = {1993}}
@Article{hansen92,
author = {K. T. Hansen},
title = {Remarks on the symbolic dynamics for the {H\'e}non map},
journal = {Phys. Lett. A},
year = {1992},
volume = {165},
pages = {100--104},
abstract = {Discuss the change of the symbol sequence of one
particular periodic orbit in the {H}{\'e}non map.}
}
@article{Hansen95,
Author = {Hansen, K. T.},
Doi = {10.1103/PhysRevE.52.2388},
Journal = {Phys. Rev. E},
Note = {\arXiv{chao-dyn/9507003}},
Pages = {2388--2391},
Title = {Alternative method to find orbits in chaotic systems},
Volume = {52},
Year = {1995},
Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevE.52.2388}}
@book{hao90b,
Address = {Singapore},
Author = {B.-L. Hao},
Publisher = {World Scientific},
Title = {Chaos {II}},
Year = {1990}}
@book{hao98,
Address = {Singapore},
Author = {B.-L. Hao and W.-M. Zheng},
Publisher = {World Scientific},
Title = {Applied Symbolic Dynamics and Chaos},
Year = {1998}}
@article{Harter69,
Author = {Harter, W. G.},
Journal = {J. Math. Phys.},
Pages = {739--752},
Title = {Algebraic theory of ray representations of finite groups},
Volume = {10},
Year = {1969}}
@Article{Harter78,
author = {Harter, W. G. and Dos Santos, N.},
title = {Double-group theory on the half-shell and the two-level system. {I. Rotation} and half-integral spin states},
journal = {Am. J. Phys.},
year = {1978},
volume = {46},
pages = {251--263}
}
@book{Harter93,
Address = {New York},
Author = {Harter, W. G.},
Publisher = {Wiley},
Title = {Principles of Symmetry, Dynamics, and Spectroscopy},
Year = {1993}}
@book{hartmanb,
Address = {New York},
Author = {P. Hartman},
Publisher = {Wiley-Interscience},
Title = {Ordinary Differential Equations},
Year = {1964}}
@inproceedings{HDB92,
Address = {New York},
Author = {J. Hegseth and F. Daviaud and P. Berge},
Booktitle = {Ordered and Turbulent Patterns in Taylor-{Couette} Flow},
Editor = {C. D. Andereck and F. Hayot},
Pages = {159--166},
Publisher = {Plenum},
Title = {Intermittent turbulence in plane and circular {Couette} flow},
Year = {1992}}
@article{Hei24,
Author = {W. Heisenberg},
Journal = {Ann. Phys. Lpz.},
Note = {fix volume number},
Pages = {577},
Title = {{\"U}ber Stabilit{\"a}t und Turbulenz von Fl{\"a}teitsstr{\"a}ser},
Volume = {74 44},
Year = {1924}}
@article{Heisen48,
Author = {Heisenberg, W.},
Journal = {Z. Phys.},
Note = {{On the statistical theory of turbulence, translation by M. L. Mahler, National Advisory Committee for Aeronautics, Technical memorandum, no. 1431}},
Pages = {628--657},
Title = {{Zur statistischen Theorie der Turbulenz}},
Volume = 124,
Year = 1948}
@article{Heisen48a,
Author = {Heisenberg, W.},
Journal = {Proc. Roy. Soc. A},
Pages = {402--406},
Title = {On the theory of statistical and isotropic turbulence},
Volume = 195,
Year = 1948}
@Article{HeiZha07,
author = {J. Heidel and F. Zhang},
title = {Nonchaotic and chaotic behavior in three-dimensional quadratic systems: {Five}-one conservative cases},
journal = {Int. J. Bifur. Chaos},
year = {2007},
volume = {17},
pages = {2049--2072},
doi = {10.1142/S021812740701821X},
eprint = {http://www.worldscientific.com/doi/pdf/10.1142/S021812740701821X}
}
@article{HeKaPo08,
Author = {E. J. Heller and L. Kaplan and F. Pollmann},
Journal = {Proc. Natl. Acad. Sci. USA},
Note = {\arXiv{0712.4093}},
Pages = {7631},
Title = {Inflationary dynamics for matrix eigenvalue problems},
Volume = {105},
Year = {2008}}
@article{hell78,
Author = {R. Helleman and T. Bountis},
Journal = {LNP},
Pages = {353},
Title = {Periodic solutions of arbitrary period, variational methods},
Volume = {93},
Year = {1978}}
@Article{HenCviLip14,
author = {J. M. Heninger and D. Lippolis and P. Cvitanovi\'c},
title = {Neighborhoods of periodic orbits and the stationary distribution of a noisy chaotic system},
journal = {Phys. Rev. E},
year = {2015},
volume = {92},
pages = {062922},
note = {\arXiv{1507.00462}},
doi = {10.1103/PhysRevE.92.062922}
}
@Misc{HenLipCvi15,
author = {J. M. Heninger and D. Lippolis and P. Cvitanovi\'c},
title = {Perturbation theory for the {Fokker-Planck} operator in chaos},
year = {2016},
note = {to be submitted to Comm. Nonlinear Sci. Numer. Simul.},
url = {http://arxiv.org/abs/1602.03044}
}
@article{HenningsonPF94,
Author = {D. Henningson and S. Reddy},
Journal = {Phys. Fluids},
Pages = {1396--1398},
Title = {On the role of linear mechanisims in transition to turbulence},
Volume = {6},
Year = {1994}}
@book{henonrtb2,
Address = {New York},
Author = {M. H{\'e}non},
Publisher = {Springer},
Title = {Generating Families in the Restricted Three-Body Problem: {II}. Quantitative Study of Bifurcations},
Year = {2001}}
@article{heron87,
Author = {J. M. Ghidaglia and B. H{\'e}non},
Journal = {Physica D},
Pages = {282--304},
Title = {Dimension of the attractors associated to the {Ginzburg-Landau} partial differential equation},
Volume = {28},
Year = {1987}}
@article{HGC08,
Author = {J. F. Gibson and J. Halcrow and P. Cvitanovi{\'c}},
Journal = {J. Fluid Mech.},
Note = {\arXiv{0808.3375}},
Pages = {1--24},
Title = {Equilibrium and traveling-wave solutions of plane {Couette} flow},
Volume = {638},
Year = {2009},
DOI = {10.1017/S0022112009990863}
}
@article{hhrugh92,
Author = {H. H. Rugh},
Journal = {Nonlinearity},
Pages = {1237},
Title = {The correlation spectrum for hyperbolic analytic maps},
Volume = {5},
Year = {1992}}
@Article{hhsys,
author = {M. H{\'e}non and C. Heiles},
title = {The applicability of the third integral of motion: {Some} numerical experiments},
journal = {Astron. J.},
year = {1964},
volume = {69},
pages = {73}
}
@Article{Hilbert93,
author = {Hilbert, D.},
title = {\"Uber die vollen Invariantensysteme},
journal = {Math. Ann.},
year = {1893},
volume = {42},
pages = {313--373},
doi = {10.1007/BF01444162}
}
@book{hille,
Address = {New York},
Author = {E. Hille},
Publisher = {Dover},
Title = {Ordinary Differential Equations in the Complex Domain},
Year = {1997}}
@article{HJM03,
Author = {B. Hof and A. Juel and T. Mullin},
Journal = {Phys. Rev. Lett.},
Pages = {244502},
Title = {Scaling of the turbulence transition threshold in a pipe},
Volume = {91},
Year = {2003}}
@Article{HK82,
author = {Hofbauer, F. and Keller, G.},
title = {Ergodic properties of invariant measures for piecewise monotonic transformations},
journal = {Math. Z.},
year = {1982},
volume = {180},
pages = {119--140},
doi = {10.1007/BF01215004},
bdsk-url-1 = {http://dx.doi.org/10.1007/BF01215004}
}
@article{HNks86,
Abstract = {Numerical work on the transition to chaos of the KSe.
The attracting solution manifold undergoes a complex bifurcation
sequence including fixed points, limit cycles, travelling waves,
homoclinic orbits, invariant tori.},
Author = {J. M. Hyman and B. Nicolaenko},
Journal = {Physica D},
Pages = {113--126},
Title = {The {Kuramoto-Sivashinsky} equation: a bridge between {PDE}'s and dynamical systems},
Volume = {18},
Year = {1986}}
@article{HNZks86,
Abstract = {The properties of the KSe is discussed in even larger
domains. Still the dynamical behavior is dominated by the skeleton
of low dimensional vector fields. More complex dynamics happens in
large domain where intermittency is accurately identified only by
high-precision numerical method. The presence of strange fixed
points overrun the cellular fixed points in large scales which
generates complex spatial structure to the solutions. The Large
scale statistical behavior of the system is computed.},
Author = {J. M. Hyman and B. Nicolaenko and S. Zaleski},
Journal = {Physica D},
Pages = {265--292},
Title = {Order and complexity in the {Kuramoto-Sivashinsky} model of weakly turbulent interfaces},
Volume = {23},
Year = {1986}}
@article{hof2006flt,
Author = {Hof, B. and Westerweel, J. and Schneider, T. M. and Eckhardt, B.},
Journal = {Nature},
Pages = {59--62},
Title = {Finite lifetime of turbulence in shear flows},
Volume = {443},
Year = {2006}}
@article{hof2008,
Author = {Hof, B. and De Lozar, A. and Kuik, D.J. and Westerweel, J.},
Journal = {Phys. Rev. Lett.},
Pages = {214501},
Title = {Repeller or attractor? {Selecting} the dynamical model for
the onset of turbulence in pipe flow},
Volume = {101},
Year = {2008}}
@book{hoffman:kunze,
Address = {Englewood Cliffs, NJ},
Author = {K. Hoffman and R. Kunze},
Edition = {Second},
Publisher = {Prentice-Hall},
Title = {Linear Algebra},
Year = 1971}
@book{Holmes96,
Address = {Cambridge},
Author = {P. Holmes and J. L. Lumley and G. Berkooz},
Publisher = {Cambridge Univ. Press},
Title = {Turbulence, Coherent Structures, Dynamical Systems and Symmetry},
Year = {1996}}
@article{hoopks85,
Author = {A. P. Hooper and R. Grimshaw},
Journal = {Phys. Fluids},
Pages = {37--45},
Title = {Nonlinear instability at the interface between two viscous fluids},
Volume = {28},
Year = {1985}}
@Article{hopf48,
author = {E. Hopf},
title = {A mathematical example displaying features of turbulence},
journal = {Commun. Pure Appl. Math.},
year = {1948},
volume = {1},
pages = {303--322},
doi = {10.1002/cpa.3160010401}
}
@article{hopf50,
Author = {E. Hopf},
Journal = {Commun. Pure Appl. Math.},
Pages = {201--230},
Title = {The partial differential equation $u_t + uu_x = u_{xx}$},
Volume = {3},
Year = {1950}}
@Unpublished{HopKan14,
author = "J. Hopcroft and R. Kannan",
title = "Foundations of Data Science",
year = "2014",
note = "
\HREF{http://research.microsoft.com/en-US/people/kannan/book-no-solutions-aug-21-2014.pdf}
{In preparation}"
}
@article{Horiuti,
Author = {K. Horiuti},
Journal = {J. Comput. Physics},
Pages = {343--370},
Title = {Comparison of conservative and rotational forms in large eddy simulation of turbulent channel flow},
Volume = {71},
Year = {1987}}
@book{hoyll06,
Address = {Cambridge},
Author = {R. Hoyle},
Publisher = {Cambridge Univ. Press},
Title = {Pattern Formation: An Introduction to Methods},
Year = {2006}}
@article{HRCLOL11,
Author = {J. Herault and F. Rincon and C. Cossu and G. I. Ogilvie and P.-Y. Longaretti},
Journal = {Phys. Rev. E},
Pages = {036321},
Title = {Periodic magnetorotational dynamo action as a prototype of nonlinear magnetic-field generation in shear flows},
Volume = {84},
Year = {2011}}
@article{HSGsize98,
Author = {M. C. Strain and H. S. Greenside},
Journal = {Phys. Rev. Lett.},
Pages = {2306--2309},
Title = {Size-dependent transition to high-dimensional chaotic dynamics in a two-dimensional excitable media},
Volume = {80},
Year = {1998}}
@article{Hurt93,
Author = {Hurt, N. E.},
Journal = {Results Math.},
Pages = {55},
Title = {Zeta functions and periodic orbit theory: {A} review},
Volume = {23},
Year = {1993}}
@article{Hurt93a,
Author = {Hurt, N. E.},
Journal = {Rev. Math. Phys.},
Pages = {713--733},
Title = {Three topics on periodic orbit theory: {A} review},
Volume = {5},
Year = {1993}}
@book{Huyg1673,
Address = {Amsterdam},
Author = {C. Huygens},
Publisher = {Swets \& Zeitlinger},
Title = {L'Horloge \`a pendule},
Year = {1673, 1967}}
@book{ince,
Address = {New York},
Author = {E. L. Ince},
Publisher = {Dover},
Title = {Ordinary Differential Equations},
Year = {1956}}
@book{infdymnon,
Address = {New York},
Author = {R. Temam},
Publisher = {Springer},
Title = {Infinite-Dimensional Dynamical Systems in Mechanics and Physics},
Year = {1988}}
@book{isham99,
Address = {Singapore},
Author = {C. J. Isham},
Publisher = {World Scientific},
Title = {Modern Differential Geometry for Physicists},
Year = {1999}}
@Article{Ishimura02,
author = {N. Ishimura},
title = {Remarks on third-order {ODEs} relevant to the {Kuramoto-Sivashinsky} equation},
journal = {J. Diff. Eqn.},
year = {2002},
volume = {178},
pages = {466--477},
doi = {10.1006/jdeq.2001.4018},
issn = {0022-0396}
}
@Article{IshNak00,
author = {Ishimura, N. and Nakamura, M.},
title = {Nonexistence of monotonic solutions of some third-order ode relevant to the {Kuramoto-Sivashinsky} equation},
journal = {Taiwanese J. Math.},
year = {2000},
volume = {4},
pages = {621--625},
url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-0000750860&partnerID=40&md5=8161d76f70fde737bab918b507546ff6}
}
@article{IT01,
Author = {T. Itano and S. Toh},
Journal = {J. Phys. Soc. Japan},
Pages = {701--714},
Title = {The dynamics of bursting process in wall turbulence},
Volume = {70},
Year = {2001}}
@article{ItGe09,
Author = {Itano, T. and Generalis, S. C.},
Journal = {Phys. Rev. Lett.},
Pages = {114501},
Title = {Hairpin vortex solution in planar {Couette} flow: A tapestry of knotted vortices},
Volume = {102},
Year = {2009}}
@book{jackson89,
Address = {Cambridge},
Author = {Jackson, E. Atlee},
Publisher = {Cambridge Univ. Press},
Title = {Perspectives of nonlinear dynamics: Vol. 1 and 2},
Year = 1989}
@article{Javier_shear,
Author = {F. J. Beron-Vera and M. J. Olascoaga and M. G. Brown and H. Kocak and I. I. Rypina},
Journal = {Chaos},
Pages = {017514},
Title = {Invariant-tori-like {Lagrangian} coherent structures in geophysical flows},
Volume = {20},
Year = {2010}}
@article{JBB83,
Author = {Jensen, M. H. and Bak, P. and Bohr, T.},
Journal = {Phys. Rev. Lett.},
Pages = {1637--1639},
Title = {Complete devil's staircase fractal dimension and universality of mode-locking structure in the circle map},
Volume = {50},
Year = {1983}}
@article{JBB84,
Author = {Jensen, M. H. and Bak, P. and Bohr, T.},
Journal = {Phys. Rev. A},
Pages = {1960--1969},
Title = {Transition to chaos by interaction of resonances in dissipative systems. {I. Circle} maps},
Volume = {30},
Year = {1984}}
@book{JBG97,
Address = {Providence R.I.},
Author = {J. Barrow-Green},
Publisher = {AMS},
Title = {Poincar{\'e} and the Tree Body Problem},
Year = {1997}}
@book{jeffb,
Address = {Cambridge},
Author = {H. Jeffreys and B. Jeffreys},
Publisher = {Cambridge Univ. Press},
Title = {Methods of Mathematical Physics},
Year = {1999}}
@article{JeongJFM97,
Author = {J. Jeong and F. Hussain and W. Schoppa and J. Kim},
Journal = {J. Fluid Mech.},
Pages = {185--214},
Title = {Coherent structures near the wall in a turbulent boundary layer},
Volume = {332},
Year = {1997}}
@article{JGB86,
Author = {P. S. Jang and R. L. Gran and D. J. Benney},
Journal = {J. Fluid Mech.},
Pages = {109--123},
Title = {On the origin of streamwise vortices in a turbulent boundary layer},
Volume = {169},
Year = {1986}}
@book{jhos,
Address = {New York},
Author = {J. K. Hale},
Publisher = {McGraw-Hill},
Title = {Oscillations in Nonlinear Systems},
Year = {1963}}
@article{jimenez07,
Author = {Jim\'{e}nez, J. and Moser, R. D.},
Journal = {Phil. Trans. Royal Soc. A},
Pages = {715},
Title = {What are we learning from simulating wall turbulence?},
Volume = {367},
Year = {2007}}
@article{JimenezJFM01,
Author = {J. Jim{\'e}nez and M. Simens},
Journal = {J. Fluid Mech},
Pages = {81--91},
Title = {Low-dimensional dynamics of a turbulent wall flow},
Volume = {435},
Year = {2001}}
@article{JKSNS05,
Author = {J. Jim{\'e}nez and G. Kawahara and M. P. Simens and M. Nagata and M. Shiba},
Journal = {Phys. Fluids},
Pages = {015105},
Title = {Characterization of near-wall turbulence in terms of equilibrium and bursting solutions},
Volume = {17},
Year = {2005}}
@article{JM91,
Author = {Jim{\'e}nez, J. and Moin, P.},
Journal = {J. Fluid Mech.},
Pages = {213--240},
Title = {The minimal flow unit in near-wall turbulence},
Volume = {225},
Year = {1991},
doi = {10.1017/S0022112091002033}
}
@Article{johnston04,
author = {H. Johnston and J.-G. Liu},
title = {Accurate, stable and efficient {Navier-Stokes} solvers based on explicit treatment of the pressure term},
journal = {J. Comput. Physics},
year = {2004},
volume = {199},
pages = {221--259}
}
@article{JoPro87,
Author = {C. A. Jones and M. R. E. Proctor},
Doi = {10.1016/0375-9601(87)90008-9},
Journal = {Phys. Lett. A},
Pages = {224--228},
Title = {Strong spatial resonance and travelling waves in {Benard} convection},
Volume = {121},
Year = {1987},
Bdsk-Url-1 = {http://dx.doi.org/10.1016/0375-9601(87)90008-9}}
@article{JT63,
Author = {D. D. Joseph and L. N. Tao},
Journal = {J. Appl. Mech.},
Pages = {147--148},
Title = {Transverse velocity components in fully developed unsteady flows},
Volume = {30},
Year = {1963}}
@Article{K31,
author = {B. O. Koopman},
title = {Hamiltonian systems and transformations in {Hilbert} space},
journal = {Proc. Natl. Acad. Sci.},
year = {1931},
volume = {17},
pages = {315--318}
}
@Incollection{KapYor79,
author = {Kaplan, J. L. and Yorke, J. A.},
title = {Numerical solution of a generalized eigenvalue problem for even mappings},
booktitle = {Functional Differential Equations and Approximation of Fixed Points},
publisher = {Springer},
year = {1979},
editor = {Peitgen, H.-O. and Walther, H.-O.},
volume = {730},
pages = {228--237},
address = {New York},
doi = {10.1007/BFb0064320}
}
@article{Kato03,
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.},
Author = {S. Kato and M. Yamada},
Journal = {Phys. Rev. E},
Pages = {025302},
Title = {Unstable periodic solutions embedded in a shell model turbulence},
Volume = {68},
Year = {2003}}
@book{Katok95,
Address = {Cambridge},
Author = {A. Katok and B. Hasselblatt},
Publisher = {Cambridge Univ. Press},
Title = {Introduction to the Modern Theory of Dynamical Systems},
Year = {1995}}
@article{KaUhlVeen12,
Author = {G. Kawahara and M. Uhlmann and L. van Veen},
Journal = {Annu. Rev. Fluid Mech},
Pages = {203--225},
Title = {The significance of simple invariant solutions in turbulent flows},
Volume = {44},
Year = {2012}}
@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. Comp. Phys.},
Keywords = {Coarse projective {integration,Continuous} {symmetry,Dynamic} {renormalization,Multiscale} {computation,Projective} integration},
Pages = {382--407},
Volume = {225},
Title = {Projective and coarse projective integration for problems
with continuous symmetries},
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.},
Year = {2007}}
@article{Kaw05,
Author = {G. Kawahara},
Journal = {Phys. Fluids},
Pages = {041702},
Title = {Laminarization of minimal plane {Couette} flow: {Going} beyond the basin of attraction of turbulence},
Volume = {17},
Year = {2005}}
@article{KawKida01,
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.},
Author = {G. Kawahara and S. Kida},
Journal = {J. Fluid Mech.},
Pages = {291--300},
Title = {Periodic motion embedded in plane {Couette} turbulence: {Regeneration} cycle and burst},
Volume = {449},
Year = {2001}}
@article{KawKida05,
Author = {G. Kawahara and S. Kida},
Journal = {Parity},
Pages = {12},
Title = {Elementary coherent structures in turbulent flows},
Volume = {20},
Year = {2005}}
@article{KawKiVeen06,
Author = {G. Kawahara and S. Kida and L. van Veen},
Journal = {Nonlinear Process in Geophysics},
Pages = {499--507},
Title = {Unstable periodic motion in turbulent flows},
Volume = {13},
Year = {2006}}
@article{Kaz83,
Author = {Kazantsev, A. P. and Pokrovsky, V. L. and Bergou, J.},
Doi = {10.1103/PhysRevA.28.3659},
Journal = {Phys. Rev. A},
Pages = {3659--3662},
Title = {Rydberg atoms in weak magnetic fields},
Volume = {28},
Year = {1983},
Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevA.28.3659}}
@article{Kazantsev01,
Author = {Kazantsev, E.},
Doi = {10.5194/npg-8-281-2001},
Journal = {Nonlinear Processes in Geophysics},
Pages = {281--300},
Title = {Sensitivity of the attractor of the barotropic ocean model to external influences: {Approach} by unstable periodic orbits},
Url = {http://www.nonlin-processes-geophys.net/8/281/2001/},
Volume = {8},
Year = {2001},
doi = {10.5194/npg-8-281-2001}}
@article{Kazantsev01a,
Abstract = {A description of a deterministic chaotic system in terms of unstable
periodic orbits (UPOs) is used to develop a method of an a priori
estimate of the sensitivity of statistical averages of the solution
to small external influences. This method allows us to determine
the forcing perturbation which maximizes the norm of the perturbation
of a statistical moment of the solution on the attractor. The method
was applied to the Lorenz model. The estimates of perturbations of
two statistical moments were compared with directly calculated values.
The comparison shows that some 100 UPOs are sufficient to realize
this approach and get a good accuracy. The linear approach remains
valid up to rather high norms of the forcing perturbation.},
Author = {Kazantsev, E.},
Doi = {10.1016/S0960-0779(00)00154-5},
Journal = {Chaos, Solitons \&\ Fractals},
Pages = {1989--2005},
Title = {Sensitivity of attractor to external influences: {Approach} by unstable periodic orbits},
Volume = {12},
Year = {2001},
Bdsk-Url-1 = {http://dx.doi.org/10.1016/S0960-0779(00)00154-5}}
@article{Kazantsev98,
Author = {Kazantsev, E.},
Doi = {10.5194/npg-5-193-1998},
Journal = {Nonlinear Processes in Geophysics},
Pages = {193--208},
Title = {Unstable periodic orbits and attractor of the barotropic ocean model},
Volume = {5},
Year = {1998},
Bdsk-Url-1 = {http://dx.doi.org/10.5194/npg-5-193-1998}}
@book{kellbv,
Address = {New York},
Author = {H. B. Keller},
Publisher = {Dover},
Title = {Numerical Methods for Two-Point Boundary-Value Problems},
Year = {1992}}
@inproceedings{Keller77,
Address = {New York},
Author = {H. B. Keller},
Booktitle = {Applications of Bifurcation Theory},
Editor = {P. H. Rabinowitz},
Pages = {359--384},
Publisher = {Academic Press},
Title = {Numerical solution of bifurcation and nonlinear eigenvalue problems},
Year = {1977}}
@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 Press}
}
@Book{Kelley,
title = {Solving Nonlinear Equations with {Newton}'s Method},
publisher = {SIAM},
year = {2003},
author = {C. T. Kelley},
isbn = {0898715466},
address = {Philadelphia},
pages = {118}
}
@Incollection{KerOli11,
author = {Kerr, R. M. and Oliver, M.},
title = {The ever-elusive blowup in the mathematical description of fluids},
booktitle = {An Invitation to Mathematics},
publisher = {Springer},
year = {2011},
editor = {Schleicher, D. and Lackmann, M.},
isbn = {978-3-642-19532-7},
pages = {137--164},
address = {New York},
doi = {10.1007/978-3-642-19533-4_10}
}
@article{Kerswell05,
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.},
Author = {R. R. Kerswell},
Journal = {Nonlinearity},
Pages = {R17--R44},
Title = {Recent progress in understanding the transition to turbulence in a pipe},
Url = {http://stacks.iop.org/0951-7715/18/R17},
Volume = {18},
Year = {2005}
}
@article{KeTu06,
Abstract = { We find that travelling waves with low wall shear
stresses (lower branch solutions) are on a surface which separates initial
conditions which uneventfully relaminarise and those which lead to a
turbulent evolution. Evidence for recurrent travelling wave visits is
found in both 5D and 10D long periodic pipes but only for those travelling
waves with low-to-intermediate wall shear stress and for less than about
10\% of the time in turbulent flow. Dynamical structures such as periodic
orbits need to be isolated and included in any such expansion.},
Author = {Kerswell, R. R. and Tutty, O.R.},
Journal = {J. Fluid Mech.},
Note = {\arXiv{physics/0611009}},
Pages = {69--102},
Title = {Recurrence of travelling waves in transitional pipe flow},
Volume = {584},
Year = {2007},
doi = {10.1017/S0022112007006301}
}
@article{kev01ks,
Author = {M. E. Johnson and M. S. Jolly and I. G. Kevrekidis},
Journal = {Int. J. Bifurc. Chaos},
Pages = {1--18},
Title = {The {O}seberg transition: visualization of global bifurcations for the {Kuramoto-Sivashinsky} equation},
Volume = {11},
Year = {2001}}
@article{Kim03,
Author = {J. Kim},
Journal = {Phys. Fluids},
Pages = {1093--1105},
Title = {Control of turbulent boundary layers},
Volume = {15},
Year = {2003}}
@article{kim10,
Author = {J. H. Kim and M. Nishihara and I. V. Adamovich and M. Samimy and S. V. Gorbatov and F. V. Pliavaka},
Journal = {Exp. Fluids},
Pages = {497},
Title = {Development of localized arc-filament {RF} plasma actuators for high-speed and high {Reynolds} number flow control},
Volume = {49},
Year = {2010}}
@Article{kim99,
author = {K. C. Kim and R. J. Adrian},
title = {Very large-scale motion in the outer layer},
journal = {Phys. Fluids},
year = {1999},
volume = {11},
pages = {417--422}
}
@article{kirby_reconstructing_1992,
Author = {M. Kirby and D. Armbruster},
Journal = {Z. Angew. Math. Phys.},
Pages = {999--1022},
Title = {Reconstructing phase space from {PDE} simulations},
Volume = {43},
Year = {1992},
Abstract = {We propose the {Karhunen-Lo\'eve} {(K-L)} decomposition as a tool
to analyze complex spatio-temporal structures in {PDE} simulations
in terms of concepts from dynamical systems theory. Taking the {Kuramoto-Sivashinsky}
equation as a model problem we discuss the {K-L} decomposition for
4 different values of its bifurcation parameter a. We distinguish
two modes of using the {K-L} decomposition: As an analytic and synthetic
tool respectively. Using the analytic mode we find unstable fixed
points and stable and unstable manifolds in a parameter regime with
structurally stable homoclinic orbits (a=17.75). Choosing the data
for a {K-L} analysis carefully by restricting them to certain burst
events, we can analyze a more complicated intermittent regime at
a=68. We establish that the spatially localized oscillations around
a so called `strange' fixed point which are considered as fore-runners
of spatially concentrated zones of turbulence are in fact created
by a very specific limit cycle (a=83.75) which, for a=87, bifurcates
into a modulated traveling wave. Using the {K-L} decomposition synthetically
by determining an optimal Galerkin system, we present evidence that
the {K-L} decomposition systematically destroys dissipation and leads
to blow up solutions.},
}
@inproceedings{KK05,
Address = {Dordrecht},
Author = {G. Kawahara and S. Kida and M. Nagata},
Booktitle = {One hundred years of boundary layer research},
Publisher = {Kluwer},
Title = {Unstable periodic motion in plane {Couette} system: {The} skeleton of turbulence},
Year = {2005}}
@article{KKR71,
Author = {Kim, H.T. and Kline, S.J. and Reynolds, W.C.},
Journal = {J. Fluid Mech.},
Pages = {133--160},
Title = {The production of turbulence near a smooth wall in a turbulent boundary layer},
Volume = {50},
Year = {1971}}
@article{kleinman10,
Author = {R. R. Kleinman and D. J. Bodony and J. B. Freund},
Journal = {Phys. Fluids},
Pages = {116103},
Title = {Shear-flow excitation mechanisms of recessed localized arc-filament plasma actuators},
Volume = {22},
Year = {2010}}
@Inproceedings{Kleiser80,
author = {L. Kleiser and U. Schumann},
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{klewicki95,
Author = {Klewicki, J. C. and Metzger, M. M. and Kelner, E. and E. M. Thurlow},
Journal = {Phys. Fluids},
Pages = {857},
Title = {Viscous sublayer flow visualizations at {$R_{\theta} \sim 1500000$}},
Volume = 7,
Year = 1995}
@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{KMM87,
Author = {J. Kim and P. Moin and R. Moser},
Journal = {J. Fluid Mech.},
Pages = {133--166},
Title = {Turbulence statistics in fully developed channel flow at low {Reynolds} number},
Volume = {177},
Year = {1987}}
@article{KNSks90,
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.},
Author = {I. G. Kevrekidis and B. Nicolaenko and J. C. Scovel},
Journal = {SIAM J. Appl. Math.},
Pages = {760--790},
Title = {Back in the saddle again: a computer assisted study of the {Kuramoto-Sivashinsky} equation},
Volume = {50},
Year = {1990}}
@article{Koch89,
Author = {W. Koch},
Journal = {Acta Mechanica},
Pages = {11},
Title = {Direct resonances in {Orr-Sommerfeld} problems},
Volume = {59},
Year = {1986}}
@article{kolm91,
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.},
Author = {A. N. Kolmogorov},
Journal = {Proc. R. Soc. Lond. A},
Pages = {9--13},
Title = {The local structure of turbulence in incompressible viscous fluid for very large {Reynolds} numbers},
Volume = {434},
Year = {1991}}
@article{kook89,
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.},
Author = {H-T. Kook and J. D. Meiss},
Journal = {Physica D},
Pages = {65--86},
Title = {Periodic orbits for reversible symplectic mappings},
Volume = {35},
Year = {1989}}
@article{kooknewt,
Abstract = {An algorithm of {N}ewton's method is presented to find
periodic orbits for {L}agrangian mappings. The method is based on
block-diagonalization of the Hessian Matrix of the action function.},
Author = {H-T Kook and J. D. Meiss},
Journal = {Physica D},
Pages = {317--326},
Title = {Application of {N}ewton's method to {L}agrangian mappings},
Volume = {36},
Year = {1989}}
@misc{kornyak-2007,
Author = {Vladimir V. Kornyak},
Note = {\arXiv{0706.3646}},
Title = {Symmetries and dynamics of discrete systems},
Year = {2007}}
@inproceedings{KraOsi00,
Address = {New York},
Author = {B. Krauskopf and H. M. Osinga},
Booktitle = {Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems},
Editor = {E. Doedel and L. Tuckerman},
Pages = {199--208},
Publisher = {Springer},
Series = {The IMA Volumes in Mathematics and its Applications},
Title = {Investigating torus bifurcations in the forced {Van der Pol} oscillator},
Volume = {119},
Year = 2000}
@article{Kras04,
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.},
Author = {I. Krasikov and G. J. Rodgers and C. E. Tripp},
Journal = {J. Phys. A},
Pages = {2365--2370},
Title = {Growing random sequences},
Volume = {37},
Year = {2004}}
@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}
}
@phdthesis{Kreilos14,
Author = {Kreilos, T.},
Note = {http://archiv.ub.uni-marburg.de/diss/z2014/0356/pdf/dtk.pdf},
School = {Philipps-Universit{\"a}t Marburg},
Title = {Turbulence Transition in Shear Flows and Dynamical Systems Theory},
Year = {2014}}
@Incollection{KREROSLIDE13,
author = {Krauss, J. and Ertun\c{c}, \"{O}. and Ostwald, C. and Lienhart, H. and Delgado, A.},
title = {Probability of occurrence of the turbulent flow structures in transitional pipe flows},
publisher = {Springer},
year = {2013},
pages = {271--278},
address = {New York},
journal = {New Results in Numerical and Experimental Fluid Mechanics VIII }
}
@article{KRSR67,
Author = {Kline, S. J. and {R}eynolds, W. C. and Schraub, F. A. and Runstadler, P. W.},
Journal = {J. Fluid Mech.},
Pages = {741--773},
Title = {The structure of turbulent boundary layers},
Volume = {30},
Year = {1967}}
@article{kruskal62,
Abstract = {Asymptotic expansion to all orders. A systematic way to construct the adiabatic invariants.},
Author = {M. Kruskal},
Journal = {J. Math. Phys.},
Pages = {806},
Title = {Asymptotic theory of {Hamiltonian} and other systems with all solutons nearly periodic},
Volume = {3},
Year = {1962}}
@Article{kruskal65,
author = {N. J. Zabusky and M. D. Kruskal},
title = {Interaction of ``Solitons'' in a Collisionless Plasma and the Recurrence of Initial States},
journal = {Phys. Rev. Lett.},
year = {1965},
volume = {15},
number = {6},
pages = {240--243},
month = aug,
doi = {10.1103/PhysRevLett.15.240},
bdsk-url-1 = {http://dx.doi.org/10.1103/PhysRevLett.15.240},
numpages = {3},
publisher = {American Physical Society}
}
@article{KrVeSchEck12,
Author = {Kreilos, T. and Veble, G. and Schneider, T. M. and Eckhardt, B.},
Doi = {10.1017/jfm.2013.212},
Journal = {J. Fluid M.},
Note = {\arXiv{1209.0593}},
Pages = {100--122},
Title = {Edge states for the turbulence transition in the asymptotic suction boundary layer},
Volume = {726},
Year = {2013},
Bdsk-Url-1 = {http://dx.doi.org/10.1017/jfm.2013.212}}
@Article{KrZaEc14,
author = {T. Kreilos and S. Zammert and B. Eckhardt},
title = {Comoving frames and symmetry-related motions in parallel shear flows},
journal = {J. Fluid Mech.},
year = {2014},
volume = {751},
pages = {685--697},
addendum = {\arXiv{1309.4590}},
doi = {10.1017/jfm.2014.305},
issn = {1469-7645}
}
@article{ks04com,
Abstract = {A contour integral method is presented to evaluate
accurately the matrix functions with removable singularities.},
Author = {A.-K. Kassam and L. N. Trefethen},
Journal = {SIAM J. Sci. Comp.},
Pages = {1214--1233},
Title = {Fourth-order time stepping for stiff {PDE}s},
Volume = {26},
Year = {2005}}
@article{kschang86,
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.},
Author = {H.-C. Chang},
Journal = {Phys. Fluids},
Pages = {3142},
Title = {Travelling waves on fluid interfaces: {N}ormal form analysis of the {K}uramoto-{S}ivashisky equation},
Volume = {29},
Year = {1986}}
@article{ksgreene88,
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.},
Author = {J. M. Greene and J.-S. Kim},
Journal = {Physica D},
Pages = {99--120},
Title = {The steady states of the {Kuramoto-Sivashinsky} equation},
Volume = {33},
Year = {1988}}
@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,
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.},
Author = {S. Bouquet},
Journal = {J. Math. Phys.},
Pages = {1242},
Title = {Hamiltonian structure and integrability of the stationary {Kuramoto-Sivashinsky} equation},
Volume = {36},
Year = {1995}}
@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,
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.},
Author = {P. Kent and J. Elgin},
Journal = {Nonlinearity},
Pages = {899--919},
Title = {Travelling-waves of the {Kuramoto-Sivashinsky} equation: period-multiplying bifurcations},
Volume = {5},
Year = {1992}}
@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}
}
@article{KT84,
Author = {L. Kadanoff and C. Tang},
Journal = {Proc. Natl. Acad. Sci. USA},
Pages = {1276},
Title = {Escape rate from strange repellers},
Volume = {81},
Year = {1984}}
@article{ku,
Abstract = {The KSe is derived through the CGLe near the bifurcation
point. Some scaling argument is used to sort out the domiant terms.},
Author = {Y. Kuramoto and T. Tsuzuki},
Journal = {Progr. Theor. Phys.},
Pages = {365},
Title = {Persistent propagation of concentration waves in dissipative media far from thermal equilibrium},
Volume = {55},
Year = {1976}}
@article{KuKuKnMo04,
Author = {A. Yu. Kuznetsova and A. P. Kuznetsov and C. Knudsen and E. Mosekilde},
Doi = {10.1142/S0218127404009995},
Journal = {Int. J. Bif. Chaos},
Pages = {1241--1266},
Title = {Catastrophe theoretic classification of nonlinear oscillators},
Volume = {14},
Year = {2004}
}
@book{kura84tur,
Address = {Berlin},
Author = {Y. Kuramoto},
Publisher = {Springer},
Title = {Chemical Oscillations, Waves and Turbulence},
Year = {1984}}
@article{kuturb78,
Abstract = {Phase turbulence and amplitude turbulence
are named and distinguished from a dynamical systems point
of view. The prototyped equations are derived.},
Author = {Y. Kuramoto},
Journal = {Suppl. Progr. Theor. Phys.},
Pages = {346--367},
Title = {Diffusion-induced chaos in reaction systems},
Volume = {64},
Year = {1978}}
@article{KZ,
Author = {L. Kleiser and T. A. Zang},
Journal = {Annu. Rev. Fluid Mech.},
Pages = {495--537},
Title = {Numerical simulation of transition in wall-bounded shear flows},
Volume = {23},
Year = {1991}}
@article{Lagha07,
Author = {Lagha, M.},
Doi = {10.1063/1.2793143},
Journal = {Phys. Fluids},
Pages = {4108},
Title = {Modeling of plane {Couette} flow. {II. On} the spreading of a turbulent spot},
Volume = 19,
Year = 2007,
Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2793143}}
@Article{Lagha07b,
author = {M. Lagha and P. Manneville},
title = {Modeling transitional plane {Couette} flow},
journal = {Eur. Phys. J. B},
year = {2007},
volume = {58},
pages = {433--447},
doi = {10.1140/epjb/e2007-00243-y},
bdsk-url-1 = {http://dx.doi.org/10.1140/epjb/e2007-00243-y}
}
@article{LagMann07,
Author = {Lagha, M. and {Manneville}, P.},
Doi = {10.1063/1.2768946},
Journal = {Phys. Fluids},
Pages = {4105},
Title = {Modeling of plane {Couette} flow. {I. Large} scale flow around turbulent spots},
Volume = 19,
Year = 2007,
Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2768946}}
@article{LaKo89,
Author = {D. P. Lathrop and E. J. Kostelich},
Journal = {Phys. Rev. A},
Pages = {4028--4031},
Title = {Characterization of an experimental strange attractor by periodic orbits},
Volume = {40},
Year = {1989}}
@article{lamb98,
Author = {J. S. W. Lamb and J. A. G. Roberts},
Journal = {Physica D},
Pages = {1},
Title = {Time reversal symmetry in dynamical systems: {A} survey},
Volume = {112},
Year = {1998}}
@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}
}
@BOOK{Landau59c,
title = {Quantum Mechanics},
publisher = {Pergamon Press},
year = {1959},
author = {L.D. Landau and E.M. Lifshitz},
address = {Oxford}
}
@article{lanmaw03,
Author = {Y. Lan and N. Garnier and P. Cvitanovi{\'c}},
Journal = {Physica D},
Pages = {193--212},
Title = {Stationary modulated-amplitude waves in the 1{D} complex {Ginzburg-Landau} equation},
Volume = {188},
Year = {2004}}
@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},
note = {{\tt ChaosBook.org/projects/theses.html}}
}
@article{lanVar1,
Author = {Y. Lan and P. Cvitanovi{\'c}},
Journal = {Phys. Rev. E},
Note = {\arXiv{nlin.CD/0308008}},
Pages = {016217},
Title = {Variational method for finding periodic orbits in a general flow},
Volume = {69},
Year = {2004}}
@book{lapack,
Address = {Philadelphia},
Author = {Anderson, E. and Bai, Z. and Bischof, C. and Blackford, S. and Demmel, J. and Dongarra, J. and Du Croz, J. and Greenbaum, A. and Hammarling, S. and McKenney, A. and Sorensen, D.},
Edition = {Third},
Isbn = {0-89871-447-8 (paperback)},
Publisher = {Society for Industrial and Applied Mathematics},
Title = {{LAPACK} Users' Guide},
Year = {1999}}
@article{laquey74,
Author = {R. E. La Quey and S. M. Mahajan and P. H. Rutherford and W. M. Tang},
Journal = {Phys. Rev. Lett.},
Pages = {391--394},
Title = {Nonlinear saturation of the trapped-ion mode},
Volume = {34},
Year = {1974}}
@article{lathrop92,
Author = {D. P. Lathrop and J. Fineberg and H. L. Swinney},
Journal = {Phys. Rev. A.},
Pages = {6390},
Title = {Transition to shear-driven turbulence in {Couette-Taylor} flow},
Volume = {46},
Year = {1992}}
@book{latt,
Address = {Singapore},
Author = {M. Creutz},
Publisher = {World Scientific},
Title = {Quantum Fields on the Computer},
Year = {1992}}
@article{lattref,
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.},
Author = {B. I. Henry and S. D. Watt and S. L. Wearne},
Journal = {Anziam J.},
Pages = {C735--C751},
Title = {A lattice refinement scheme for finding periodic orbits},
Volume = {42E},
Year = {2000}}
@book{Lau05,
Address = {Boca Raton, FL},
Author = {Lautrup, B.},
Publisher = {{CRC Press}},
Title = {Physics of Continuous Matter: Exotic and Everyday Phenomena in the Macroscopic World},
Year = {2011}}
@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}
}
@Incollection{lauw86,
author = {H. A. Lauwerier},
title = {Two-dimensional iterative maps},
booktitle = {Chaos},
publisher = {Princeton Univ. Press},
year = {1986},
editor = {V. Holden},
isbn = {9780691610535},
pages = {58--96},
address = {Princeton}
}
@article{LCC06,
Author = {Y. Lan and C. Chandre and P. Cvitanovi{\'c}},
Journal = {Phys. Rev. E},
Note = {\arXiv{nlin.CD/0508026}},
Pages = {046206},
Title = {Variational method for locating invariant tori},
Volume = {74},
Year = {2006}}
@article{lee01,
Author = {K. Lee and L. Cortelezzi and J. Kim and J. L. Speyer},
Journal = {Phys. Fluids},
Pages = {1321},
Title = {Application of reduced order controller to turbulent flows for drag reduction},
Volume = {13},
Year = {2001}}
@article{Lega:holes,
Author = {J. Lega},
Journal = {Physica D},
Pages = {152--153},
Title = {Traveling hole solutions of the complex {Ginzburg Landau} equation: a review},
Volume = {269},
Year = {2001}}
@article{levenberg44,
author = {Levenberg, K},
journal = {Quart. Appl. Math.},
number = {2},
pages = {164--168},
title = {A method for the solution of certain non-linear problems in least squares},
volume = {2},
year = {1944}
}
@misc{LevnMezi08,
Abstract = {We present a computational study of the
invariant sets visualization method based on ergodic
partition theory. The algorithms
for computation of the time averages of many L1 functions
are developed and employed producing the approximation of
the phase space ergodic partitioning. The method is
exposed in the context of discrete-time dynamical systems
producing a graphical representation of the phase space
in terms of the invariant set structure that gives a
substantial insight into the global and local properties
of the dynamics. We use the Chirikov standard map in
order to show the implementation of the method, followed
by applications to other multi-dimensional maps. We examine
the visualization of periodic sets using harmonic time
averages.},
Author = {Levnaji\'c, Z. and Mezi\'c, I.},
Note = {\arXiv{0805.4221}},
Title = {Ergodic theory and visualization {I}: {Visualization} of ergodic partition and invariant sets},
Year = {2008}}
@article{lfind,
Author = {F. K. Diakonos and P. Schmelcher and O. Biham},
Journal = {Phys. Rev. Lett.},
Note = {\arXiv{chao-dyn/9810022}},
Pages = {4349},
Title = {Systematic computation of the least unstable periodic orbits in chaotic attractors},
Volume = {81},
Year = {1998}}
@incollection{LHR94,
Address = {Dordrecht},
Author = {A. Lundbladh and and S. Henningson and S. C. Reddy},
Booktitle = {Turbulence and Combustion},
Editor = {Hussaini, M.Y. and Gatski, T.B. and Jackson, T.L.},
Publisher = {Kluwer},
Title = {Threshold amplitudes for transition in channel flows},
Year = {1994}}
@Article{Li08,
author = {Li, Y. C.},
title = {Persistence of steady {3D Euler} solutions for {3D Navier-Stokes} equations},
note = {\arXiv{0811.0383}}
}
@article{Lichtner2007,
Author = {M. Lichtner and M. Radziunas and L. Recke},
Journal = {Math. Methods Appl. Sci.},
Pages = {931--960},
Title = {Well-posedness, smooth dependence and center manifold reduction for a semilinear hyperbolic system from laser dynamics},
Volume = {30},
Year = {2007}}
@book{liede,
Address = {New York},
Author = {P. J. Olver},
Publisher = {Spring-Verlag},
Series = {GTM},
Title = {Applications of {Lie} Groups to Differential Equations},
Volume = {107},
Year = {1998}}
@misc{LiHuCa10,
Author = {Libert, A.-S. and Hubaux, C. and Carletti, T.},
Note = {\arXiv{arXiv:1005.5611}},
Title = {Symplectic integration of deviation vectors and chaos determination. Application to the {H\'enon-Heiles} model and to the restricted three-body problem},
Year = {2010}}
@article{LiMa83,
Author = {Li, T.-P. and Ma, Y.-Q.},
Doi = {10.1086/161295},
Journal = {Astrophys. J.},
Pages = {317--324},
Title = {Analysis methods for results in gamma-ray astronomy},
Volume = {272},
Year = {1983}
}
@Article{LiMoRo01,
author = {Lim, C. and Montaldi, J. and {Roberts}, M.},
title = {Relative equilibria of point vortices on the sphere},
journal = {Physica D},
year = {2001},
volume = {148},
pages = {97--135}
}
@book{Lin55,
Address = {Cambridge},
Author = {Lin, C. C.},
Publisher = {Cambridge Univ. Press},
Title = {The Theory of Hydrodynamic Stability},
Year = {1955}}
@unpublished{LipCvi07,
Author = {D. Lippolis and P. Cvitanovi\'c},
Note = {In preparation, 2011},
Title = {Optimal resolution of the state space of a chaotic flow in presence of noise}}
@article{LipCvi08,
Author = {D. Lippolis and P. Cvitanovi\'c},
Journal = {Phys. Rev. Lett.},
Pages = {014101},
Title = {How well can one resolve the state space of a chaotic map?},
Volume = {104},
Year = {2010},
doi={10.1103/PhysRevLett.104.014101},
note = {\arXiv{0902.4269}},
}
@Article{liuliupego05,
author = {J.-G. Liu and J. Liu and R. L. Pego},
title = {Divorcing pressure from viscosity in incompressible {Navier-Stokes} dynamics},
year = {2005},
url = {ar
Xiv:math.AP/0502549}
}
@Article{lixigraham06,
author = {W. Li and L. Xi and M.D. Graham},
title = {Nonlinear travelling waves as a framework for understanding turbulent drag reduction},
journal = {J. Fluid Mech},
year = {2006},
volume = {565},
pages = {353--362}
}
@Article{lixigraham07,
author = {W. Li and M.D. Graham},
title = {Polymer induced drag reduction in exact coherent structures of plane Poiseuille flow},
journal = {Phys. Fluids},
year = {2007},
volume = {19},
number = {083101},
pages = {1--15}
}
@Incollection{Llibre15,
author = {Llibre, J.},
title = {The averaging theory for computing periodic orbits},
booktitle = {Central Configurations, Periodic Orbits, and Hamiltonian Systems},
publisher = {Springer},
year = {2015},
isbn = {978-3-0348-0932-0},
series = {Advanced Courses in Mathematics - CRM Barcelona},
pages = {1--104},
address = {Basel},
doi = {10.1007/978-3-0348-0933-7_1},
language = {English}
}
@book{lls,
Address = {Providence, RI},
Author = {B. L. Ro\v{z}destvenski\v{i} and N. N. Janenko},
Publisher = {AMS},
Title = {Systems of Quasilinear Equations and Their Applications to Gas Dynamics},
Year = {1983}}
@book{LM94,
Address = {Berlin},
Author = {A. Lasota and M. MacKey},
Publisher = {Springer},
Title = {Chaos, Fractals, and Noise; Stochastic Aspects of Dynamics},
Year = {1994}}
@Article{LoCaCoPeGo11,
author = {Lombardi, M. and Caulfield, C.P. and Cossu, C. and Pesci, A.I. and Goldstein, R.E.},
title = {Growth and instability of a laminar plume in a strongly stratified environment},
journal = {J. Fluid Mech.},
year = {2011},
volume = {671},
pages = {184--206},
doi = {10.1017/S0022112010005574},
bdsk-url-1 = {http://dx.doi.org/10.1017/S0022112010005574}
}
@Article{lop05rel,
author = {V. L{\'o}pez and P. Boyland and M. T. Heath and R. D. Moser},
title = {Relative periodic solutions of the {Complex Ginzburg-Landau} equation},
journal = {SIAM J. Appl. Dyn. Syst.},
year = {2006},
volume = {4},
pages = {1042--1075},
note = {\arXiv{nlin/0408018}},
doi = {10.1137/040618977},
abstract = {Define relative periodic orbits and use Fourier modes
to find them.},
bdsk-url-1 = {http://dx.doi.org/10.1137/040618977}
}
@Article{LopMar00,
author = {Lopez, J. M. and Marques, F.},
title = {Determining the self-rotation number following a {Neimark-Sacker} bifurcation in the periodically forced {Taylor-Couette} flow},
journal = {Z. Angew. Math. Phys.},
year = {2000},
volume = {51},
pages = {61--74},
doi = {10.1007/PL00001507},
keywords = {Key words. Floquet theory, self-rotation number, parametric excitation, quasiperiodic flow, Taylor--Couette flow.}
}
@Article{LopMarq05,
author = {J. M. Lopez and F. Marques},
title = {Finite aspect ratio {Taylor-Couette} flow: {Shil'nikov} dynamics of 2-tori},
journal = {Physica D},
year = {2005},
volume = {211},
pages = {168--191}
}
@article{LorentzDiff,
author = {Cvitanovi\'c, P. and Eckmann, J.-P. and Gaspard, P.},
title = {Transport properties of the {Lorentz} gas
in terms of periodic orbits},
journal = {Chaos Solit. Fract.},
volume = {6},
pages = {113--120},
year={1995},
note = {\arXiv{chao-dyn/9312003}},
doi = {10.1016/0960-0779(95)80018-C}
}
@article{lorenz63,
Author = {E. N. Lorenz},
Journal = {J. Atmos. Sci.},
Pages = {130--141},
Title = {Deterministic nonperiodic flow},
Volume = {20},
Year = {1963}}
@Article{LoTh10,
author = {G. J. Lord and V. Th{\"u}mmler},
title = {Computing stochastic travelling waves},
journal = {SIAM J. Sci. Comput.},
year = {2012},
volume = {34},
pages = {B24--B43},
addendum = {\arXiv{1006.0428}},
doi = {10.1137/100784734}
}
@article{Low58,
Abstract = {A variational principle is found for the {Boltzmann-Vlasov} equation
for an ionized gas in an electromagnetic field. The principle leads
to a new formulation of the problem of small oscillations about equilibrium.},
Author = {F. E. Low},
Journal = {Proc. R. Soc. London A},
Pages = {282--287},
Title = {A {{Lagrangian}} formulation of the {{Boltzmann-Vlasov}} equation for plasmas},
Volume = {248},
Year = {1958}}
@article{Luce95,
Author = {B. P. Luce},
Journal = {Physica D},
Pages = {553--581},
Title = {Homoclinic explosions in the complex {Ginzburg-Landau} equation},
Volume = {84},
Year = {1995}}
@Article{LucKer12,
author = {Lucas, D. and Kerswell, R. R.},
title = {Spatiotemporal dynamics in {2D Kolmogorov} flow over large domains},
journal = {J. Fluid Mech.},
year = {2014},
volume = {750},
pages = {518--554},
note = {\arXiv{1308.3356}},
doi = {10.1017/jfm.2014.270},
bdsk-url-1 = {http://dx.doi.org/10.1017/jfm.2014.270}
}
@Article{LucKer14,
author = {Lucas, D. and Kerswell, R. R.},
title = {Recurrent flow analysis in spatiotemporally chaotic 2-dimensional {Kolmogorov} flow},
journal = {Phys. Fluids},
year = {2015},
volume = {27},
pages = {518--554},
note = {\arXiv{1406.1820}},
doi = {10.1063/1.4917279}
}
@Proceedings{Lumley_1989,
title = {Whither Turbulence? Turbulence at the Crossroads: Proceedings, Workshop Cornell U., {March} 22-24, 1989},
year = {1990},
editor = {Lumley, J. L.},
volume = {357},
address = {New York},
publisher = {Springer},
doi = {10.1007/3-540-52535-1}
}
@article{lumley67,
Author = {{Bakewell}, P. and {Lumley}, J. L.},
Journal = {Phys. Fluids},
Pages = {1880},
Title = {Viscous sublayer and adjacent wall region in turbulent pipe flow},
Volume = 10,
Year = 1967}
@article{lumley98,
Author = {Lumley, J. L. and Blossey, P.},
Journal = {Ann. Rev. Fluid Mech.},
Pages = {311},
Title = {Control of turbulence},
Volume = 30,
Year = 1998}
@article{LundJohan91,
Author = {A. Lundbladh and A. Johansson},
Journal = {J. Fluid Mech.},
Pages = {499--516},
Title = {Direct simulation of turbulent spots in plane {Couette} flow},
Volume = {229},
Year = {1991}}
@misc{lvov-2007,
Author = {V. S. L'vov and I. Procaccia and O. Rudenko},
Note = {\arXiv{0705.4592}},
Title = {Universal model of finite-{Reynolds} number turbulent flow in channels and pipes},
Year = {2007}}
@Article{LY75,
author = {T. Li and J. A. Yorke},
title = {Period 3 implies chaos},
journal = {Amer. Math. Monthly},
year = {1975},
volume = {82},
pages = {985}
}
@article{lyaos,
Abstract = {The Liapunov characteristic numbers are defined by investigating the limiting volume of infinitesimal size. Their existence is proved by construction and application of the ergodic theorem. At a.e. every point, the tangent space is divided into stable and unstable subspace corresponding the positive and negative Liapunov numbers. If the eigenvalues are non-degenerate, the Liyapunov vectors are well defined a.e.},
Author = {V. I. Oseledec},
Journal = {Trans. Moscow Math. Soc.},
Pages = {197--221},
Title = {A multiplicative ergodic theorem. {Liapunov} characteristic numbers for dynamical systems},
Volume = {19},
Year = {1968}}
@Article{Lyons13,
author = {Lyons, L.},
title = {{Bayes and Frequentism}: a particle physicist's perspective},
journal = {Contemporary Physics},
year = {2013},
volume = {54},
pages = {1--16},
doi = {10.1080/00107514.2012.756312},
abstract = { In almost every scientific field, an experiment involves
collecting data and then analysing it. The analysis stage will often
consist in trying to extract some physical parameter and estimating
its uncertainty; this is known as Parameter Determination. An example
would be the determination of the mass of the top quark, from data
collected from high energy proton--proton collisions. A different aim
is to choose between two possible hypotheses. For example, are data
on the recession speed s of distant galaxies proportional to their
distance d, or do they fit better to a model where the expansion of
the Universe is accelerating? There are two fundamental approaches to
such statistical analyses -- Bayesian and Frequentist. This article
discusses the way they differ in their approach to probability, and
then goes on to consider how this affects the way they deal with
Parameter Determination and Hypothesis Testing. The examples are
taken from everyday life and from particle physics. }
}
@Book{lyschm,
title = {{Lyapunov-Schmidt} Methods in Nonlinear Analysis and Applications},
publisher = {Kluwer},
year = {2002},
author = {N. Sidorov and B. Loginov and A. Sinitsyn and M. Falaleev},
address = {Dordrecht}
}
@article{M86,
Author = {S. A. Maslowe},
Journal = {Ann. Rev. Fluid Mech.},
Pages = {405--432},
Title = {Critical layers in shear flows},
Volume = {18},
Year = {1986}}
@article{Ma02,
Author = {X. Ma and G.E. Karniadakis},
Journal = {J. Fluid Mech.},
Pages = {181--190},
Title = {A low-dimensional model for simulating 3-dimensional cylinder flow},
Volume = {458},
Year = {2002}}
@ARTICLE{MacSch12,
author = {R. MacPherson and B. Schweinhart},
title = {Measuring shape with topology},
journal = {J. Math. Phys.},
year = {2012},
volume = {53},
pages = {073516},
doi = {10.1063/1.4737391}
}
@book{mallet02,
Address = {Oxford},
Author = {Mallet, J.-L.},
Publisher = {Oxford Univ. Press},
Title = {Geomodeling},
Year = {2002}}
@book{man90b,
Address = {Boston},
Author = {P. Manneville},
Publisher = {Academic Press},
Title = {Dissipative structures and weak turbulence},
Year = {1990}}
@book{mandel,
Address = {New York},
Author = {B. B. Mandelbrot},
Publisher = {WH Freeman Co.},
Title = {The Fractal Geometry of Nature},
Year = {1982}}
@Article{Mann04,
author = {P. Manneville},
title = {Spots and turbulent domains in a model of transitional plane {Couette} flow},
journal = {Theor. Comp. Fluid Dyn.},
year = {2004},
volume = {18},
pages = {169--181},
doi = {10.1007/s00162-004-0142-4}
}
@Article{Mann08,
author = {P. Manneville},
title = {Understanding the sub-critical transition to turbulence in wall flows},
journal = {Pramana},
year = {2008},
volume = {70},
pages = {1009--1021},
doi = {10.1007/s12043-008-0105-1}
}
@unpublished{Mann14,
Author = {Manneville, P.},
Note = {\arXiv{1403.6374}},
Title = {On the transition to turbulence of wall-bounded flows in general, and plane {Couette} flow in particular},
Year = 2014}
@Article{marcio03,
author = {M. Gameiro and W. Kalies and K. Mischaikow},
title = {Topological characterization of spatial-temporal chaos},
journal = {Phys. Rev. E},
year = {2004},
volume = {70},
pages = {035203},
doi = {10.1103/PhysRevE.70.035203},
abstract = {Use the Betti number to characterize the spatiotemporal
dynamics of a system. A lyapunov exponent is suggested to measure the
space as well as temporal complexity of the evolution.}
}
@Article{Marcus84,
Author = {Marcus, P. S.},
Title = {Simulation of {Taylor-Couette} flow. {Part} 1.
{Numerical} methods and comparison with experiment},
Journal = {J. Fluid Mech.},
Volume = {146},
Pages = {45--64},
doi = {10.1017/S0022112084001762},
year = 1984
}
@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}
}
@book{MarRat99,
Address = {New York},
Author = {Marsden, J. E. and Ratiu, T. S.},
Publisher = {Springer},
Title = {Introduction to Mechanics and Symmetry},
Year = 1999}
@book{Marsd92,
Address = {Cambridge},
Author = {Marsden, J. E.},
Publisher = {Cambridge Univ. Press},
Title = {Lectures on Mechanics},
Year = {1992}}
@book{marsdenbb,
Author = {J. E. Marsden and M. McCracken},
Publisher = {Springer},
Title = {The {H}opf Bifurcation and its Applications},
Year = {1976}}
@article{mart87lfa,
Abstract = {local frequency analysis is used and action-angle
coordinate is parametrized by the frequency ratio. The transport
zones and partial barriers are identified according to the },
Author = {C. C. Martens and M. J. Davis and G. S. Ezra},
Journal = {Chem. Phys. Lett.},
Pages = {519},
Title = {Local Frequency analysis of chaotic motion in multidimensional systems: energy transport and bottlenecks in planar {OCS}},
Volume = {142},
Year = {1987}}
@Article{Marw08,
author = {N. Marwan},
title = {A historical review of recurrence plots},
journal = {Eur. Phys. J. Spec. Top.},
year = {2008},
volume = {164},
pages = {3--12},
doi = {10.1140/epjst/e2008-00829-1}
}
@article{MaSiRo73,
Author = {P. C. Martin and E. D. Siggia and H. A. Rose},
Journal = {Phys. Rev. A},
Pages = {423--437},
Title = {Statistical mechanics of classical systems},
Volume = {8},
Year = {1973}}
@article{Mather91,
Author = {J. N. Mather},
Journal = {J. Am. Math. Soc.},
Pages = {207--263},
Title = {Variational construction of orbits of twist difeomorphisms},
Volume = {4},
Year = {1991}}
@Book{MathWalk73,
title = {Mathematical Methods of Physics},
publisher = {W. A. Benjamin},
year = {1970},
author = {J. Mathews and R. L. Walker},
isbn = {9780805370027},
address = {Reading, MA}
}
@Article{May76,
author = {R. May},
title = {Simple mathematical models with very complicated dynamics},
journal = {Nature},
year = {1976},
volume = {261},
pages = {459--467}
}
@article{MC96pha,
Abtract = {instability of plane waves},
Author = {P. Manneville and H. Chat{\'e}},
Journal = {Physica D},
Pages = {30--46},
Title = {Phase turbulence in the two-dimensional complex {Ginzburg-Landau} equation},
Volume = {96},
Year = {1996}}
@article{McKean74,
Author = {McKean, H. P.},
Journal = {Commun. Pure Appl. Math.},
Pages = {225--246},
Title = {Selberg's trace formula as applied to a compact {Riemann} surface},
Volume = {25},
Year = {1972}}
@article{McKeon04,
Author = {B. J. McKeon and J. Li and W. Jiang and J. F. Morrison and A. J. Smits},
Journal = {J. Fluid Mech.},
Pages = {135--147},
Title = {Further observations on the mean velocity in fully-developed pipe flow},
Volume = {501},
Year = {2004}}
@Article{Mcoh03,
author = {M. van Hecke},
title = {Coherent and incoherent structures in systems described by the 1{D CGLE}: {Experiments} and identification},
journal = {Physica D},
year = {2003},
volume = {174},
pages = {134},
abstract = {Give an overview of the most important properties of the
coherent structures and address the question how such structures may
be identified in the experimental space-time data set.}
}
@article{mcord86,
Author = {C. K. McCord},
Journal = {J. Math. Anal. Appl.},
Pages = {584--592},
Title = {Uniqueness of connection orbits in the equation $y^{(3)}=y^2-1$},
Volume = {114},
Year = {1986}}
@book{Mees81,
Address = {New York},
Author = {Mees, A. I.},
Publisher = {Wiley},
Title = {Dynamics of Feedback Systems},
Year = {1981}}
@article{MEF04,
Author = {J. Moehlis and B. Eckhardt and H. Faisst},
Journal = {Chaos},
Pages = {S11},
Title = {Fractal lifetimes in the transition to turbulence},
Volume = {14},
Year = {2004}}
@article{Mehta05,
Abstract = {We obtain numerical solutions for extremely
small symmetry-preserving perturbations of the nonlinear
equations governing 2D {Couette} flow.},
Author = {Mehta, P.~G. and {Healey}, T.~J.},
Journal = {Phys. Fluids},
Pages = {4108},
Title = {On steady solutions of symmetry-preserving perturbations of the two-dimensional {Couette} flow problem},
Volume = 17,
Year = 2005}
@article{mellibovsky11,
Author = {Mellibovsky, F. and Eckhardt, B.},
Journal = {J. Fluid Mech.},
Pages = {96--129},
Title = {{Takens--Bogdanov} bifurcation of
travelling-wave solutions in pipe flow},
Volume = {670},
Year = {2011}}
@article{mellibovsky12,
Author = {Mellibovsky, F. and Eckhardt, B.},
Journal = {J. Fluid Mech.},
Note = {\arXiv{1110.1222}},
Pages = {149--190},
Title = {From travelling waves to mild chaos:
{A} supercritical bifurcation cascade in pipe flow},
Volume = {709},
Year = {2011}}
@Article{MeseguerANM07,
author = {A. Meseguer and F. Mellibovsky},
title = {On a solenoidal {Fourier-Chebyshev} spectral method for stability analysis of the {Hagen--Poiseuille} flow},
journal = {Appl. Num. Math.},
year = {2007},
volume = {57},
pages = {920--938}
}
@article{MesTre03,
title={Linearized pipe flow to {Reynolds} number $10^7$},
author={Meseguer, A and Trefethen, L. N},
journal={J. Comput. Phys.},
volume={186},
pages={178--197},
year={2003}
}
@article{MeWi94,
Author = {I. Mezi\'c and S. Wiggins},
Journal = {J. Nonlin. Sci.},
Pages = {157--194},
Title = {On the integrability and perturbation of three-dimensional fluid flows with symmetry},
Abstract = {The purpose of this paper is to develop analytical methods for studyingparticle
paths in a class of three-dimensional incompressible fluid flows.
In this paper we study three-dimensionalvolume preserving vector
fields that are invariant under the action of a one-parameter symmetry
group whose infinitesimal generator is autonomous and volume-preserving.
We show that there exists a coordinate system in which the vector
field assumes a simple form. In particular, the evolution of two
of the coordinates is governed by a time-dependent, one-degree-of-freedom
Hamiltonian system with the evolution of the remaining coordinate
being governed by a first-order differential equation that depends
only on the other two coordinates and time. The new coordinates depend
only on the symmetry group of the vector field. Therefore they arefield-independent.
The coordinate transformation is constructive. If the vector field
is time-independent, then it possesses an integral of motion. Moreover,
we show that the system can be further reduced toaction-angle-angle
coordinates. These are analogous to the familiar action-angle variables
from Hamiltonian mechanics and are quite useful for perturbative
studies of the class of systems we consider. In fact, we show how
our coordinate transformation puts us in a position to apply recent
extensions of the {Kolmogorov-Arnold-Moser} {(KAM)} theorem for three-dimensional,
volume-preserving maps as well as three-dimensional versions of Melnikov's
method. We discuss the integrability of the class of flows considered,
and draw an analogy with Clebsch variables in fluid mechanics.},
Volume = {4},
Year = {1994}}
@Article{Mezic05,
author = {Mezi\'c, Igor},
title = {Spectral properties of dynamical systems, model reduction and decompositions},
journal = {Nonlinear Dyn.},
year = {2005},
volume = {41},
pages = {309--325},
doi = {10.1007/s11071-005-2824-x}
}
@article{MFE04,
Author = {J. Moehlis and H. Faisst and B. Eckhardt},
Journal = {New Journal of Physics},
Pages = {56},
Title = {A low-dimensional model for turbulent shear flows},
Volume = {6},
Year = {2004}}
@Article{MFE04b,
author = {J. Moehlis and H. Faisst and B. Eckhardt},
title = {Periodic orbits and chaotic sets in a low-dimensional model for shear flows},
journal = {SIAM J. Appl. Dyn. Sys.},
year = {2005},
volume = {4},
pages = {352--376},
doi = {10.1137/040606144}
}
@article{mfind,
Author = {R. L. Davidchack and Y.-C. Lai},
Journal = {Phys. Rev. E},
Pages = {6172},
Title = {Efficient algorithm for detecting unstable periodic orbits in chaotic systems},
Volume = {60},
Year = {1999}}
@article{MFKM10,
Abstract = {A method for estimating a period of unstable periodic solutions
is suggested in continuous dissipative chaotic dynamical
systems. The measurement of a minimum distance between a
reference state and an image of transformation of it exhibits a
characteristic structure of the system, and the local minima of
the structure give candidates of period and state of
corresponding symmetric solutions. Appropriate periods and
initial states for the Newton method are chosen efficiently by
setting a threshold to the range of the minimum distance and
the period.},
Author = {Morita, Y. and Fujiwara, N. and Kobayashi, M. U. and Mizuguchi, T.},
Journal = {Chaos},
Pages = {013126},
Title = {Scytale decodes chaos: {A} method for estimating unstable symmetric solutions},
Volume = {20},
Year = {2010}}
@book{MH,
Address = {New York},
Author = {K. R. Meyer and G. R. Hall},
Publisher = {Springer},
Title = {Introduction to Hamiltonian Dynamical Systems and the {N}-body Problem},
Year = {1992}}
@article{MHMwind96,
Abstract = {There exist a band of winding numbers associated with
stable states in the phase turbulent regime. The states with large
winding numbers decay to the ones with smaller winding numbers.
Defect chaos happens when the range of stable winding numbers
vanishes.},
Author = {R. Montagne and E. Hern{\'a}ndez-Garc{\'\i}a and M. S. Miguel},
Journal = {Phys. Rev. Lett.},
Pages = {267},
Title = {Winding number instability in the phase-turbulence regime of the complex {Ginzburg-Landau} equation},
Volume = {77},
Year = {1996}}
@article{michsiv77,
Abstract = {Consider the effect of hydrodynamics instability and diffusion thermal instability through the numerical calculation of the KSe.},
Author = {D. M. Michelson and G. I. Sivashinsky},
Journal = {Acta Astronaut.},
Pages = {1207--1221},
Title = {Nonlinear analysis of hydrodynamic instability in laminar flames - {II}. Numerical experiments},
Volume = {4},
Year = {1977}}
@Misc{MiPlSt11,
author = {Mini\'c, Dj. and Pleimling, M. and Staples, A. E.},
title = {On the steady state distributions for turbulence},
year = {2011},
note = {\arXiv{1105.2941}}
}
@book{MiraGum80,
Address = {Berlin},
Author = {I. Gumowski and C. Mira},
Publisher = {Springer},
Title = {Recurrances and Discrete Dynamical Systems},
Year = {1980}}
@article{mis01ks,
Author = {P. Zgliczynski and K. Mischaikow},
Journal = {Foundations of Comp. Math.},
Pages = {255--288},
Title = {Rigorous Numerics for Partial Differential Equations: the {Kuramoto-Sivashinsky} equation},
Volume = {1},
Year = {2001}}
@article{misdis,
Author = {K. Mischaikow and M. Mrozek and J. Reiss and A. Szymczak},
Journal = {Phys. Rev. Lett.},
Pages = {1144},
Title = {Construction of Symbolic Dynamics from Experimental Time Series},
Volume = {82},
Year = {1999}}
@article{mislor1,
Author = {K. Mischaikow and M. Mrozek},
Journal = {Math. Comp.},
Pages = {1023--1046},
Title = {Chaos in the {Lorenz} equations: {A} computer assisted proof {Part II}: {Details}},
Volume = {67},
Year = {1998}}
@phdthesis{MitchellThesis13,
Address = {Atlanta},
Author = {Mitchell Jr., R.},
Note = {{\tt ChaosBook.org/projects/theses.html}},
School = {School of Physics, Georgia Inst. of Technology},
Title = {Transition to turbulence and mixing in a quasi-two-dimensional
{Lorentz} force-driven {Kolmogorov} flow},
Year = {2013}}
@Article{Mks86,
author = {D. Michelson},
title = {Steady solutions of the {Kuramoto-Sivashinsky} equation},
journal = {Physica D},
year = {1986},
volume = {19},
pages = {89--111},
doi = {10.1016/0167-2789(86)90055-2}
}
@article{MM,
Author = {P. Moin and K. Mahesh},
Journal = {Annu. Rev. Fluid Mech.},
Pages = {539--78},
Title = {Direct numerical simulation: a tool in turbulence research},
Volume = {30},
Year = {1998}}
@article{MMG95,
Author = {S. Malerud and K. J. M\aa{}l\o{}y and W. I. Goldburg},
Journal = {Phys. Fluids},
Pages = {1949--1955},
Title = {Measurements of turbulent velocity fluctuations in a planar {Couette} cell},
Volume = {7},
Year = {1995}}
@inproceedings{Moffat90,
Address = {New York},
Author = {H. K. Moffatt},
Booktitle = {Whither Turbulence? Turbulence at the Crossroads},
Editor = {J. L. Lumley},
Organization = {Springer Lect. Notes in Physics},
Pages = {250--257},
Publisher = {Springer},
Title = {Fixed points of turbulent dynamical systems and suppression of nonlinearity},
Year = {1990}}
@article{Moffatt69,
Author = {Moffatt, H. K.},
Journal = {J. Fluid Mech},
Pages = {117--129},
Title = {The degree of knottedness of tangled vortex lines},
Volume = {35},
Year = {1969}}
@incollection{Moffatt90,
Address = {New York},
Author = {H. K. Moffatt},
Booktitle = {Whither Turbulence? Turbulence at the Crossroads},
Editor = {Lumley, J. L.},
Pages = {250--257},
Publisher = {Springer},
Title = {Fixed points of turbulent dynamical systems and suppression of nonlinearity, {Comment} 1},
Year = {1990}}
@article{MoMeCa84,
Abstract = {The Lagrangian density for the regularized-long-wave equation
(also known as the BBM equation) is presented. Using the trial
function technique, ordinary differential equations that describe
the time dependence of the position of the peaks, amplitudes,
and widths for the collision of two solitary waves are obtained.
These equations are analyzed in the Born and ``equal-width''
approximations and compared with numerical results obtained
by direct integration utilizing the split-step fast Fourier-transform
method. The computations show that collisions are inelastic and
that production of solitary waves may occur.},
Author = {P. J. Morrison and J. D. Meiss and J. R. Cary},
Journal = {Physica D},
Pages = {324--336},
Title = {Scattering of regularized-long-wave solitary waves},
Volume = {11},
Year = {1984}}
@book{MonYag71,
Address = {Cambridge},
Author = {Monin, A.S. and Yaglom, A.M.},
Publisher = {MIT Press},
Title = {Statistical Fluid Mechanics},
Year = {1971}}
@book{Moon87,
Address = {New York},
Author = {F. C. Moon},
Publisher = {Wiley},
Title = {Chaotic Vibrations: An Introduction for Applied Scientists and Engineers},
Year = 1987}
@inproceedings{Morkovin69,
Address = {New York},
Author = {M. V Morkovin},
Booktitle = {Viscous Drag Reduction},
Editor = {C. S. Wells},
Pages = {1--31},
Publisher = {Plenum},
Title = {On the many faces of transition},
Year = {1969}}
@Article{MoRo99,
author = {Montaldi, J.~A. and {Roberts}, R.~M.},
title = {Relative equilibria of molecules},
journal = {J. Nonlinear Science},
year = {1999},
volume = {9},
pages = {53--88}
}
@Article{MoRoSt88,
author = {Montaldi, J.~A. and {Roberts}, R.~M. and {Stewart}, I.~N.},
title = {Periodic solutions near equilibria of symmetric {Hamiltonian} systems},
journal = {Phil. Trans. Royal Soc. A},
year = {1988},
volume = {325},
pages = {237--293}
}
@article{Morr80,
Author = {Morrison, P. J.},
Journal = {Phys. Lett. A},
Pages = {383--386},
Title = {The {Maxwell-Vlasov} equations as a continuous {Hamiltonian} system},
Volume = {80},
Year = {1980}}
@article{Morr98,
Author = {Morrison, P. J.},
Journal = {Rev. Mod. Phys.},
Pages = {467--521},
Title = {Hamiltonian description of the ideal fluid},
Volume = {70},
Year = {1998}}
@article{MorrGree80,
Author = {Morrison, P. J. and J. M. Greene},
Journal = {Phys. Rev. Lett.},
Note = {see also Phys. Rev. Lett. 48, 569 (1982)},
Pages = {790--794},
Title = {Noncanonical {Hamiltonian} Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics},
Volume = {45},
Year = {1980}}
@article{Morrison02,
Author = {J. F. Morrison and W. Jiang and B. J. McKeon and A. J. Smits},
Journal = {Phys. Rev. Lett.},
Pages = {214501},
Title = {Reynolds number dependence of streamwise velocity spectra in turbulent pipe flow},
Volume = {88},
Year = {2002}}
@article{Morrison04,
Author = {J. F. Morrison and B. J. McKeon and W. Jiang and A. J. Smits},
Journal = {J. Fluid Mech.},
Pages = {99--131},
Title = {Scaling of the streamwise velocity component in turbulent pipe flow},
Volume = {508},
Year = {2004}}
@article{Moser83,
Author = {R. Moser and P. Moin and A. Leonard},
Journal = {J. Comp. Phys.},
Pages = {524--544},
Title = {A spectral method for the {Navier-Stokes} equations with applications to {Taylor-Couette} flow},
Volume = {52},
Year = {1983}}
@article{moser96,
Author = {J. Moser},
Journal = {Ann. Scuola Norm. Super. Pisa},
Pages = {265--315},
Title = {A rapidly converging iteration method and nonlinear partial differential equations},
Volume = {20},
Year = {1966}}
@article{moser96a,
Author = {J. Moser},
Journal = {Ann. Scuola Norm. Super. Pisa},
Pages = {499--535},
Volume = {20},
Year = {1966}}
@article{MoserPF99,
Author = {R. Moser and J. Kim and N. N. Mansour},
Journal = {Phys. Fluids},
Pages = {943--946},
Title = {Direct numerical simulation of turbulent channel flow up to {$Re_{\tau} = 590$}},
Volume = {11},
Year = {1999}}
@book{mosersr,
Address = {Princeton},
Author = {J. Moser},
Publisher = {Princeton Univ. Press},
Title = {Stable and Random Motions in Dynamical Systems},
Year = {1973}}
@article{Mostow57,
Author = {G. D. Mostow},
Journal = {Ann. Math.},
Pages = {432--446},
Title = {Equivariant embeddings in {Euclidean} space},
Volume = {65},
Year = {1957}}
@book{mpc,
Address = {New York},
Author = {E. F. Camacho and C. Bordons},
Publisher = {Springer},
Title = {Model Predictive Control},
Year = {2004}}
@article{MR94,
Author = {G. P. Morriss and L. Rondoni},
Journal = {J. Stat. Phys.},
Pages = {553},
Volume = {75},
Year = {1994}}
@article{MRTK07,
Author = {Marwan, N. and Romano, M. C. and Thiel, M. and Kurths, J.},
Journal = {Phys. Rep.},
Pages = {237},
Title = {Recurrence plots for the analysis of complex systems},
Volume = {438},
Year = {2007}}
@article{MS51,
Author = {D. Meksyn and J. T. Stuart},
Journal = {Proc. Roy. Soc. A},
Pages = {517},
Title = {Stability of viscous motion between parallel planes for finite disturbances},
Volume = {208},
Year = {1951}}
@article{MS66,
Author = {D. W. Moore and E. A. Spiegel},
Journal = {Astrophys. J.},
Pages = {871},
Title = {A thermally excited nonlinear oscillator},
Volume = {143},
Year = {1966}}
@book{MuKe05,
Address = {Dordrecht},
Author = {T. Mullin and R. R. Kerswell},
Publisher = {Kluwer},
Title = {Non-uniqueness of Solutions to the Navier-Stokes Equations
and their Connection with Laminar-Turbulent Transition},
Year = {2005}}
@article{Mul10,
Author = {Mullin, T.},
Journal = {Ann. Rev. Fluid Mech.},
Pages = {1--24},
Title = {Experimental studies of transition to turbulence in a pipe},
Volume = {43},
Year = {2011}}
@Incollection{MuPe06,
author = {T. Mullin and J. Peixinho},
title = {Recent observations of the transition to turbulence in a pipe},
booktitle = {IUTAM Symposium on Laminar-Turbulent Transition},
publisher = {Springer},
year = {2006},
editor = {Govindarajan, R.},
volume = {78},
series = {Fluid Mechanics and Its Applications},
pages = {45--55},
address = {New York},
doi = {10.1007/1-4020-4159-4_5}
}
@article{MvH98,
Author = {M. van Hecke},
Journal = {Phys. Rev. Lett.},
Pages = {1896},
Title = {Building blocks of spatiotemporal intermittency},
Volume = {80},
Year = {1998}}
@article{MvHhole01,
Abtract = {Ordered hole-defect dynamics happens when an unstable hole invade a plane wave state and periodically nucleates defects from which new holes are born. The interaction between the holes and a self-disordered background are essential for the occurrence of spatiotemporal chaos in hole-defect states.},
Author = {M. van Hecke and M. Howard},
Journal = {Phys. Rev. Lett.},
Pages = {2018},
Title = {Ordered and self-disordered dynamics of holes and defects in the one-dimensional complex {Ginzburg-Landau} equation},
Volume = {86},
Year = {2001}}
@article{N90,
Author = {M. Nagata},
Journal = {J. Fluid Mech.},
Pages = {519--527},
Title = {Three-dimensional finite-amplitude solutions in plane {Couette} flow: {Bifurcation} from infinity},
Volume = {217},
Year = {1990},
DOI = {10.1017/S0022112090000829}
}
@Article{N96,
author = {M. Nagata},
title = {Nonlinear solutions of modified plane {Couette} flow in the presence of a transverse magnetic field},
journal = {J. Fluid Mech.},
year = {1996},
volume = {307},
pages = {231--243}
}
@article{N97,
Author = {Nagata, M.},
Journal = {Phys. Rev. E},
Pages = {2023--2025},
Title = {Three-dimensional traveling-wave solutions in plane {Couette} flow},
Volume = {55},
Year = {1997}}
@article{NaBu83,
Author = {Nagata, M. and Busse, F. H.},
Journal = {J. Fluid Mech.},
Pages = {1--26},
Title = {Three-dimensional tertiary motions in a plane shear layers},
Volume = {135},
Year = {1983}}
@article{NagDeg13,
Author = {Nagata, M. and Deguchi, K.},
Doi = {10.1017/jfm.2013.515},
Journal = {J. Fluid Mech.},
Title = {Mirror-symmetric exact coherent states in plane {Poiseuille} flow},
Volume = {735},
Year = {2013},
Bdsk-Url-1 = {http://dx.doi.org/10.1017/jfm.2013.515}}
@ARTICLE{Nambu73,
author = {Nambu, Y.},
title = {Generalized {Hamiltonian} dynamics},
journal = {Phys. Rev. D},
year = {1973},
volume = {7},
pages = {2405--2414},
doi = {10.1103/PhysRevD.7.2405}
}
@article{name,
Author = {M. Giona and A. Adrover},
Journal = {Phys. Rev. Lett.},
Pages = {3864},
Title = {Nonuniform stationary measure of the invariant unstable
foliation in {Hamiltonian} and fluid mixing systems},
volume = {81},
Year = {1998}}
@Incollection{Narasimha89,
author = {R. Narasimha},
title = {The utility and drawbacks of traditional approaches},
booktitle = {Whither Turbulence? Turbulence at the Cross-Road},
publisher = {Springer},
year = {1989},
editor = {J. Lumley},
pages = {13--48},
address = {Berlin}
}
@article{Navier1823,
Author = {Navier, C. L. M. H.},
Journal = {M{\'e}moires de l'Acad{\'e}mie Royale des Sciences de l'Institut de France},
Pages = {389--440},
Title = {M{\'e}moire sur les lois du mouvement des fluides},
Volume = {6},
Year = {1823}}
@ARTICLE{Neimark59,
title={On some cases of periodic motions depending on parameters},
author={Neimark, J.},
journal={Dokl. Akad. Nauk SSSR},
volume={129},
pages={736--739},
year={1959}
}
@article{newcgl1,
Author = {P. K. Newton and L. Sirovich},
Journal = {Quart. Appl. Math.},
Pages = {49--58},
Title = {Instabilities of {Ginzburg-Landau} equation: periodic solutions},
Volume = {XLIV},
Year = {1986}}
@Article{newcgl2,
author = {P. K. Newton and L. Sirovich},
title = {Instabilities of {Ginzburg-{L}andau} equation: secondary bifurcation},
journal = {Quart. Appl. Math.},
year = {1986},
volume = {XLIV},
pages = {367--374}
}
@article{newenv74,
Author = {A. C. Newell},
Journal = {Lect. Appl. Math.},
Pages = {157--163},
Title = {Envelope equations},
Volume = {15},
Year = {1974}}
@article{newton1687pnpm,
Author = {Newton, I.},
Title = {Philosophiae Naturalis Principia Mathematica},
Year = {1687}}
@Article{nhouse78,
author = {S. E. Newhouse and D. Ruelle and F. Takens},
title = {Occurrence of strange {A}xiom {A} attractors near quasi-periodic flows on ${T}^m(m=3$ or more)},
journal = {Commun. Math. Phys.},
year = {1978},
volume = {64},
pages = {35}
}
@article{NiHa91,
Author = {Ning, C. Z. and Haken, H.},
Doi = {10.1103/PhysRevA.43.6410},
Journal = {Phys. Rev. A},
Pages = {6410--6413},
Title = {Phase anholonomy in dissipative optical-systems with periodic oscillations},
Volume = {43},
Year = {1991}
}
@article{NiHa92,
Abstract = {We show that the geometrical phase defined for the dissipative systems
[1] is invariant under a unitary transformation. This implies for
lasers an invariance of the geometrical phase for different choices
of the reference frequency.},
Author = {Ning, C. Z. and Haken, H.},
Doi = {10.1007/BF01318153},
Journal = {Z. Phys. B},
Pages = {261--262},
Title = {An invariance property of the geometrical phase and its consequence in detuned lasers},
Volume = {{89}},
Year = {{1992}}
}
@Article{NiHa92a,
author = {C. Z. Ning and H. Haken},
title = {Geometrical phase and amplitude accumulations in dissipative systems with cyclic attractors},
journal = {Phys. Rev. Lett.},
year = {1992},
volume = {68},
pages = {2109--2112},
abstract = {We show that the geometrical {(Berry)} phases discovered in Hamiltonian
systems can also be defined as resulting from parallel transportation
of vectors for nonlinear dissipative systems with cyclic attractors.
If the nonlinear dissipative systems possess a certain kind of asymptotic
solution defined in this Letter, the phase and amplitude accumulation
of a geometrical type can be defined. Detuned one- and two-photon
lasers showing periodic intensity pulsations are taken as examples
of such systems.}
}
@article{NingHakenCLE90,
Author = {Ning, C. Z. and Haken, H.},
Doi = {10.1103/PhysRevA.41.3826},
Journal = {Phys. Rev.},
Pages = {3826},
Title = {Detuned lasers and the complex {Lorenz} equations:
{Subcritical} and supercritical {Hopf} bifurcations},
Volume = {41},
Year = {1990}
}
@article{NISHI08,
Author = {Nishi, M. and B\"{u}lent, \"{U}. and Durst, F. and Biswas, G.},
Journal = {J. Fluid Mech.},
Pages = {425--446},
Title = {Laminar-to-turbulent transition of pipe flows through puffs and slugs},
Volume = {614},
Year = {2008}}
@article{Nizette2001,
Author = {M. Nizette and T. Erneux and A. Gavrielides and V. Kovanis},
Journal = {Phys. Rev. E},
Pages = {026212},
Title = {Stability and bifurcations of periodically modulated, optically injected laser diodes},
Volume = {63},
Year = {2001}}
@inproceedings{NK04,
Author = {M. Nagata and G. Kawahara},
Booktitle = {Advances in Turbulence X},
Editor = {H. I. Andersson and P.-A. Krogstad},
Organization = {CIMNE},
Series = {Proceedings of the Tenth European Turbulence Conference},
Title = {Three-dimensional periodic solutions in rotating/non-rotating plane {Couette} flow},
Year = {2004}}
@article{NLSmal91,
Abtract = {A Hamiltonian equation is proposed for the generalized nonlinear {Schr\"odinger} equation. Soliton solutions are found and the stability of the plane waves is investigated.},
Author = {B. A. Malomed and L. Stenflo},
Journal = {J. Phys. A},
Pages = {L1149--L1153},
Title = {Modulational instabilities and soliton solutions of a generalized nonlinear {Schr\"odinger} equation},
Volume = {24},
Year = {1991}}
@article{NLSper77,
Abstract = {A nonlinear {Schr\"odinger} equation is found with growth and damping terms in the weakly nonlinear regime. An exact solitary solution is found for the equation.},
Author = {N. R. Pereira and L. Stenflo},
Journal = {Phys. Fluid},
Pages = {1733--1734},
Title = {Nonlinear {{Schr\"odinger}} equation including growth and damping},
Volume = {20},
Year = {1977}}
@article{NLSstn03,
Abstract = {Under the external driving force, an oscillon solutions is found.},
Author = {L. Stenflo and M. Y. Yu},
Journal = {Phys. Plasmas},
Pages = {912--913},
Title = {Oscillons at a plasma surface},
Volume = {10},
Year = {2003}}
@Article{noack03,
author = {B. R. Noack and K. Afanasiev and M. Morzynski and F. Thiele},
title = {A hierarchy of low-dimensional models for the transient and post-transient cylinder wake},
journal = {J. Fluid Mech.},
year = {2003},
volume = {497},
pages = {335--363}
}
@Article{Noether15,
author = {Noether, E.},
title = {{Der Endlichkeitssatz der Invarianten endlicher Gruppen}},
journal = {Math. Ann.},
year = {1915},
volume = {77},
pages = {89--92},
doi = {10.1007/BF01456821}
}
@article{noisy_Fred,
Author = {P. Cvitanovi\'{c} and C. P. Dettmann and R. Mainieri and G. Vattay},
Journal = {J. Stat. Phys.},
Note = {\arXiv{chao-dyn/9807034}},
Pages = {981},
Title = {Trace formulas for stochastic evolution operators: {Weak} noise perturbation theory},
Volume = {93},
Year = {1998}}
@Article{nongal89,
author = {M. Martine and R. Temam},
title = {Nonlinear {G\"a}lerkin Methods},
journal = {SIAM J. Numer. Anal.},
year = {1989},
volume = {26},
pages = {1139--1157}
}
@Article{NP,
author = {K. Lust and D. Roose},
title = {An adaptive {Newton-Picard} algorithm with subspace iteration for computing periodic solutions},
journal = {SIAM J. Sci. Comput.},
year = {1998},
volume = {19},
pages = {1188--1209},
doi = {10.1137/S1064827594277673}
}
@Book{nrc,
title = {Physics in a new era: an overview},
publisher = {National Academy Press},
year = {2001},
author = {{National Research Council report}},
address = {Washington, D.C.},
note = {{\tt www.nap.edu/books/0309073421/html}}
}
@book{ns,
Address = {New York},
Author = {J. W. Thomas},
Publisher = {Spring-Verlag},
Title = {Numerical Partial Differential Equations},
Year = {1995}}
@article{NSTks85,
Abstract = {The KSe shows coherent spatial patterns but temporal chaos. The anti-diffusion term destablizes the solutin while stability is achived by the collective action of all modes through the nonlinear convection term. An estimate of the L^2 norm of the solution in the antisymmetric periodic function space is given. The number of determining Fourier modes is calculated together with fractal dimension of the strange attractor.},
Author = {B. Nicolaenko and B. Scheurer and R. Temam},
Journal = {Physica D},
Pages = {155--183},
Title = {Some global dynamical properties of the {Kuramoto-Sivashinsky} equations: nonlinear stability and attractors},
Volume = {16},
Year = {1985}}
@techreport{octave,
Author = {J. Eaton},
Institution = {University of Wisconsin},
Publisher = {J. Eaton},
Title = {{GNU Octave}: a high-level language for numerical computation},
Url = {http://www.octave.org},
Year = {2007}
}
@article{OK96,
Author = {S. A. Orszag and L. C. Kells},
Journal = {J. Fluid Mech.},
Pages = {159--205},
Title = {Transition to turbulence in plane {Poiseuille} and plane {Couette} flow},
Volume = {96},
Year = {1980}}
@unpublished{OleMog07,
Abstract = {derive exact analytical solutions of the GOY shell model
of turbulence. In the absence of forcing and viscosity obtain
closed form solutions in terms of Jacobi elliptic functions. With
three shells the model is integrable. Obtain a Kolmogorov solution
in the limit of infinitely many shells. Also identify cases where
the models exhibits transitions to chaotic states lying on strange
attractors or ergodic energy surfaces.},
Author = {Olesen, P. and Jensen, M.s H.},
Note = {\arXiv{0705.3123}},
Title = {Exact periodic solutions of shells models of turbulence},
Year = {2007}}
@book{OlverInv,
Address = {Cambridge},
Author = {P. J. Olver},
Publisher = {Cambridge Univ. Press},
Title = {Classical Invariant Theory},
Year = {1999}}
@book{OlvSha06,
Address = {Upper Saddle River, NJ},
Author = {Olver, P. J. and Shakiban, C.},
Publisher = {Prentice-Hall},
Title = {Applied Linear Algebra},
Year = {2006}}
@Article{Onsager53,
author = {L. Onsager and S. Machlup},
title = {Fluctuations and irreversible processes},
journal = {Phys. Rev.},
year = {1953},
volume = {91},
pages = {1505--1512}
}
@article{OonoTaka80,
Author = {Oono, Y. and Takahashi, Y.},
Journal = {Prog. Theor. Phys},
Pages = {1804--1807},
Title = {Chaos, external noise and {Fredholm} theory},
Volume = {63},
Year = {1980}}
@techreport{openpipeflow,
Author = {Willis, A. P.},
Institution = {U. Sheffield},
Note = {{\tt {Openpipeflow.org}}},
Publisher = {A. P. Willis},
Title = {{Channelflow}: {Pipe} flow code for incompressible flow},
Year = {2014}}
@article{optp1,
Author = {B. R. Hunt and E. Ott},
Journal = {Phys. Rev. Lett.},
Pages = {2254},
Title = {Optimal periodic orbits of chaotic systems},
Volume = {76},
Year = {1996}}
@article{optp2,
Author = {T.-H. Yang and B. R. Hunt and E. Ott},
Journal = {Phys. Rev. E},
Pages = {1950},
Title = {Optimal periodic orbits of continuous time chaotic systems},
Volume = {62},
Year = {2000}}
@article{Orr1907,
Author = {Orr, W. M. F.},
Journal = {Proc. Roy. Irish Acad. A},
Pages = {69--138},
Title = {The stability or instability of the steady motions of a liquid. {Part II: A} viscous liquid},
Volume = {27},
Year = {1907}}
@article{Orszag71,
Author = {S. A. Orszag},
Journal = {J. Fluid Mech.},
Pages = {689--703},
Title = {Accurate solution of the {Orr-Sommerfeld} stability equation},
Volume = {50},
Year = {1971},
Abstract = {The Orr-Sommerfeld equation is solved numerically using
expansions in Chebyshev polynomials and the QR matrix eigenvalue
algorithm. It is shown that results of great accuracy are obtained
very economically. The method is applied to the stability of plane
Poiseuille flow. Expansions in Chebyshev polynomials are better
suited to the solution of hydrodynamic stability problems than
expansions in other, seemingly more relevant, sets of orthogonal
functions.},
}
@article{Orszag86,
Author = {S. A. Orszag and M. Israeli and M. Deville},
Journal = {J. Sci. Comput.},
Pages = {75--111},
Title = {Boundary conditions for incompressible flows},
Volume = {1},
Year = {1986}}
@article{OsSchn62,
Author = {A. Ostrowski and H. Schneider},
Journal = {J. Math. Anal. Appl.},
Pages = {72--84},
Title = {Some theorems on the inertia of general matrices},
Volume = {4},
Year = {1962}}
@article{ott90a,
Author = {E. Ott and C. Grebogi and J. A. Yorke},
Journal = {Phys. Rev. Lett},
Pages = {1196},
Title = {Controlling chaos},
Volume = {64},
Year = {1990}}
@article{Pal61,
Author = {R. S. Palais},
Journal = {Ann. Math.},
Pages = {295--323},
Title = {On the existence of slices for actions of non-compact {Lie} groups},
Volume = {73},
Year = {1961}}
@article{Palacios07,
Author = {A. Palacios},
Journal = {Scholarpedia},
Pages = {2352},
Title = {Heteroclinic cycles},
Volume = {2},
Year = {2007}}
@book{Pan96,
Address = {New York},
Author = {R. L. Panton},
Publisher = {Wiley},
Title = {Incompressible flow},
Year = {1996}}
@book{Pan97,
Address = {Southhampton},
Editor = {R. L. Panton},
Publisher = {Computational Mechanics Publications},
Title = {Self-sustaining mechanisms of wall turbulence},
Year = {1997}}
@article{ParLau86,
title={Resonances and torsion numbers of driven dissipative nonlinear oscillators},
author={U. Parlitz and W. Lauterborn},
journal={Z. Naturforsch. A},
volume={41},
number={4},
pages={605--614},
year={1986}
}
@article{ParLau87,
title={Period-doubling cascades and devil's staircases of the driven
{van der Pol} oscillator},
author={U. Parlitz and W. Lauterborn},
journal={Phys. Rev. A},
volume={36},
pages={1428},
year={1987},
}
@article{Pasq87,
Author = {F. Pasquarelli and A. Quarteroni and G. Sacchi-Landriani},
Journal = {J. Comp. Sci.},
Pages = {195--226},
Title = {Spectral approximations of the {Stokes} problem by divergence-free functions},
Volume = {2},
Year = {1987}}
@Article{PC88,
author = {P. Cvitanovi{\'c}},
title = {Invariant measurement of strange sets in terms of cycles},
journal = {Phys. Rev. Lett.},
year = {1988},
volume = {61},
pages = {2729}
}
@article{PCar,
Author = {Cvitanovi\'c, P.},
Doi = {10.1103/PhysRevD.14.1536},
Journal = {Phys. Rev. D},
Pages = {1536--1553},
Title = {Group theory for {Feynman} diagrams in non-{Abelian} gauge theories},
Volume = {14},
Year = 1976,
Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevD.14.1536}}
@book{PCgr,
Address = {Princeton},
Author = {P. Cvitanovi\'c},
Note = {{\tt birdtracks.eu}},
Publisher = {Princeton Univ. Press},
Title = {{Group Theory} - {Birdtracks}, {Lie}'s, and {Exceptional Groups}},
Year = {2008}}
@article{pchaot,
Abstract = {Use near recurrence to extract periodic orbits
and their stability eigenvalues.},
Author = {D. Auerbach and P. Cvitanovi{\'c} and J.-P. Eckmann and G. Gunaratne and I. Procaccia},
Journal = {Phys. Rev. Lett.},
Pages = {23},
Title = {Exploring chaotic motion through periodic orbits},
Volume = {58},
Year = {1987}}
@Article{PeaHal13,
author = {T. Peacock and G. Haller},
title = {Lagrangian coherent structures: {The} hidden skeleton of fluid flows},
journal = {Physics Today},
year = {2013},
volume = {66},
number = {2},
pages = {41--47},
doi = {10.1063/PT.3.1886},
bdsk-url-1 = {http://dx.doi.org/10.1063/PT.3.1886}
}
@article{peckham90,
title={The necessity of the {Hopf} bifurcation for periodically
forced oscillators},
author={Peckham, B. B.},
journal={Nonlinearity},
volume={3},
pages={261},
year={1990}
}
@article{PenDai07,
Author = {J. Peng and J. O. Dabiri},
Journal = {Experiments in Fluids, to appear},
Title = {A potential-flow, deformable-body model for fluid-structure interactions with compact vorticity: application to animal swimming measurements},
Url = {http://dabiri.caltech.edu/people/dabiri.html},
Year = {2007}
}
@book{Penr04,
Address = {New York},
Author = {R. Penrose},
Publisher = {A. A. Knopf},
Title = {The Road to Reality: A Complete Guide to the Laws of the Universe},
Year = {2005}}
@article{percgl,
Author = {A. V. Porubov and M. G. Velarde},
Journal = {J. Math. Phys.},
Pages = {884},
Title = {Exact periodic solutions of the complex {Ginzburg-Landau} equation},
Volume = {40},
Year = {1999}}
@book{perg,
Address = {New York},
Author = {P. Walters},
Publisher = {Speringer-Verlag},
Title = {An introduction to ergodic theory},
Year = {1981}}
@book{perkob,
Address = {New York},
Author = {L. Perko},
Publisher = {Springer},
Title = {Differential Equations and Dynamical Systems},
Year = {1991}}
@Article{PetCorBol06,
author = {Pethel, S. D. and Corron, N. J. and Bollt, E.},
title = {Symbolic dynamics of coupled map lattices},
journal = {Phys. Rev. Lett.},
year = {2006},
volume = {96},
pages = {034105},
doi = {10.1103/PhysRevLett.96.034105}
}
@Article{PetCorBol07,
author = {Pethel, S. D. and Corron, N. J. and Bollt, E.},
title = {Deconstructing spatiotemporal chaos using local symbolic dynamics},
journal = {Phys. Rev. Lett.},
year = {2007},
volume = {99},
pages = {214101},
doi = {10.1103/PhysRevLett.99.214101}
}
@article{Peterhof1999,
Author = {D. Peterhof and B. Sandstede},
Journal = {J. Nonlinear Sci.},
Pages = {575--613},
Title = {All-optical clock recovery using multisection distributed-feedback lasers},
Volume = {9},
Year = {1999}}
@article{Petr40,
Author = {Petrov, G.I.},
Journal = {Prikl. Mat. Mekh.},
Pages = {3--12},
Title = {Application of {Galerkin}'s method to the problem of the stability of the flow of a viscous fluid},
Volume = {4},
Year = {1940}}
@inproceedings{petsc-efficient,
Author = {S. Balay and W. D. Gropp and L. C. McInnes and B. F. Smith},
Booktitle = {Modern Software Tools in Scientific Computing},
Editor = {E. Arge and A. M. Bruaset and H. P. Langtangen},
Pages = {163--202},
Publisher = {Birkh{\"{a}}user Press},
Title = {Efficient management of parallelism in object oriented numerical software libraries},
Year = {1997}}
@techreport{petsc-user-ref,
Author = {S. Balay and K. Buschelman and V. Eijkhout and W. D. Gropp and D. Kaushik and M. G. Knepley and L. C. McInnes and B. F. Smith and H. Zhang},
Institution = {Argonne National Laboratory},
Number = {ANL-95/11 - Revision 2.1.5},
Title = {{PETS}c Users Manual},
Year = {2004}}
@misc{petsc-web-page,
Author = {S. Balay and K. Buschelman and W. D. Gropp and D. Kaushik and M. G. Knepley and L. C. McInnes and B. F. Smith and H. Zhang},
Note = {www.mcs.anl.gov/petsc},
Title = {{PETSc} web page},
Year = {2001}}
@article{PexMull06,
Author = {J. Peixinho and T. Mullin},
Journal = {Phys. Rev. Lett.},
Pages = {094501},
Title = {Decay of turbulence in pipe flow},
Volume = {96},
Year = {2006}}
@Article{PexMull07,
author = {J. Peixinho and T. Mullin},
title = {Finite-amplitude thresholds for transition in pipe flow},
journal = {J. Fluid Mech.},
year = {2007},
volume = {582},
pages = {169--178}
}
@book{Peyret02,
Address = {New York},
Author = {R. Peyret},
Publisher = {Springer},
Title = {Spectral Methods for Incompressible Flows},
Year = {2002}}
@Incollection{Pfenninger61,
author = {W. Pfenninger},
title = {Transition in the inlet length of tubes at high Reynolds numbers},
booktitle = {Boundary Layer and Flow Control},
publisher = {Pergamon},
year = {1961},
editor = {G. V. Lachmann},
pages = {970--980},
address = {Oxford, UK},
doi = {10.1016/B978-1-4832-1323-1.50013-0}
}
@book{PG97,
Address = {Cambridge},
Author = {P. Gaspard},
Publisher = {Cambridge Univ. Press},
Title = {Chaos, Scattering and Statistical Mechanics},
Year = {1997}}
@article{PhysToday04,
Author = {R. Fitzgerald},
Journal = {Physics Today},
Number = {2},
Pages = {21--23},
Title = {New experiments set the scale for the onset of turbulence in pipe flow},
Volume = {57},
Year = {2004}}
@article{PhysWorld04,
Author = {C. Barenghi},
Journal = {Physics World},
Number = {12},
Title = {Turbulent transition for fluids},
Volume = {17},
Year = {2004}}
@article{pimsimp,
Abstract = {They generalized 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.},
Author = {P. Moresco and S. P. Dawson},
Journal = {Physica D},
Pages = {38},
Title = {The {PIM}-simplex method: an extension of the {PIM}-triple
method to saddles with an arbitrary number of expanding directions},
Volume = {126},
Year = {1999}}
@article{pimstag,
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.},
Author = {D. Sweet and H. E. Nusse and J. A. Yorke},
Journal = {Phys. Rev. Lett.},
Pages = {2261},
Title = {Stagger-and-Step Method: Detecting and Computing Chaotic Saddles in Higher Dimensions},
Volume = {86},
Year = {2001}}
@article{pimyk,
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.},
Author = {H. E. Nusse and J. A. Yorke},
Journal = {Physica D},
Pages = {137},
Title = {A procedure for finding numerical trajectories on chaotic saddles},
Volume = {36},
Year = {1989}}
@article{pizza2012chaotic,
Author = {Pizza, G. and Frouzakis, C. E. and Mantzaras, J.},
Journal = {Combustion Theory and Modelling},
Pages = {275--299},
Title = {Chaotic dynamics in premixed hydrogen/air channel flow combustion},
Volume = {16},
Year = {2012}}
@article{PoBaGr88,
Author = {A. Politi and R. Badii and P. Grassberger},
Journal = {J. Phys. A},
Pages = {L763},
Title = {On the geometric structure of non-hyperbolic attractors},
Volume = {15},
Year = {1988}}
@article{Poinc1896,
Author = {H. Poincar\'e},
Journal = {C. R. Acad. Sci. Paris},
Pages = {915--918},
Title = {Sur les solutions p\'eriodiques et le principe de moindre action},
Volume = {123},
Year = {1896}}
@book{poincare,
Address = {Paris},
Author = {H. Poincar{\'e}},
Note = {For a very readable exposition of Poincar{\'e}'s work and the development of the dynamical systems theory up to 1920's see \refref{JBG97}.},
Publisher = {Guthier-Villars},
Title = {Les m{\'e}thodes nouvelles de la m{\'e}chanique c{\'e}leste},
Year = {1899}}
@book{poinper,
Address = {New York},
Author = {H. Poincar{\'e}},
Publisher = {Springer},
Series = {History of Modern Physics and Astronomy},
Title = {New Methods in Celestial Mechanics},
Volume = {13},
Year = {1992}}
@ARTICLE{Poiseuille1844,
author={J. L. Poiseuille},
title={Recherches exp{\'e}rimentales sur le mouvement des liquides dans
les tubes de tr{\`e}s-petits diam{\`e}tres},
year={1840},
journal={C. R. Acad. Sci.},
volume={11},
pages={961}
}
@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},
bdsk-url-1 = {http://dx.doi.org/10.1016/j.physd.2005.01.001}
}
@article{pomeau80,
Author = {Y. Pomeau and P. Manneville},
Journal = {Commun. Math. Phys.},
Pages = {189},
Title = {Intermittent transition to turbulence in dissipative dynamical systems},
Volume = {74},
Year = {1980}}
@book{popebook,
Address = {Cambridge},
Author = {S. B. Pope},
Publisher = {Cambridge Univ. Press},
Title = {Turbulent Flows},
Year = {2000}}
@article{postgal89,
Author = {B. Garc{\'\i}a-Archilla and J. Novo and E. S. Titi},
Journal = {SIAM J. Numer. Anal.},
Pages = {941--972},
Title = {Postprocessing the {G \"alerkin} method: a novel approach to approximate inertial manifolds},
Volume = {35},
Year = {1998}}
@misc{Postl08,
Author = {C. M. Postlethwaite},
Note = {\arXiv{0804.2903}},
Title = {Stabilisation of long-period periodic orbits using time-delayed feedback control},
Year = {2008}}
@Article{PP83,
author = {W. Parry and M. Pollicott},
title = {An analogue of the prime number theorem for closed orbits of Axiom {A} flows},
journal = {Ann. Math.},
year = {1983},
volume = {118},
pages = {573--591}
}
@article{PP90,
Author = {W. Parry and M. Pollicott},
Journal = {Ast\'erisque},
Pages = {268},
Title = {Zeta functions and the periodic orbit structure of hyperbolic dynamics},
Volume = {187-188},
Year = {1990}}
@Article{Prants14,
author = {Prants, S. V.},
title = {Chaotic {Lagrangian} transport and mixing in the ocean},
journal = {Eur. Phys. J. Spec. Top.},
year = {2014},
volume = {223},
pages = {2723--2743},
doi = {10.1140/epjst/e2014-02288-5}
}
@article{pre88top,
Author = {P. Cvitanovi\'{c} and G. H. Gunaratne and I. Procaccia},
Journal = {Phys. Rev. A},
Pages = {1503},
Title = {Topological and metric properties of {H\'enon}-type strange attractors},
Volume = {38},
Year = {1988}}
@book{Press:96,
Address = {Cambridge},
Author = {W. H. Press and B. P. Flannery and S. A. Teukolsky and W. T. Vetterling},
Publisher = {Cambridge Univ. Press},
Title = {Numerical Recipes in Fortran},
Year = {1996}}
@article{Pringle07,
Author = {Pringle, C. C. T. and Kerswell, R. R.},
Journal = {Phys. Rev. Lett.},
Pages = {074502},
Title = {Asymmetric, helical, and mirror-symmetric traveling waves in pipe flow},
Volume = {99},
Year = {2007}}
@Article{Pringle09,
author = {Pringle, C. C. T. and Duguet, Y. and Kerswell, R. R.},
title = {Highly symmetric travelling waves in pipe flow},
journal = {Phil. Trans. Royal Soc. A},
year = {2009},
volume = {367},
pages = {457--472},
note = {\arXiv{0804.4854}}
}
@article{PSD01,
Abstract = {Change the stability of the fixed points on the
Poincar{\'e} section to find periodic orbits.},
Author = {D. Pingel and P. Schmelcher and F. K. Diakonos},
Journal = {Phys. Rev. E},
Pages = {026214},
Title = {Detecting unstable periodic orbits in chaotic continuous-time dynamical systems},
Volume = {64},
Year = {2001}}
@article{PSDB00,
Author = {D. Pingel and P. Schmelcher and F. K. Diakonos and O. Biham},
Journal = {Phys. Rev. E},
Note = {\arXiv{nlin.CD/0006011}},
Pages = {2119},
Title = {Theory and applications of the systematic detection of unstable periodic orbits in dynamical systems},
Volume = {62},
Year = {2000}}
@Article{pugh67cl,
author = {C. Pugh},
title = {An improved closing lemma and a general density theorem},
journal = {Amer. J. Math.},
year = {1967},
volume = {89}
}
@article{PWK15,
author = {Pringle, C.C.T. and Willis, A.P. and Kerswell, R.R.},
title = {Fully localised nonlinear energy growth optimals in pipe flow},
journal = {Phys. Fluids},
year = {2015},
volume = {27},
pages = {064102},
}
@article{PXBFB14,
Author = {Pumir, A. and Xu, H. and Boffetta, G. and Falkovich, G. and Bodenschatz, E.},
Doi = {10.1103/PhysRevX.4.041006},
Journal = {Phys. Rev. X},
Pages = {041006},
Title = {Redistribution of kinetic energy in turbulent flows},
Volume = {4},
Year = {2014}
}
@article{R1883,
Author = {O. Reynolds},
Journal = {Proc. Roy. Soc. Lond. Ser A},
Pages = {935--982},
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},
Volume = {174},
Year = {1883}}
@Article{R1894,
author = {O. 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{RaCoRi14,
author = {S. Rawat and C. Cossu and F. Rincon},
title = {Relative periodic orbits in plane {Poiseuille} flow},
journal = {Compt. Rend. M\'ecanique},
year = {2014},
volume = {342},
pages = {485--489},
doi = {10.1016/j.crme.2014.05.008},
abstract = {A branch of relative periodic orbits is found in plane
Poiseuille flow in a periodic domain at Reynolds numbers ranging from Re
= 3000 to Re = 5000 . These solutions consist in sinuous quasi-streamwise
streaks periodically forced by quasi-streamwise vortices in a
self-sustained process. The streaks and the vortices are located in the
bulk of the flow. Only the amplitude, but not the shape, of the averaged
velocity components does change as the Reynolds number is increased from
3000 to 5000. We conjecture that these solutions could therefore be
related to large- and very large-scale structures observed in the bulk of
fully developed turbulent channel flows. },
bdsk-url-1 = {http://dx.doi.org/10.1016/j.crme.2014.05.008}
}
@article{Radziunas2006,
Author = {M. Radziunas},
Journal = {Physica D},
Pages = {98--112},
Title = {Numerical bifurcation analysis of the traveling wave model of multisection semiconductor lasers},
Volume = {213},
Year = {2006}}
@article{RaHaAb96,
Author = {Rauh, A. and Hannibal, L. and Abraham, N. B.},
Journal = {Physica D},
Pages = {45--58},
Title = {Global stability properties of the complex {Lorenz} model},
Volume = {99},
Year = {1996}}
@article{Rand82,
Author = {Rand, D.},
Journal = {Arch. Rational Mech. Anal.},
Pages = {1--3},
Title = {Dynamics and symmetry - predictions for modulated waves in rotating fluids},
Volume = {79},
Year = {1982}}
@article{RaOsSeSi82,
Author = {D. Rand and S. Ostlund and J. Sethna and E. Siggia},
Doi = {10.1103/PhysRevLett.49.132},
Journal = {Phys. Rev. Lett.},
Pages = {132--135},
Title = {Universal transition from quasiperiodicity to chaos in dissipative systems},
Volume = {49},
Year = {1982},
Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevLett.49.132}}
@article{Recke1998,
Author = {L. Recke and D. Peterhof},
Journal = {J. Diff. Eqn.},
Pages = {233--262},
Title = {Abstract forced symmetry breaking and forced frequency locking of modulated waves},
Volume = {144},
Year = {1998}}
@article{Recke1998a,
Author = {L. Recke},
Journal = {Ukrain. Math. J.},
Pages = {94--101},
Title = {Forced frequency locking of rotating waves},
Volume = {50},
Year = {1998}}
@article{Recke2010,
Author = {L. Recke and A. Samoilenko and A. Teplinsky and V. Tkachenko and S. Yanchuk},
Journal = {Discrete Contin. Dynam. Systems},
Note = {\arXiv{1108.5990}},
Pages = {847--875},
Title = {Frequency locking of modulated waves},
Volume = {31},
Year = {2011}}
@Article{reimann1,
author = {P. Reimann},
title = {Noisy one-dimensional maps near a crisis 1: {Weak {Gauss}ian} white and colored noise},
journal = {J. Stat. Phys.},
year = {1996},
volume = {82},
pages = {1467--1501}
}
@article{reimann2,
Author = {P. Reimann},
Journal = {J. Stat. Phys.},
Pages = {403--425},
Title = {Noisy one-dimensional maps near a crisis 2: {General} uncorrelated weak noise},
Volume = {85},
Year = {1996}}
@article{Rem03,
Author = {D. Rempfer},
Journal = {Review of Fluid Mechanics},
Pages = {229},
Title = {Low-dimensional modeling and numerical simulation
of transition in simple shear flows},
Volume = {35},
Year = {2003}}
@Article{Rem06,
Author = {Rempfer, D.},
Title = {On boundary conditions for incompressible
{Navier-Stokes} problems},
Journal = {Appl. Mech. Rev.},
Volume = {59},
Pages = {107--125},
doi = {10.1115/1.2177683},
year = {2006}
}
@misc{ReSaTkYa11,
Author = {Recke, L. and {Samoilenko}, A. and {Tkachenko}, V. and {Yanchuk}, S.},
Note = {\arXiv{1108.5990}},
Title = {Frequency locking by external forcing in systems with rotational symmetry},
Year = {2011}}
@Book{Richter10,
title = {Semiclassical Theory of Mesoscopic Quantum Systems},
publisher = {Springer},
year = {2000},
author = {Richter, K.},
isbn = {9783662156506},
address = {Berlin},
__markedentry = {[pc77:]},
doi = {10.1007/BFb0109634},
url = {http://www.physik.uni-regensburg.de/forschung/richter/richter/pages/research/springer-tracts-161.pdf}
}
@Article{Rincon07,
author = {Rincon, F.},
title = {On the existence of two-dimensional nonlinear steady states in plane {Couette} flow},
journal = {Phys. Fluids},
year = {2007},
volume = {19},
pages = {4105},
note = {\arXiv{0706.1165}},
abstract = {It is not possible to obtain the nonlinear plane
Couette flow solutions reported by Cherhabili and
Ehrenstein [Eur. J. Mech. B/Fluids 14, 667 (1995)] using
their Poiseuille-Couette homotopy. We also demonstrate that
the steady solutions obtained by Mehta and Healey [Phys.
Fluids 17, 094108 (2005)] for small symmetry-preserving
perturbations are influenced by an artefact of the modified
system of equations used in their paper.}
}
@book{Risken96,
Address = {New York},
Author = {H. Risken and H. Haken},
Publisher = {Springer},
Title = {The {F}okker-{P}lanck Equation},
Year = {1996}}
@misc{RMChaosHist,
Author = {R. Mainieri},
Note = {Appendix {\em ``A brief history of chaos''}, in \refref{DasBuch}}}
@article{Robinson91,
Author = {S. K. Robinson},
Journal = {Ann. Rev. Fluid Mech.},
Pages = {601--639},
Title = {{Coherent motions in the turbulent boundary layer}},
Volume = {23},
Year = {1991}}
@techreport{Robinson91b,
Author = {S. K. Robinson},
Institution = {NASA},
Note = {103859},
Title = {The Kinematics of turbulent boundary layer structure},
Year = {1991}}
@Article{ROC07,
author = {F. Rincon and G. I. Ogilvie and C. Cossu},
title = {On self-sustaining processes in {Rayleigh}-stable rotating plane Couette flows and subcritical transition to turbulence in accretion disks},
journal = {Astron. Astrophys.},
year = {2007},
volume = {463},
pages = {817--832}
}
@article{Rolland14,
Author = {Rolland, J.},
Journal = {Fluid Dyn. Res.},
Note = {\arXiv{1401.3586}},
Pages = {015512},
Title = {Turbulent spot growth in plane {Couette} flow: statistical
study and formation of spanwise vorticity},
Volume = {46},
Year = {2014}}
@article{Romanov73,
Author = {V. A. Romanov},
Journal = {Functional Anal. Applics.},
Pages = {137--146},
Title = {Stability of plane-parallel {Couette} flow},
Volume = {7},
Year = {1973}}
@article{Rome69,
Author = {Rome, H.},
Journal = {IEEE Trans. Automatic Control},
Pages = {592--593},
Title = {A direct solution to the linear variance equation of a time-invariant linear system},
Volume = {14},
Year = {1969}}
@article{RoMeBaSchHe09,
Author = {Rowley, C. W. and Mezi\'c, I. and Bagheri, S. and Schlatter, P. and Henningson, D. S.},
Journal = {J. Fluid Mech.},
Pages = {115},
Title = {Spectral analysis of nonlinear flows},
Volume = {641},
Year = {2009}}
@article{RoSa00,
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.},
Author = {Roweis, S. T. and Saul, L. K.},
Doi = {10.1126/science.290.5500.2323},
Journal = {Science},
Pages = {2323--2326},
Title = {Nonlinear dimensionality reduction by locally linear embedding},
Volume = {290},
Year = {2000}
}
@article{ross,
Author = {O. E. R{\"o}ssler},
Journal = {Phys. Lett. A},
Pages = {397},
Title = {An equation for continuous chaos},
Volume = {57},
Year = {1976}}
@article{roth99,
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.},
Author = {D. Rothstein and E. Henry and J. P. Gollub},
Journal = {Nature},
Pages = {770--772},
Title = {Persistent patterns in transient chaotic fluid mixing},
Volume = {401},
Year = {1999}}
@ARTICLE{Roupas11,
author = {Z. Roupas},
title = {Phase space geometry and chaotic attractors in dissipative {Nambu}
mechanics},
journal = {J. Phys. A},
year = {2012},
volume = {45},
pages = {195101},
note = {\arXiv{1110.0766}}
}
@article{rowley_reconstruction_2000,
Author = {Rowley, C. W. and Marsden, J. E.},
Title = {Reconstruction equations and the {Karhunen-Lo\'eve} expansion
for systems with symmetry},
Abstract = {We present a method for applying the {Karhunen-Lo\'eve} decomposition
to systems with continuous symmetry. The techniques in this paper
contribute to the general procedure of removing variables associated
with the symmetry of a problem, and related ideas have been used
in previous works both to identify coherent structures in solutions
of {PDEs,} and to derive low-order models via Galerkin projection.
The main result of this paper is to derive a simple and easily implementable
set of reconstruction equations which close the system of {ODEs}
produced by Galerkin projection. The geometric interpretation of
the method closely parallels techniques used in geometric phases
and reconstruction techniques in geometric mechanics. We apply the
method to the {Kuramoto-Sivashinsky} equation and are able to derive
accurate models of considerably lower dimension than are possible
with the traditional {Karhunen-Lo\'eve} expansion.},
Journal = {Physica D},
Pages = {1--19},
Volume = {142},
Year = {2000},
doi = {10.1016/S0167-2789(00)00042-7}
}
@article{rowley_reduction_2003,
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},
Author = {Rowley, C. W. and Kevrekidis, I. G. and Marsden, J. E. and Lust, K.},
Journal = {Nonlinearity},
Pages = {1257--1275},
Title = {Reduction and reconstruction for self-similar dynamical systems},
Volume = {16},
Year = {2003}}
@article{rowley04,
Author = {C. W. Rowley},
Journal = {Int. J. Bif. Chaos},
Pages = {997},
Title = {Model reduction for fluids, using balanced proper orthogonal decomposition},
Volume = {15},
Year = {2005}}
@article{RP67,
Author = {W. C. Reynolds and M. C. Potter},
Journal = {J. Fluid Mech.},
Pages = {465},
Title = {Finite-amplitude instability of parallel shear flows},
Volume = {27},
Year = {1967}}
@Article{RPM,
author = {G. M. Shroff and H. B. Keller},
title = {Stabilization of unstable procedures: the recursive projection method},
journal = {SIAM J. Numer. Anal.},
year = {1993},
volume = {30},
pages = {1099--1120},
doi = {10.1137/0730057},
address = {Philadelphia},
bdsk-url-1 = {http://dx.doi.org/10.1137/0730057},
issn = {0036-1429}
}
@article{RSBH98,
Author = {S. C. Reddy and P. J. Schmid and J. S. Baggett and D. S. Henningson},
Journal = {J. Fluid Mech.},
Pages = {269--303},
Title = {On stability of streamwise streaks and transition thresholds in plane channel flows},
Volume = {365},
Year = {1998}}
@book{rtb,
Address = {New York},
Author = {V. Szebehely},
Publisher = {Academic Press},
Title = {Theory of Orbits},
Year = {1967}}
@article{rue04ne,
Abstract = {New ideas of nonequilibrium physics are introduced. Emphasize the SRB measure approach and energy fluctuation theorem.},
Author = {D. Ruelle},
Journal = {Physics Today},
Number = {5},
Pages = {48--53},
Title = {Conversations on nonequilibrium physics with an extraterrestrial},
Volume = {57},
Year = {2004}}
@article{ruelext,
Author = {D. Ruelle},
Journal = {Commun. Math. Phys.},
Pages = {287--302},
Title = {Large volume limit of the distribution of characteristic exponents in turbulence},
Volume = {87},
Year = {1982}}
@article{ruell71,
Author = {D. Ruelle and F. Takens},
Journal = {Commun. Math. Phys.},
Pages = {167},
Title = {On the nature of turbulence},
Volume = {20},
Year = {1971}}
@book{ruelle,
Address = {Reading, MA},
Author = {D. Ruelle},
Publisher = {Addison-Wesley},
Title = {Statistical Mechanics, Thermodynamic Formalism},
Year = {1978}}
@article{Ruelle76,
Author = {D. Ruelle},
Journal = {Inv. Math.},
Pages = {231--242},
Title = {Zeta-functions for expanding maps and {Anosov} flows},
Volume = {34},
Year = {1976}}
@article{ruelle86,
Abstract = {For a class of differentiable dynamical systems (called Axiom
A systems) it has been shown by Pollicott and the author that correlation
functions have Fourier transforms which are meromorphic in a strip. The
poles (or resonances) are, however, not easy to locate. This note reviews
the results which are known and discusses a simple model where the
position of resonances can be estimated. The effect of noise is also
discussed.},
Author = {D. Ruelle},
Journal = {J. Stat. Phys.},
Pages = {281--292},
Title = {Locating resonances for {Axiom A} dynamical systems},
Volume = {44},
Year = {1986}}
@article{ruelle86a,
Author = {D. Ruelle},
Journal = {Phys. Rev. Lett.},
Pages = {405--407},
Title = {Resonances of chaotic dynamical systems},
Volume = {56},
Year = {1986}}
@techreport{Ruelle95,
Address = {Bures-sur-Yvette},
Author = {Ruelle, D},
Institution = {Inst. Hautes Etud. Sci.},
Number = {IHES-P-95-30},
Title = {Functional determinants related to dynamical systems and the thermodynamic formalism},
Year = {1995}}
@article{RVVCMG97,
Adsurl = {http://adsabs.harvard.edu/abs/1997QuSOp...9R...1R},
Author = {{Rold{\'a}n}, E. and {de Valc{\'a}rcel}, G.~J. and {Vilaseca}, R. and {Corbal{\'a}n}, R. and {Mart{\'{\i}}nez}, V.~J. and {Gilmore}, R.},
Journal = {Quantum and Semiclassical Optics},
Pages = {1-+},
Title = {{REVIEW ARTICLE: The dynamics of optically pumped molecular lasers. On its relation with the Lorenz - Haken model}},
Volume = 9,
Year = 1997}
@article {Sacker65,
author = {Sacker, R. J.},
title = {A new approach to the perturbation theory of invariant surfaces},
journal = {Commun. Pure Appl. Math.},
volume = {18},
doi = {10.1002/cpa.3160180409},
pages = {717--732},
year = {1965},
}
@article{saka90br,
Author = {H. Sakaguchi},
Journal = {Prog. Ther. Phys.},
Pages = {792--800},
Title = {Breakdown of the phase dynamics},
Volume = {84},
Year = {1990}}
@article{Sakhr1,
Author = {Sakhr, J.},
Doi = {10.1103/PhysRevA.64.044102},
Journal = {Phys. Rev. A},
Pages = {044102},
Title = {Amplitude ordering of the trace formula for the two-particle disk billiard},
Volume = {64},
Year = {2001},
Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevA.64.044102}}
@ARTICLE{SalEck15,
author={Salewski, M. and Eckhardt, B.},
title={Turbulent states in plane {Couette} flow with rotation},
journal={Phys. Fluids},
year={2015},
volume={27},
pages={045109},
doi={10.1063/1.4917281},
}
@article{SALOHO11,
Author = {Samanta, D. and De Lozar, A. and Hof, B.},
Journal = {J. Fluid Mech.},
Pages = {193--204},
Title = {Experimental investigation of laminar turbulent intermittency in pipe flow},
Volume = {681},
Year = {2011}}
@article{Samoilenko2005,
Author = {A. M. Samoilenko and L. Recke},
Journal = {Ukrain. Math. J.},
Pages = {1089--1119},
Title = {Conditions for synchronization of one oscillation system},
Volume = {57},
Year = {2005}}
@article{SaRo02,
Author = {Saul, L. K. and Roweis, S. T.},
Journal = {J. Mach. Learn. Res.},
Pages = {119--155},
Title = {Think Globally, Fit Locally: Unsupervised Learning of Low Dimensional Manifolds},
Volume = {4},
Year = {2003}}
@Article{Sartori91,
author = {Sartori, G.},
title = {Geometric invariant theory: a model-independent approach to spontaneous symmetry and/or supersymmetry breaking},
journal = {La Rivista del Nuovo Cimento},
year = {1991},
volume = {14},
pages = {1--120},
doi = {10.1007/BF02810048},
bdsk-url-1 = {http://dx.doi.org/10.1007/BF02810048}
}
@article{Sartorius1998,
Author = {B. Sartorius and C. Bornholdt and O. Brox and H. J. Ehrke and D. Hoffmann and R. Ludwig and M. M{\"o}hrle},
Journal = {Electronics Letters, 34},
Pages = {1664--1665},
Title = {All-optical clock recovery module based on self-pulsating dfb laser},
Year = {1998}}
@Article{SB95,
author = {R. Sureshkumar and A. Beris},
title = {Linear stability analysis of viscoelastic {P}oiseuille flow using an {A}rnoldi-based orthogonalization algorithm},
journal = {J. Non-Newton. Fluid Mech.},
year = {1995},
volume = {56},
pages = {151--182}
}
@article{SCD07,
Author = {Cvitanovi{\'c}, P. and Davidchack, R. L. and Siminos, E.},
Journal = {SIAM J. Appl. Dyn. Syst.},
Note = {\arXiv{0709.2944}},
Pages = {1--33},
Title = {On the state space geometry of the {Kuramoto-Sivashinsky} flow in a periodic domain},
Volume = {9},
Year = {2009}}
@unpublished{SCD09b,
Author = {Siminos, E. and Cvitanovi{\'c}, P. and Davidchack, R. L.},
Note = {In preparation},
Title = {Recurrent spatio-temporal structures of translationally invariant {Kuramoto-Sivashinsky} flow},
Year = {2011}}
@article{SchEck06,
Author = {T. M. Schneider and B. Eckhardt},
Journal = {Chaos},
Pages = {041103},
Title = {Edge of chaos in pipe flow},
Volume = {16},
Year = {2006}}
@article{SchEckVoll07,
Author = {Schneider, T. M. and Eckhardt, B. and Vollmer, J.},
Journal = {Phys. Rev. E},
Pages = {066313},
Title = {Statistical analysis of coherent structures in transitional pipe flow},
Volume = {75},
Year = {2007}}
@article{SchEckYor07,
Author = {T. M. Schneider and B. Eckhardt and J. Yorke},
Journal = {Phys. Rev. Lett.},
Pages = {034502},
Title = {Turbulence, transition, and the edge of chaos in pipe flow},
Volume = {99},
Year = {2007}}
@book{SchGer00,
Address = {New York},
Author = {Schlichting, H. and Gersten, K.},
Keywords = {Boundary-Layer Theory},
Publisher = {Springer},
Title = {Boundary-Layer Theory},
Year = {2000}}
@phdthesis{Schmi99,
Author = {A. Schmiegel},
Note = {Available on {\tt archiv.ub.uni-marburg.de/diss/z2000/0062}},
School = {Philipps-Universit{\"a}t Marburg},
Title = {Transition to turbulence in linearly stable shear flows},
Year = {1999}}
@article{Schmid10,
author = {P. J. Schmid},
title = {Dynamic mode decomposition of numerical and experimental data},
journal = {J. Fluid Mech.},
volume = {656},
year = {2010},
pages = {5--28},
numpages = {24},
doi = {10.1017/S0022112010001217}
}
@article{Schneider2005,
Author = {K. R. Schneider},
Journal = {Int. J. Bifurc. Chaos},
Pages = {3579--3588},
Title = {Entrainment of modulation frequency: {A} case study},
Volume = {15},
Year = {2005}}
@book{Schouten89,
Address = {New York},
Author = {Schouten, J. A.},
Publisher = {Dover},
Title = {Tensor Analysis for Physicists},
Year = {1989}}
@book{Schutz80,
Address = {Cambridge},
Author = {Schutz, B. F.},
Publisher = {Cambridge Univ. Press},
Title = {Geometrical Methods of Mathematical Physics},
Year = {1980}}
@article{science04,
Abstract = {Transition to turbulence in pipe flow is one of the most
fundamental and longest-standing problems in fluid dynamics.
Stability theory suggests that the flow remains laminar for all flow
rates, but in practice pipe flow becomes turbulent even at moderate
speeds. This transition drastically affects the transport efficiency
of mass, momentum, and heat. On the basis of the recent discovery of
unstable traveling waves in computational studies of the
Navier-Stokes equations and ideas from dynamical systems theory, a
model for the transition process has been suggested. We report
experimental observation of these traveling waves in pipe flow,
confirming the proposed transition scenario and suggesting that the
dynamics associated with these unstable states may indeed capture the
nature of fluid turbulence.},
Author = {B. Hof and C. W. H. van Doorne and J. Westerweel and
F. T. M. Nieuwstadt and H. Faisst and B. Eckhardt and
H. Wedin and R. R. Kerswell and F. Waleffe},
Journal = {Science},
Pages = {1594--1598},
Title = {Experimental observation of nonlinear traveling waves in turbulent pipe flow},
Volume = {305},
Year = {2004},
DOI = {10.1126/science.1100393}
}
@article{SD98,
Abstract = {Change stability of maps to locate periodic orbits.},
Author = {P. Schmelcher and F. K. Diakonos},
Journal = {Phys. Rev. E},
Pages = {2739},
Title = {General appoach to the localization of unstable periodic orbits in chaotic dynamical systems},
Volume = {57},
Year = {1998}}
@Article{SE00,
author = {A. Schmiegel and B. Eckhardt},
title = {Persistent turbulence in annealed plane {Couette} flow},
journal = {Europhys. Lett.},
year = {2000},
volume = {51},
pages = {395--400}
}
@article{SE97,
Abstract = {plane {Couette} flow: For Re > 200,
chaotic transients appear with lifetimes longer than the viscous one.
Depending on the type of the perturbation, isolated initial conditions
with infinite lifetime appear for Reynolds numbers
larger than about 270--305. In this third regime, the lifetime
as a function of Reynolds number and amplitude is fractal.
These results suggest that in the transition region
the turbulent dynamics is characterized by a chaotic repeller
rather than an attractor.},
Author = {A. Schmiegel and B. Eckhardt},
Journal = {Phys. Rev. Lett.},
Note = {\arXiv{chao-dyn/9704018}},
Pages = {5250},
Title = {Fractal stability border in plane {Couette} flow},
Volume = {79},
Year = {1997}}
@article{Selberg56,
Author = {Selberg, A.},
Journal = {J. Indian Math. Soc. (N.S.)},
Pages = {47--87},
Title = {Harmonic analysis and discontinuous groups in weakly symmetric {Riemannian} spaces with applications to {Dirichlet} series},
Volume = {20},
Year = {1956}}
@article{semwogerere04,
Author = {D. Semwogerere and M. F. Schatz},
Journal = {Phys. Rev. Lett.},
Pages = {124502},
Title = {Secondary instabilities of hexagonal patterns in a {B}enard-{M}arangoni convection experiment},
Volume = {93},
Year = {2004}}
@article{SeOsRaSi83,
Author = {Sethna, J. and Ostlund, S. and Rand, D. A. and Siggia, E.},
Journal = {Physica D},
Pages = {303},
Title = {Universal properties of the transition from quasi-periodicity to chaos in dissipative systems},
Volume = {8},
Year = {1983}}
@book{seydel,
Address = {New York},
Author = {R. Seydel},
Publisher = {Elsevier},
Title = {From equilibrium to chaos: {Practical} bifurcation and stability analysis},
Year = 1988}
@article{SGB10,
Author = {T. M. Schneider and J. F. Gibson and J. Burke},
Journal = {Phys. Rev. Lett.},
Pages = {104501},
Title = {Snakes and ladders: {L}ocalized solutions of plane {Couette} flow},
Volume = {104},
Year = {2010}}
@article{SGLDE08,
Author = {T. M. Schneider and J. F. Gibson and M. Lagha and F. De Lillo and B. Eckhardt},
Journal = {Phys. Rev. E.},
Note = {\arXiv{0805.1015}},
Pages = {037301},
Title = {Laminar-turbulent boundary in plane {Couette} flow},
Volume = {78},
Year = {2008}}
@article{SHcgl90,
Author = {W. van Saarloos and P. C. Hohenberg},
Journal = {Phys. Rev. Lett.},
Pages = {749},
Title = {Pulses and Fronts in the Complex {Ginzburg-Landau} Equation near a Subcritical Bifurcation},
Volume = {64},
Year = {1990}}
@article{shch04fli,
Abstract = {Fast Lyapunov indicator method is used to identify the
chaotic and regular regions in the phase space. Periodic orbits are
determined to characterized the resonance properties of these
regions.},
Author = {E. Shchekinova and C. Chandre and Y. Lan and T. Uzer},
Journal = {J. Chem. Phys.},
Pages = {1--7},
Title = {Analyzing intramolecular dynamics by fast {Lyapunov} indicators},
Volume = {121},
Year = {2004}}
@article{SHcoh92,
Abstract = {Propose the concept of coherent structures and study the
competition between them in chaotic dynamics. Specifically, the
invasion of front to another state is discussed.},
Author = {W. van Saarloos and P. C. Hohenberg},
Journal = {Physica D},
Pages = {303--367},
Title = {Fronts, pulses, sources and sinks in generalized complex {Ginzburg-Landau} equations},
Volume = {56},
Year = {1992}}
@Article{shearless,
author = {M. Farazmand and D. Blazevski and G. Haller},
title = {Shearless transport barriers in unsteady two-dimensional flows and maps},
journal = {Physica D},
year = {2014},
volume = {278-279},
pages = {44--57}
}
@article{ShenKad82,
Author = {S. J. Shenker and L. P. Kadanoff},
Journal = {J. Stat. Phys.},
Pages = {631},
Title = {Critical behavior of a {KAM} surface: I. {Empirical} results},
Volume = {27},
Year = {1982}}
@article{Shenker82,
Author = {S.J. Shenker},
Doi = {10.1016/0167-2789(82)90033-1},
Journal = {Physica D},
Pages = {405--411},
Title = {Scaling behavior in a map of a circle onto itself: {Empirical} results},
Volume = {5},
Year = {1982},
Bdsk-Url-1 = {http://dx.doi.org/10.1016/0167-2789(82)90033-1}}
@misc{ShiAvHof2015,
author = {{Shi}, L. and {Avila}, M. and {Hof}, B.},
title = "The universality class of the transition to turbulence",
note = {\arXiv{1504.03304}},
year = 2015,
}
@Article{Shte69,
author = {Shtern, V. N.},
title = {Stability of plane {Couette} flow},
journal = {J. Appl. Mechanics and Tech. Physics},
year = {1969},
volume = {10},
pages = {793--796}
}
@article{ShWi06,
Author = {Shapere, A. and Wilczek, F.},
Journal = {J. Fluid Mech.},
Pages = {557},
Title = {Geometry of self-propulsion at low {Reynolds} number},
Volume = {198},
Year = {2006}}
@unpublished{ShWi16,
Author = {Short, K. Y. and Willis, A. P.},
Title = {Bifurcation structure of relative periodic orbits in pipe flow},
Note = {In preparation.},
Year = {2016}}
@article{ShWi87,
Author = {Shapere, A. and Wilczek, F.},
Journal = {Phys. Rev. Lett.},
Pages = {2051--2054},
Title = {Self-propulsion at low {Reynolds} number},
Volume = {58},
Year = {1987}}
@article{ShWi89,
Author = {Shapere, A. and Wilczek, F.},
Journal = {J. Fluid Mech.},
Pages = {587--599},
Title = {Efficiencies of self-propulsion at low {Reynolds} number},
Volume = {198},
Year = {1989}}
@article{ShWi89a,
Author = {Shapere, A. and Wilczek, F.},
Journal = {Amer. J. Physics},
Pages = {514--518},
Title = {Gauge kinematics of deformable bodies},
Volume = {57},
Year = {1989}}
@article{SiCvi10,
Author = {Siminos, E. and Cvitanovi{\'c}, P.},
Journal = {Physica D},
Pages = {187--198},
Title = {Continuous symmetry reduction and return maps for high-dimensional flows},
Volume = {240},
Year = {2011}}
@article{Sieber2002,
Author = {J. Sieber},
Journal = {SIAM J. Appl. Math.},
Pages = {248--270},
Title = {Numerical bifurcation analysis for multisection semiconductor lasers},
Volume = {1},
Year = {2002}}
@Article{siegel42,
author = {C. L. Siegel},
title = {Iteration of analytic functions},
journal = {Ann. Math.},
year = {1942},
volume = {43},
pages = {607--612}
}
@article{SieStei90,
Author = {Sieber, M. and Steiner, F.},
Doi = {10.1016/0167-2789(90)90058-W},
Journal = {Physica D},
Pages = {248--266},
Title = {Classical and quantum mechanics of a strongly chaotic billiard system},
Volume = {44},
Year = {1990}
}
@article{SieStei91,
Author = {Sieber, M. and Steiner, F.},
Journal = {Phys. Rev. Lett.},
Pages = {1941--1944},
Title = {Quantization of chaos},
Volume = {67},
Year = {1991}}
@phdthesis{SiminosThesis,
Address = {Atlanta},
Author = {Siminos, E.},
Note = {\\ \wwwcb{/projects/theses.html}},
School = {School of Physics, Georgia Inst. of Technology},
Title = {Recurrent Spatio-temporal Structures in Presence of Continuous Symmetries},
Year = {2009}}
@Inproceedings{simo89,
author = {C. Simo},
title = {On the analytical and numerical approximation of invariant manifolds},
booktitle = {Les M\'{e}thodes Modernes de la M\'{e}canique C\'{e}leste (Goutelas '89)},
year = {1989},
editor = {D. Benest and C. Froeschl\'{e}},
pages = {285--329},
address = {Gif-sur-Yvette},
publisher = {Editions Frontieres}
}
@article{sinai,
Author = {Ya. G. Sinai},
Journal = {Russian Mathematical Surveys},
Pages = {21},
Title = {Gibbs measures in ergodic theory},
Volume = {166},
Year = {1972}}
@book{sinai76,
Address = {Princeton},
Author = {Ya. G. Sinai},
Publisher = {Princeton Univ. Press},
Title = {Introduction to Ergodic Theory},
Year = {1976}}
@article{sirovcgl90,
Author = {J. D. Rodriguez and L. Sirovich},
Journal = {Physica D},
Pages = {77--86},
Title = {Low-dimensional dynamics for the complex {Ginzburg-Landau} equation},
Volume = {43},
Year = {1990}}
@article{Sirovich94,
Author = {L. Sirovich and X. Zhou},
Journal = {Phys. Fluids},
Pages = {1579--1582},
Title = {Reply to ``{O}bservations regarding '{C}oherence and chaos in a model of turbulent boundary layer' by {X}. {Z}hou and {L}. {S}irovich''},
Volume = {6},
Year = {1994}}
@article{siv,
Abstract = {A integro-differential equation (KSe) is derived directly from the flame propagation equations near the bifurcation point.},
Author = {G. I. Sivashinsky},
Journal = {Acta Astronaut.},
Pages = {1177},
Title = {Nonlinear analysis of hydrodynamical instability in laminar flames - {I}. {D}erivation of basic equations},
Volume = {4},
Year = {1977}}
@Article{sivmich80,
author = {G. I. Sivashinsky and D. M. Michelson},
title = {On irregular wavy flow of a liquid down a vertical plane},
journal = {Progr. Theor. Phys. Suppl.},
year = {1980},
volume = {66},
pages = {2112--2114},
doi = {On irregular wavy flow of a liquid down a vertical plane}
}
@article{skeleton,
Author = {P. Cvitanovi{\'c}},
Journal = {Physica D},
Pages = {138},
Title = {Periodic orbits as the skeleton of classical and quantum chaos},
Volume = {51},
Year = {1991}}
@phdthesis{Skuf05,
Author = {J. D. Skufca},
School = {U. Maryland},
Title = {Understanding the chaotic saddle with focus on a 9-variable model of planar {Couette} flow},
Url = {https://drum.umd.edu/dspace/handle/1903/2897},
Year = {2005}}
@article{SM,
Author = {C. R. Smith and S. P. Metzler},
Journal = {J. Fluid Mech.},
Pages = {27},
Title = {The study of hairpin vortices in a laminar boundary layer},
Volume = {129},
Year = {1983}}
@article{SM83,
Author = {Smith, CR and Metzler, SP},
Journal = {J. Fluid Mech.},
Pages = {27--54},
Title = {The characteristics of low-speed streaks in the near-wall region of a turbulent boundary layer},
Volume = {129},
Year = {1983}}
@article{smale,
Author = {S. Smale},
Journal = {Bull. Amer. Math. Soc.},
Pages = {747--817},
Title = {Differentiable dynamical systems},
Volume = {73},
Year = {1967}}
@book{Smale00,
Address = {Singapore},
Author = {S. Smale and F. Cucker and R. Wong},
Publisher = {World Scientific},
Title = {The collected papers of {Stephen Smale}},
Year = {2000}}
@article{Smale70I,
Author = {S. Smale},
Journal = {Invent. Math.},
Pages = {305--331},
Title = {Topology and mechanics, {I}.},
Volume = {10},
Year = {1970}}
@article{Smale70II,
Author = {S. Smale},
Journal = {Invent. Math.},
Pages = {45--64},
Title = {Topology and mechanics, {II}.},
Volume = {11},
Year = {1970}}
@book{Smale80,
Address = {New York},
Annote = {Subtitle: Essays on Dynamical Systems, Economic Processes and related
topics},
Author = {S. Smale},
Publisher = {Springer},
Title = {The Mathematics of Time},
Year = {1980}}
@article{SMcgl02,
Abstract = {The behavior of the solutions of the CGLe without
diffusion is studied with a periodic boundary condition.},
Author = {J. Scheuer and B. A. Malomed},
Journal = {Physica D},
Pages = {102--115},
Title = {Stable and chaotic solutions of the complex {Ginzburg-Landau} equation with periodic boundary conditions},
Volume = {161},
Year = {2002}}
@article{SMH05,
Author = {T. R. Smith and J. Moehlis and P. Holmes},
Journal = {J. Fluid Mech.},
Pages = {71--110},
Title = {Low-dimensional models for turbulent plane {Couette} flow in a minimal flow unit},
Volume = {538},
Year = {2005}}
@article{smith2005POD,
title={Low-dimensional modelling of turbulence using the proper orthogonal
decomposition: a tutorial},
author={Smith, T. R. and Moehlis, J. and Holmes, P.},
journal={Nonlinear Dynamics},
volume={41},
pages={275--307},
year={2005}
}
@Article{SNGS,
author = {J. Sanche\'{z} and M. Net and B. Gar{\'c}ia-Archilla and C. Sim{\'o}},
title = {{Newton-Krylov} continuation of periodic orbits for {Navier-Stokes} flows},
journal = {J. Comput. Physics},
year = {2004},
volume = {201},
pages = {13--33}
}
@book{Sou70,
Address = {Paris},
Author = {J.-M. Souriau},
Publisher = {Dunod},
Title = {Structure des syst\`emes dynamiques},
Year = {1970}}
@book{Sou97,
Address = {Boston},
Author = {J.-M. Souriau},
Publisher = {Birkh\"auser},
Title = {Structure of Dynamical Systems},
Year = {1997}}
@article{Spalart91,
Author = {P. R. Spalart and R. D. Moser and M. M. Rogers},
Journal = {J. Comp. Phys.},
Pages = {297--324},
Title = {Spectral Methods for the {Navier-Stokes} Equations with One Infinite and Two Periodic Directions},
Volume = {96},
Year = {1991}}
@article{spald91,
Abstract = {A brief survey of modelling equations of fully developed
homogeneous turbulence. Kolmogorov's original paper is translated.},
Author = {D. B. Spalding},
Journal = {Proc. R. Soc. Lond. A},
Pages = {211--216},
Title = {Kolmogorov's two-equation model of turbulence},
Volume = {434},
Year = {1991}}
@book{sparrow,
Address = {New York},
Author = {C. Sparrow},
Publisher = {Springer},
Title = {The {Lorenz} Equations: {Bifurcations}, Chaos and Strange Attractors},
Year = {1982}}
@article{spercgl,
Author = {W. Sch{\"o}pf and L. Kramer},
Journal = {Phys. Rev. Lett.},
Pages = {2316},
Title = {Small-amplitude periodic and chaotic solutions of the complex {Ginzburg-Landau} equation for a subcritical bifurcation},
Volume = {66},
Year = {1991}}
@book{spfunc1,
Address = {New York},
Author = {N. M. Temme},
Publisher = {Wiley-Interscience},
Title = {Special functions: an introduction to the classical functions of mathematical physics},
Year = {1996}}
@techreport{SpiekerBlog,
Author = {Spieker, D. W.},
Institution = {School of Physics, Georgia Inst. of Technology},
Note = {{\tt Channelflow.org/dokuwiki/doku.php/gtspring2009:spieker\_blog}},
Title = {{Gone fishin' - a research blog on fluid flows}},
Year = {2009}}
@Article{spotspreading,
author = {N. Tillmark},
title = {On the spreading mechanisms of a turbulent spot in plane {Couette} flow},
journal = {Europhys. Lett.},
year = {1995},
volume = {32},
pages = {481--485}
}
@article{Spruill07,
Author = {Spruill, M. C.},
Journal = {Elect. Commun. in Probab.},
Pages = {234--247},
Title = {Asymptotic distribution of coordinates on high dimensional spheres},
Volume = {12},
Year = {2007}}
@article{SPScgl92,
Author = {B. I. Shraiman and A. Pumir and W. van Saarloos and P. C. Hohenberg and H. Chat{\'e} and M. Holen},
Journal = {Physica D},
Pages = {241--248},
Title = {Spatiotemporal chaos in the one-dimensional complex {Ginzburg-Landau} equation},
Volume = {57},
Year = {1992}}
@article{Sq33,
Author = {H. B. Squire},
Journal = {Proc. Roy. Soc. A},
Pages = {621},
Title = {On the stability of three-dimensional distribution of viscous fluid between parallel walls},
Volume = {142},
Year = {1933}}
@Article{SS86,
author = {Saad, Y. and Schultz, M. H.},
title = {{GMRES: A} generalized minimal residual algorithm for solving nonsymmetric linear systems},
journal = {SIAM J. Sci. Stat. Comput.},
year = {1986},
volume = {7},
number = {3},
pages = {856--869}
}
@article{stabpcgl,
Author = {B. J. Matkowsky and V. Volpert},
Journal = {Quart. Appl. Math.},
Pages = {265--281},
Title = {Stability of plane wave solutions of complex {Ginzburg-Landau} equations},
Volume = {LI},
Year = {1993}}
@Article{StaSte11,
author = {M. Stanislavova and A. Stefanov},
title = {Asymptotic estimates and stability analysis of {Kuramoto-Sivashinsky} type models},
journal = {J. Evol. Eqs.},
year = {2011},
volume = {11},
pages = {605--635},
doi = {10.1007/s00028-011-0103-5}
}
@article{SteLar91,
title={The quasiperiodic route to chaos in a model of the peroxidase--oxidase reaction},
author={C. G. Steinmetz and R. Larter},
journal={J. Chem. Phys},
volume={94},
pages={1388--1396},
year={1991},
}
@article{sten94sol,
Abstract = {A nonlinear equation of the plasma surface is derived for the small deformation. Possible solitary solutions are discussed.},
Author = {L. Stenflo and O. M. Gradov},
Journal = {Phys. Plasmas},
Pages = {2804},
Title = {Solitary surface-wave deformation of a plasma boundary},
Volume = {1},
Year = {1994}}
@article{sten98osc,
Author = {L. Stenflo and M. Y. Yu},
Journal = {Phys. Plasmas},
Pages = {3122},
Title = {Plasma oscillons in spherically bounded magnetized plasmas},
Volume = {5},
Year = {1998}}
@Article{sternb57,
author = {S. Sternberg},
journal = {Amer. J. Math.},
year = {1957},
volume = {79},
pages = {809}
}
@Article{sternb58,
author = {S. Sternberg},
journal = {Amer. J. Math.},
year = {1958},
volume = {80},
pages = {623}
}
@Article{sternb59,
author = {S. Sternberg},
journal = {Amer. J. Math.},
year = {1959},
volume = {81},
pages = {578}
}
@book{StGo09,
Address = {Cambridge},
Author = {M. Stone and P. Goldbart},
Publisher = {Cambridge Univ. Press},
Title = {Mathematics for Physics: A Guided Tour for Graduate Students},
Year = {2009}}
@Article{STMEG15,
author = {Suri, B. and Tithof, J. and Mitchell, R. and R. Grigoriev and M. Schatz},
title = {Velocity profile in a two-layer {Kolmogorov}-like flow},
journal = {Phys. Fluids},
year = {2014},
volume = {26},
pages = {053601},
doi = {10.1063/1.4873417}
}
@article{StMeiss98,
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.},
Author = {D. Sterling and J. D. Meiss},
Journal = {Phys. Lett. A},
Pages = {46--52},
Title = {Computing periodic orbits using the anti-integrable limit},
Volume = {241},
Year = {1998}}
@article{Stokes1845,
Author = {Stokes, G. G.},
Journal = {Trans. Cambridge Philosophical Soc.},
Pages = {287--319},
Title = {On the theories of the internal friction of fluids in motion and of the equilibrium and motion of elastic solids},
Volume = {8},
Year = {1845}}
@Conference{STPGS15,
author = {B. Suri and J. Tithof and R. K. Pallantla and R. Grigoriev and M. Schatz},
title = {Experimental Observation of Exact Coherent Structures in a Weakly Turbulent Quasi-Two-Dimensional Flow},
booktitle = {68\textsuperscript{th} Annual Meeting of the APS Division of Fluid Dynamics},
year = {2015},
volume = {60},
series = {Bull. Am. Phys. Soc},
url = {http://meetings.aps.org/link/BAPS.2015.DFD.D29.2}
}
@book{strogb,
Address = {Cambridge, MA},
Author = {S. H. Strogatz},
Publisher = {Perseus Books},
Title = {Nonlinear Dynamics and Chaos},
Year = {2000}}
@Article{StrWan02,
author = {Strauss, W. and Wang, G.},
title = {Instability of traveling waves of the {Kuramoto-Sivashinsky} equation},
journal = {Chin. Ann. Math.},
year = {2002},
volume = {23},
number = {02},
pages = {267--276},
doi = {10.1142/S0252959902000250},
eprint = {http://www.worldscientific.com/doi/pdf/10.1142/S0252959902000250}
}
@inproceedings{sunil_ahuja_template-based_2006,
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},
Address = {Piscataway, NJ},
Author = {S. Ahuja and {I.G.} Kevrekidis and {C.W.} Rowley},
Booktitle = {2006 American Control Conference, 14-16 June 2006},
Keywords = {control system synthesis,feedback,invariance, linearisation techniques,nonlinear control systems, partial differential equations,stability},
Pages = {6},
Publisher = {{IEEE}},
Title = {Template-based stabilization of relative equilibria},
Year = {2006}}
@Article{SunKanZha11,
author = {Sun, L.-S. and Kang, X.-Y. and Zhang, Q. and Lin, L.-X.},
title = {A method of recovering the initial vectors of globally coupled map lattices based on symbolic dynamics},
journal = {Chin. Phys. B},
year = {2011},
volume = {20},
pages = {120507},
doi = {10.1088/1674-1056/20/12/120507}
}
@article{SW60,
Author = {J. T. Stuart and J. Watson},
Journal = {J. Fluid Mech.},
Pages = {353--370, 371--389},
Title = {On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows},
Volume = {9},
Year = {1960}}
@article{SWG02,
Author = {P. A. Stone and F. Waleffe and M. D. Graham},
Journal = {Phys. Rev. Lett.},
Pages = {20831},
Title = {Toward a structural understanding of turbulent drag reduction: {Nonlinear} coherent states in viscoelastic shear flows},
Volume = {89},
Year = {2002}}
@article{SwinneyPhysD83,
Author = {H. L. Swinney},
Journal = {Physica D},
Pages = {3--15},
Title = {Observations of order and chaos in nonlinear systems},
Volume = {7},
Year = {1983}}
@article{SYE05,
Abstract = {identify and follow the edge of chaos and provide evidence
that it is a surface. For low Reynolds numbers we find that the edge of chaos
coincides with the stable manifold of a periodic orbit, whereas at higher
Reynolds numbers it is the stable set of a higher-dimensional chaotic
object.},
Author = {J. D. Skufca and J. A. Yorke and B. Eckhardt},
Journal = {Phys. Rev. Lett.},
Pages = {174101},
Title = {Edge of chaos in a parallel shear flow},
Volume = {96},
Year = {2006}}
@Article{Sylvester1852,
author = {Sylvester, J. J.},
title = {A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares},
journal = {Philos. Mag.},
year = {1852},
volume = {4},
pages = {138--142}
}
@article{symmcgl,
Author = {A. Doelman},
Journal = {Physica D},
Pages = {398--428},
Title = {Breaking the hidden symmetry in the {Ginzburg-Landau} equation},
Volume = {97},
Year = {1996}}
@Article{TaGiCh09,
author = {Takeuchi, K. A. and Ginelli, F. and Chat\'{e}, H.},
title = {{Lyapunov} analysis captures the collective dynamics of large chaotic systems},
journal = {Phys. Rev. Lett.},
year = {2009},
volume = {103},
pages = {154103},
note = {\arXiv{0907.4298}},
abstract = {We show, using generic globally-coupled systems, that the collective
dynamics of large chaotic systems is encoded in their Lyapunov spectra:
most modes are typically localized on a few degrees of freedom, but
some are delocalized, acting collectively on the trajectory. For
globally-coupled maps, we show moreover a quantitative correspondence
between the collective modes and some of the so-called Perron-Frobenius
dynamics. Our results imply that the conventional definition of extensivity
must be changed as soon as collective dynamics sets in.}
}
@article{tajima02,
Author = {S. Tajima and H. S. Greenside},
Journal = {Phys. Rev. E},
Pages = {017205},
Title = {Microextensive chaos of a spatially extended system},
Volume = {66},
Year = {2002}}
@article{TakaOono84,
Author = {Takahashi, Y. and Oono, Y.},
Journal = {Prog. Theor. Phys},
Pages = {851--854},
Title = {Towards the statistical mechanics of chaos},
Volume = {71},
Year = {1984}}
@ARTICLE{Takh93,
author = {Takhtajan, L.},
title = {On foundation of the generalized {Nambu} mechanics},
journal = {Commun. Math. Phys.},
year = {1994},
volume = {160},
pages = {295--316},
note = {\arXiv{hep-th/9301111}},
doi = {10.1007/BF02103278}
}
@book{Taub61,
Address = {New York},
Editor = {A. H. Taub},
Publisher = {Pergamon Press},
Title = {{John von Neumann}: Collected Works. {Volume II}: {Operators}, Ergodic Theory and Almost Periodic Functions in a Group},
Year = {1961}}
@Article{Taylor23,
author = {Taylor, G. I.},
title = {Stability of a viscous liquid contained between two rotating cylinders},
journal = {Phil. Trans. Royal Soc. A},
year = {1923},
volume = {223},
pages = {289--343}
}
@article{Taylor38,
Author = {Taylor, G. I.},
Doi = {10.1098/rspa.1938.0032},
Journal = {Proc. R. Soc. Lond. A},
Pages = {476--490},
Title = {The spectrum of turbulence},
Volume = {164},
Year = {1938},
Bdsk-Url-1 = {http://dx.doi.org/10.1098/rspa.1938.0032}}
@article{tbohr,
Author = {M. Bazhenov and T. Bohr and K. Gorshkov and M. Rabinovich},
Journal = {Phys. Lett. A},
Pages = {104},
Title = {The diversity of steady state solutions of the complex {Ginzburg-Landau} equation},
Volume = {217},
Year = {1996}}
@book{TE05,
title={Spectra and pseudospectra: the behavior of nonnormal matrices and operators},
author={L. N. Trefethen and M. Embree},
year={2005},
publisher={Princeton Univ. Press},
Address = {Princeton},
}
@article{TeHaHe10,
Author = {Tempelmann, D. and Hanifi, A. and Henningson, D. S.},
Doi = {10.1017/S0022112009993260},
Journal = {J. Fluid Mech.},
Pages = {5--37},
Title = {Spatial optimal growth in three-dimensional boundary layers},
Volume = {646},
Year = {2010},
Bdsk-Url-1 = {http://dx.doi.org/10.1017/S0022112009993260}}
@article{temam91edd,
Abstract = {Estimates on the dimension of the attractors and the
decay rates of Fourier mode of NSe are given. By approximation of
the inertial manifold, the interaction between the large and small
eddies are derived. Such laws open the way to numerically efficient
algorithms via multiscale mthods.},
Author = {R. Temam},
Journal = {Proc. R. Soc. Lond. A},
Pages = {23--39},
Title = {Approximation of attractors, large eddy simulations and multiscale methods},
Volume = {434},
Year = {1991}}
@Article{tenenbaum2000,
author = {Tenenbaum, J. B. and Silva, V. de and Langford, J. C.},
title = {A global geometric framework for nonlinear dimensionality reduction},
journal = {Science},
year = {2000},
volume = {290},
pages = {2319--2323},
doi = {10.1126/science.290.5500.2319},
abstract = {Scientists working with large volumes of high-dimensional data, such
as global climate patterns, stellar spectra, or human gene distributions,
regularly confront the problem of dimensionality reduction: finding
meaningful low-dimensional structures hidden in their high-dimensional
observations. The human brain confronts the same problem in everyday
perception, extracting from its high-dimensional sensory inputs -30,000
auditory nerve fibers or 106 optic nerve fibers- manageably small
number of perceptually relevant features. Here we describe an approach
to solving dimensionality reduction problems that uses easily measured
local metric information to learn the underlying global geometry
of a data set. Unlike classical techniques such as principal component
analysis (PCA) and multidimensional scaling (MDS), our approach is
capable of discovering the nonlinear degrees of freedom that underlie
complex natural observations, such as human handwriting or images
of a face under different viewing conditions. In contrast to previous
algorithms for nonlinear dimensionality reduction, ours efficiently
computes a globally optimal solution, and, for an important class
of data manifolds, is guaranteed to converge asymptotically to the
true structure.}
}
@book{Tennekes72,
Addr = {Cambridge, MA},
Author = {H. Tennekes and J.L. Lumley},
Publisher = {MIT Press},
Title = {A First Course in Turbulence},
Year = {1972}}
@article{TFGphase97,
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.},
Author = {A. Torcini und H. Frauenkron and P. Grassberger},
Journal = {Phys. Rev. E},
Pages = {5073},
Title = {Studies of phase turbulence in the one dimensional complex {Ginzburg-Landau} equation},
Volume = {55},
Year = {1997}}
@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},
note = {\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{Thiffeault11,
Author = {Thiffeault, J.-L.},
Doi = {10.1088/0951-7715/25/2/R1},
Journal = {Nonlinearity},
Note = {\arXiv{1105.1101}},
Pages = {1--44},
Title = {Using multiscale norms to quantify mixing and transport},
Volume = {25},
Year = {2012},
Bdsk-Url-1 = {http://dx.doi.org/10.1088/0951-7715/25/2/R1}}
@ARTICLE{Thiffeault2001,
author = {Thiffeault, J.-L. },
title = {Derivatives and constraints in chaotic flows: asymptotic behaviour
and a numerical method},
journal = {Physica D},
year = {2002},
volume = {172},
pages = {139--161},
note = {\arXiv{nlin/0101012}},
abstract = {In a smooth flow, the leading-order response of trajectories to infinitesimal
perturbations in their initial conditions is described by the finite-time
Lyapunov exponents and associated characteristic directions of stretching.
We give a description of the second-order response to perturbations
in terms of Lagrangian derivatives of the exponents and characteristic
directions. These derivatives are related to generalised Lyapunov
exponents, which describe deformations of phase space elements beyond
ellipsoidal. When the flow is chaotic, care must be taken in evaluating
the derivatives because of the exponential discrepancy in scale along
the different characteristic directions. Two matrix decomposition
methods are used to isolate the directions of stretching, the first
appropriate in finding the asymptotic behaviour of the derivatives
analytically, the second better suited to numerical evaluation. The
derivatives are shown to satisfy differential constraints that are
realised with exponential accuracy in time. With a suitable reinterpretation,
the results of the paper are shown to apply to the Eulerian framework
as well.}
}
@Article{Thoo98,
author = {Thoo, J. B.},
title = {A picture is worth a thousand words},
journal = {College Math. J.},
year = {1998},
volume = {29},
pages = {408--411}
}
@Article{TI03,
author = {S. Toh and T. Itano},
title = {A periodic-like solution in channel flow},
journal = {J. Fluid Mech.},
year = {2003},
volume = {481},
pages = {67--76}
}
@BOOK{Tinkham,
title = {Group Theory and Quantum Mechanics},
publisher = {Dover},
year = {2003},
author = {Tinkham, M.},
address = {New York}
}
@article{ToDe94,
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.},
Author = {Toronov, V. Y. and Derbov, V. L.},
Journal = {Phys. Rev. E},
Pages = {1392--1399},
Title = {Geometric phases in lasers and liquid flows},
Volume = {49},
Year = {1994}}
@article{ToDe94a,
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.},
Author = {Toronov, V. Y. and Derbov, V. L.},
Journal = {Phys. Rev. A},
Keywords = {complex Lorenz, geometric phase, Laser, symmetry},
Pages = {878--881},
Title = {Geometric-phase effects in laser dynamics},
Volume = {50},
Year = {1994}}
@article{ToDe97,
Abstract = {An investigation is made of the geometric phases in a ring laser with
counterpropagating waves. It is shown that the frequency splitting
of the counterpropagating waves that appears in the case of radiation
pulsations or is induced by displacement of an external mirror can
be regarded as a manifestation of a geometric phase.},
Author = {Toronov, V. Y. and Derbov, V. L.},
Journal = {Quantum Electronics},
Pages = {644--648},
Title = {Geometric phases in a ring laser},
Volume = {27},
Year = {1997}}
@article{ToDe97a,
Author = {Toronov, V. Y. and Derbov, V. L.},
Journal = {Phys. Rev. E},
Pages = {3689--3692},
Title = {Boundedness of attractors in the complex {Lorenz} model},
Volume = {55},
Year = {1997}}
@article{ToDe98,
Author = {Toronov, V. Y. and Derbov, V. L.},
Journal = {J. Optical Soc. of America B},
Pages = {1282--1290},
Title = {Topological properties of laser phase},
Volume = {15},
Year = {1998}}
@article{tompaid96,
Abstract = {{N}ewton 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.},
Author = {S. Tompaidis},
Journal = {Experimental Mathematics},
Pages = {211--230},
Title = {Numerical Study of Invariant Sets of a Quasi-periodic Perturbation of a Symplectic Map},
Volume = {5},
Year = {1996}}
@article{torc96,
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.},
Author = {A. Torcini},
Journal = {Phys. Rev. Lett.},
Pages = {1047},
Title = {Order parameter for the transition from phase to amplitude turbulence},
Volume = {77},
Year = {1996}}
@article{toricgl,
Author = {P. Tak{\'a}{\v{c}}},
Journal = {Nonlinearity},
Pages = {289--321},
Title = {Invariant 2-tori in the time-dependent {Ginzburg-Landau} equation},
Volume = {5},
Year = {1992}}
@Book{Townsend76,
title = {The Structure of Turbulent Shear Flows, 2\textsuperscript{nd} Edn.},
publisher = {Cambridge Univ. Press},
year = {1976},
author = {A. A. Townsend},
address = {Cambridge}
}
@article{transgol,
Author = {R. Ramshankar and J. P. Gollub},
Journal = {Phys. Fluids A},
Pages = {1344},
Title = {Transport by capillary waves, {Part II}: {Scalar} dispersion and the structure of the concentration field},
Volume = {3},
Year = {1991}}
@article{Tre97,
Author = {L. Trefethen},
Journal = {SIAM Rev.},
Pages = {383--406},
Title = {Pseudospectra of linear operators},
Volume = {39},
Year = {1997}}
@book{Tref00,
Address = {Philadelphia},
Author = {L. N. Trefethen},
Publisher = {SIAM},
Title = {Spectral methods in {M}atlab},
Year = {2000}}
@book{Trefethen97,
Addr = {Philadelphia},
Author = {L. N. Trefethen and D. Bau},
Publisher = {SIAM},
Title = {Numerical Linear Algebra},
Year = {1997}}
@UNPUBLISHED{TrThNi14,
author = {{Tranchida}, J. and {Thibaudeau}, P. and {Nicolis}, S.},
title = "Knotted strange attractors and matrix {Lorenz} systems",
note = {\arXiv{1404.7774}},
year = 2014,
}
@article{TTRD93,
Author = {N. Trefethen and A. E. Trefethen and S. C. Reddy and T. A. Driscoll},
Journal = {Science},
Pages = {578--584},
Title = {Hydrodynamic stability without eigenvalues},
Volume = {261},
Year = {1993}}
@Article{TuckBar03,
author = {L. S. Tuckerman and D. Barkley},
title = {Symmetry breaking and turbulence in perturbed plane {Couette} flow},
journal = {Theor. Comp. Fluid Dyn.},
year = {2002},
volume = {16},
pages = {91--97},
note = {\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.}
}
@article{TuckBar07,
Author = {L. S. Tuckerman and D. Barkley},
Journal = {Springer Proc. Physics},
Pages = {224--226},
Title = {Mean flow and modeling of turbulent-laminar patterns in plane {Couette} flow},
Volume = {117},
Year = {2007}}
@Article{tucker2002ros,
author = {Tucker, W.},
title = {A rigorous {ODE} solver and Smale's 14{th} problem},
journal = {Found. Comp. Math},
year = {2002},
volume = {2},
pages = {53--117},
address = {New York},
publisher = {Springer}
}
@book{Tung85,
Address = {Singapore},
Author = {Tung, W.-K.},
Publisher = {World Scientific},
Title = {Group Theory in Physics},
Year = {1985}}
@article{umb94pat,
Author = {F. Melo and P. Umbanhowar and H. L. Swinney},
Journal = {Phys. Rev. Lett.},
Pages = {172--175},
Title = {Transition to parametric wave patterns in a vertically oscillated granular layer},
Volume = {72},
Year = {1994}}
@phdthesis{vanDoorne04,
Author = {C. W. H. van Doorne},
School = {Delft},
Title = {Stereoscopic {PIV} on transition in pipe flow},
Url = {www.ahd.tudelft.nl},
Year = {2004}
}
@article{VC08,
Author = {D. Viswanath and P. Cvitanovi\'{c}},
Journal = {J. Fluid Mech.},
Addendum = {\arXiv{0801.1918}},
Pages = {215--233},
Title = {Stable manifolds and the transition to turbulence in pipe flow},
Volume = {627},
Year = {2009}}
@incollection{Visw07a,
Address = {Berlin},
Author = {D. Viswanath},
Booktitle = {Mathematics and Computation, a Contemporary View. The Abel Symposium 2006},
Note = {\arXiv{physics/0701337}},
Publisher = {Springer},
Series = {Abel Symposia},
Title = {The dynamics of transition to turbulence in plane {Couette} flow},
Volume = {3},
Year = 2008}
@article{Visw07b,
Author = {D. Viswanath},
Journal = {J. Fluid Mech.},
Note = {\arXiv{physics/0604062}},
Pages = {339--358},
Title = {Recurrent motions within plane {Couette} turbulence},
Volume = {580},
Year = {2007}}
@article{Visw08,
Author = {D. Viswanath},
Journal = {Phil. Trans. Royal Soc. A},
Pages = {561--576},
Title = {The critical layer in pipe flow at high {R}eynolds number},
Volume = {367},
Year = {2009}}
@article{VK11,
Author = {L. van Veen and G. Kawahara},
Journal = {Phys. Rev. Lett.},
Pages = {114501},
Title = {Homoclinic tangle on the edge of shear turbulence},
Volume = {107},
Year = {2011}}
@article{VlToDe98,
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.},
Author = {Vladimirov, A. G. and Toronov, V. Y. and Derbov, V. L.},
Journal = {Int. J. Bifur. Chaos},
Pages = {723--729},
Title = {The complex {Lorenz} model: {Geometric} structure, homoclinic bifurcation and one-dimensional map},
Volume = {8},
Year = {1998}}
@article{VlToDe98a,
Author = {Vladimirov, A. G. and Toronov, V. Y. and Derbov, V. L.},
Journal = {Technical Physics},
Pages = {877--884},
Title = {Properties of the phase space and bifurcations in the complex {Lorenz} model},
Volume = {43},
Year = {1998}}
@article{vN32,
Author = {J. von Neumann},
Journal = {Ann. Math.},
Note = {See additions \cite{vonNeum32}. Reprinted in \cite{Taub61}.},
Pages = {587--642},
Title = {{Zur Operatorenmethode in der klassischen Mechanik}. ({German}) [{On} the operator methods of classical mechanics]},
Volume = {33},
Year = {1932}}
@article{vonNeum32,
Author = {J. von Neumann},
Journal = {Ann. Math.},
Pages = {789--791},
Title = {{Zus{\"a}tze zur Arbeit ``Zur Operatorenmethode in der klassischen Mechanik''}. ({German}) [{Additions} to the work ``{On} operator methods in classical mechanics'']},
Volume = {33},
Year = {1932}}
@Article{Voros87,
author = {Voros, A.},
title = {Spectral functions, special functions and the {Selberg} zeta function},
journal = {Commun. Math. Phys.},
year = {1987},
volume = {110},
pages = {439--465},
doi = {10.1007/BF01212422}
}
@Article{Voros92,
author = {Voros, A.},
title = {Spectral zeta functions},
journal = {Adv. Stud. Pure Math},
year = {1992},
volume = {21},
pages = {327--358}
}
@book{Vorst03,
Address = {Cambridge},
Author = {H. A. Vorst},
Publisher = {Cambridge Univ. Press},
Title = {Iterative Krylov Methods for Large Linear Systems},
Year = {2003}}
@article{voth02,
Abstract = {Based on the experimentally generated velocity field,
fixed points on Poincar{\'e} section are identified as well as their
stable and unstable manifolds. These information can be related to the
gradient of the stretching field and the distribution of passive scalar
quantities. It seems that on these manifolds the stretching field gains
its maximum.},
Author = {G. A. Voth and G. Haller and J. P. Gollub},
Journal = {Phys. Rev. Lett.},
Pages = {254501},
Title = {Experimental measurements of streching fields in fluid mixing},
Volume = {88},
Year = {2002}}
@article{voth07,
Author = {K.-Y. Chan and D. Stich and G. A. Voth},
Journal = {Rev. Sci. Inst.},
Pages = {02374},
Title = {Real-time image compression for high-speed particle tracking},
Volume = {78},
Year = {2007}}
@article{vp98,
Abstract = {A numerical method based on the gradient minimization
and Periodic Schur Decomposition (PSD) is devised to solve the
linear-quadratic (LQ) optimization problem the linear periodic
discrete-time control systems. It can be used as a discretized
version of the similar problem in the continuous regime.},
Author = {A. Varga and S. Pieters},
Journal = {Automatica},
Pages = {477--481},
Title = {Gradient-Based Approach to Solve Optimal Periodic Output Feedback Control Problems},
Volume = {34},
Year = {1998}}
@article{VSE98,
Author = {J. Vollmer and T. M. Schneider and B. Eckhardt},
Note = {\arXiv{0808.2636}},
Title = {Basin boundary, edge of chaos, and edge state in a two-dimensional model},
Year = {2007}}
@article{W01,
Author = {F. Waleffe},
Journal = {J. Fluid Mech.},
Pages = {93--102},
Title = {Exact coherent structures in channel flow},
Volume = {435},
Year = {2001}}
@inproceedings{W02,
Author = {F. Waleffe},
Booktitle = {Proceedings of the International Symposium on ``Dynamics and Statistics of Coherent Structures in Turbulence: Roles of Elementary Vortices''},
Editor = {Shigeo Kida},
Pages = {115--128},
Publisher = {National Center of Sciences, Tokyo, Japan},
Title = {Exact coherent structures and their instabilities: {Toward} a dynamical-system theory of shear turbulence},
Year = {2002}}
@article{W03,
Author = {F. Waleffe},
Journal = {Phys. Fluids},
Pages = {1517--1543},
Title = {Homotopy of exact coherent structures in plane shear flows},
Volume = {15},
Year = {2003}}
@inproceedings{W90b,
Author = {F. Waleffe},
Booktitle = {Annual Research Briefs},
Pages = {159--168},
Publisher = {Center for Turbulence Research, Stanford University},
Title = {On the origin of the streak spacing in turbulent shear flow},
Year = {1990}}
@article{W95a,
Author = {F. Waleffe},
Journal = {Stud. Applied Math.},
Pages = {319--343},
Title = {Hydrodynamic stability and turbulence: {Beyond} transients to a {Self-Sustaining Process}},
Volume = {95},
Year = {1995}}
@article{W95b,
Author = {F. Waleffe},
Journal = {Phys. Fluids},
Pages = {3060--3066},
Title = {Transition in shear flows: {Nonlinear} normality versus non-normal linearity},
Volume = {7},
Year = {1995}}
@article{W97,
Author = {F. Waleffe},
Journal = {Phys. Fluids},
Pages = {883--900},
Title = {On a {Self-Sustaining Process} in shear flows},
Volume = {9},
Year = {1997}}
@article{W98,
Author = {F. Waleffe},
Journal = {Phys. Rev. Lett.},
Pages = {4140--4143},
Title = {Three-dimensional coherent states in plane shear flows},
Volume = {81},
Year = {1998}}
@unpublished{Waleffe90,
Author = {F. Waleffe},
Journal = {Working paper, available at \verb+www.math.wisc.edu/~waleffe/ECS/sspctr90.pdf+},
Note = {Center for Turbulence Research, Stanford University/NASA Ames, unpublished preprint (1990)},
Title = {Proposal for a self-sustaining mechanism in shear flows},
Year = {1990}}
@incollection{WalPant97,
Address = {Southampton},
Author = {F. Waleffe and J. Kim},
Booktitle = {Self-Sustaining Mechanisms of Wall Bounded Turbulence},
Editor = {R. L. Panton},
Pages = {385--422},
Publisher = {Computational Mechanics Publications},
Title = {How streamwise Rolls and Streaks Self-Sustain in a Shear Flow},
Url = {www.math.wisc.edu/~waleffe/ECS/WK97.pdf},
Year = {1997}
}
@phdthesis{Way08,
Author = {Way, R.},
School = {Dept. of Mathematics, U. Surrey},
Title = {Dynamics in the {Hopf} bundle, the geometric phase and
implications for dynamical systems},
Year = {2008}}
@Article{WBHZ14,
author = {H. Wedin and A. Bottaro and A. Hanifi and G. Zampogna},
title = {Unstable flow structures in the {Blasius} boundary layer},
journal = {Eur. Phys. J. E},
year = {2014},
volume = {37},
pages = {34},
doi = {10.1140/epje/i2014-14034-1}
}
@InProceedings{WeBoHaZa13,
author = {Wedin, H. and Bottaro, A. and Hanifi, A. and Zampogna, G.},
title = {Three-dimensional nonlinear states in the {Blasius} boundary layer},
booktitle = {AIMETA Conference, 17-20 September 2013},
year = {2013},
volume = {XXI},
address = {Torino, Italy},
url = {http://www.aimeta.dicam.unibo.it/sites/www.aimeta.dicam.unibo.it/files/static/www.aimetatorino2013.it/cdrom/cdrom_pdf_fullpaper/000550004201.pdf}
}
@article{WegSol02,
title={On some mathematics for visualizing high dimensional data},
author={Wegman, E. J. and Solka, J. L.},
journal={Sankhy{\=a}: Indian J. Statistics, Ser. A},
pages={429--452},
year={2002},
}
@article{Weil49,
Author = {Weil, A.},
Journal = {Bull. Amer. Math. Soc},
Pages = {497--508},
Title = {Numbers of solutions of equations in finite fields},
Volume = {55},
Year = {1949}}
@article{weiss82,
Author = {J. Weiss and M. Tabor and G. Carnevale},
Journal = {J. Math. Phys.},
Pages = {522},
Title = {The {P}ainlev{\'e} property for partial differential equations},
Volume = {24},
Year = {1983}}
@Article{weiss83,
author = {J. Weiss},
title = {The {P}ainlev{\'e} for partial differential equations. {II}: {B\"a}cklund transformation, {Lax} pairs, and the {Schwarzian} derivative},
journal = {J. Math. Phys.},
year = {1983},
volume = {24},
pages = {1405}
}
@Article{weiss86,
author = {J. Weiss},
title = {{B\"a}cklund transformation and the {P}ainlev{\'e} property},
journal = {J. Math. Phys.},
year = {1986},
volume = {27},
pages = {1293}
}
@misc{WeissFeig,
Author = {Weisstein, E. W.},
Note = {mathworld.wolfram.com/FeigenbaumConstant.html},
Title = {Feigenbaum constant},
Year = {2012}}
@book{Weyl39,
title={The Classical Groups:
Their Invariants and Representations},
author={Weyl, H.},
year={1939},
publisher = {Princeton Univ. Press},
address = {Princeton, NJ},
isbn = {9780691057569}
}
@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 = {2015}}
@article{WGW07,
Author = {J. Wang and J. F. Gibson and F. Waleffe},
Journal = {Phys. Rev. Lett.},
Note = {\arXiv{physics/0703072}},
Pages = {204501},
Title = {Lower branch coherent states in shear flows: {Transition} and control},
Volume = {98},
Year = {2007}}
@article{WHC10,
Author = {A. P. Willis and Y. Hwang and C. Cossu},
Journal = {Phys. Rev. E},
Pages = {036321},
Title = {Optimally amplified large-scale streaks and drag reduction in turbulent pipe flow},
Volume = {82},
Year = {2010}}
@unpublished{WiAvCv12,
Author = {Willis, A. P. and Avila, M. and Cvitanovi{\'c}, P.},
Note = {In preparation},
Title = {Symmetries of turbulent flows: {How} to slice them},
Year = {2012}}
@article{Wieczorek2005,
Author = {S. Wieczorek and B. Krauskopf and T. B. Simpson and D. Lenstra},
Journal = {Phys. Rep.},
Pages = {1--128},
Title = {The dynamical complexity of optically injected semiconductor lasers},
Volume = {416},
Year = {2005}}
@book{wiggb,
Address = {New York},
Author = {S. Wiggins},
Publisher = {Springer},
Title = {Global Bifurcations and Chaos},
Year = {1988}}
@Article{Wilczak03,
author = {D. Wilczak},
title = {Chaos in the {Kuramoto-Sivashinsky} equations -- a computer-assisted proof},
journal = {J. Diff. Eqn.},
year = {2003},
volume = {194},
pages = {433--459},
doi = {10.1016/S0022-0396(03)00104-9}
}
@article{WillKer07,
Author = {A. P. Willis and R. R. Kerswell},
Journal = {Phys. Rev. Lett.},
Pages = {014501},
Title = {Critical behavior in the relaminarization of localized turbulence in pipe flow},
Volume = {98},
Year = {2007}}
@article{WillKer08,
Author = {A. P. Willis and R. R. Kerswell},
Journal = {Phys. Rev. Lett.},
Pages = {124501},
Title = {Coherent structures in localised and global pipe turbulence},
Volume = {100},
Year = {2008}}
@Article{WillKer09,
author = {Willis, A. P. and Kerswell, R. R.},
title = {Turbulent dynamics of pipe flow captured in a reduced model: puff relaminarisation and localised edge states},
journal = {J. Fluid Mec.},
year = {2009},
volume = {619},
pages = {213--233},
note = {\arXiv{0712.2739}}
}
@Article{Wimmer73,
author = {H. K. Wimmer},
title = {{On the Ostrowski-Schneider Inertia Theorem}},
journal = {J. Math. Anal. Appl.},
year = {1973},
volume = {41},
pages = {164--169}
}
@article{Win92,
Author = {D. Wintgen and K. Richter and G. Tanner},
Journal = {Chaos},
Pages = {53},
Title = {Quantization of chaotic systems},
Volume = {2},
Year = {1992}}
@Article{WiShCv15,
author = {Willis, A. P. and Short, K. Y. and Cvitanovi{\'c}, P.},
title = {Symmetry reduction in high dimensions, illustrated in a turbulent pipe},
journal = {Phys. Rev. E},
year = {2016},
volume = {93},
pages = {022204},
note = {\arXiv{1504.05825}},
doi = {10.1103/PhysRevE.93.022204}
}
@article{witt99hol,
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.},
Author = {R. W. Wittenberg and P. Holmes},
Journal = {Chaos},
Pages = {452},
Title = {Scale and space localization in the {Kuramoto-Sivashinsky} equation},
Volume = {9},
Year = {1999}}
@article{WK04,
Author = {H. Wedin and R. R. Kerswell},
Journal = {J. Fluid Mech.},
Pages = {333--371},
Title = {Exact coherent structures in pipe flow: {Traveling} wave solutions},
Volume = {508},
Year = {2004}}
@inproceedings{WKH93,
Address = {Berlin},
Author = {F. Waleffe and J. Kim and J. Hamilton},
Booktitle = {Turbulent Shear Flows 8: Selected papers from the Eighth International Symposium on Turbulent Shear Flows, Munich, Germany, Sept.\ 9-11, 1991},
Editor = {F. Durst and R. Friedrich and B. E. Launder and F. W. Schmidt and U. Schumann and J. H. Whitelaw},
Pages = {37--49},
Publisher = {Springer},
Title = {On the origin of streaks in turbulent shear flows},
Year = {1993}}
@article{WoSa07,
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.},
Author = {Wolfe, C. L. and Samelson, R. M.},
Journal = {Tellus A},
Pages = {355--366},
Title = {An efficient method for recovering {Lyapunov} vectors from singular vectors},
Volume = {59},
Year = {2007}}
@Article{WPKM08,
author = {A. P. Willis and J. Peixinho and R. R. Kerswell and T. Mullin},
title = {Experimental and theoretical progress in pipe flow transition},
journal = {Phil. Trans. R. Soc. A},
year = {2008},
volume = {366},
pages = {2671--2684}
}
@techreport{Wri06,
Author = {T. Write},
Institution = {Oxford Univ.},
Note = {{\tt web.comlab.ox.ac.uk/projects/pseudospectra/eigtool}},
Title = {Eigtool, a graphical tool for nonsymmetric eigenproblems},
Year = {2006}}
@article{WRT92,
author = {Wintgen, D. and Richter, K. and Tanner, G.},
title = {The semiclassical helium atom},
journal = {Chaos},
year = {1992},
volume = {2},
pages = {19--33},
doi = {10.1063/1.165920}
}
@book{Wu85,
Address = {Singapore},
Author = {Wu, K.-T.},
Publisher = {World Scientific},
Title = {Group theory in physics},
Year = {1985}}
@Article{Wulff00,
author = {C. Wulff},
title = {Transitions from relative equilibria to relative periodic orbits},
journal = {Doc. Math.},
year = {2000},
volume = {5},
pages = {227--274}
}
@article{Wulff01,
Author = {Wulff, C. and Lamb, J. S. W. and Melbourne, I.},
Journal = {Ergod. Theor. Dynam. Syst.},
Pages = {605--635},
Title = {Bifurcation from relative periodic solutions},
Volume = {21},
Year = {2001}}
@article{Wulff02,
Author = {Wulff, C.},
Doi = {10.1524/zpch.2002.216.4.535},
Journal = {Z. Phys. Chem.},
Pages = {535},
Title = {Spiral waves and {Euclidean} symmetries},
Volume = {216},
Year = {2002}
}
@article{Wulff03,
Author = {C. Wulff},
Journal = {J. Geometry and Physics},
Pages = {309--338},
Title = {Persistence of {Hamiltonian} relative periodic orbits},
Volume = {48},
Year = {2003}}
@techreport{Wulff94,
Author = {C. Wulff and A. Hohmann and P. Deuflhard},
Institution = {Konrad-Zuse-Zentrum Berlin},
Number = {SC 94-12},
Title = {Numerical continuation of periodic orbits with symmetry},
Url = {http://personal.maths.surrey.ac.uk/st/C.Wulff/preprints/Wulff_Hoh_Deuf_94-12.ps.gz},
Year = {1994}
}
@inproceedings{WW05,
Address = {Dordrecht},
Author = {F. Waleffe and J. Wang},
Booktitle = {Non-uniqueness of Solutions to the Navier-Stokes
Equations and their Connection with Laminar-Turbulent Transition},
Editor = {T. Mullin and R. R. Kerswell},
Pages = {85--106},
Publisher = {Kluwer},
Title = {Transition threshold and the {Self-Sustaining Process}},
Year = {2005}}
@article{wygnanski2006tpp,
Author = {Wygnanski, I. and Sokolov, M. and Friedman, D.},
Journal = {J. Fluid Mech.},
Pages = {283--304},
Title = {On transition in a pipe. {Part 2. The} equilibrium puff},
Volume = {69},
Year = {1975}}
@article{Xia92,
Author = {Xia, Z.},
Journal = {Ergod. Theor. Dynam. Syst.},
Pages = {621--31},
Title = {Existence of invariant tori in volume-preserving diffeomorphisms},
Volume = {12},
Year = {1992}}
@article{YaTaGiChRa08,
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.},
Author = {Yang, H.-l. and Takeuchi, K. A. and Ginelli, F. and Chat\'{e}, H. and Radons, G.},
Journal = {Phys. Rev. Lett.},
Note = {\arXiv{0807.5073}},
Pages = {074102},
Title = {Hyperbolicity and the effective dimension of spatially-extended dissipative systems},
Volume = {102},
Year = {2009}}
@unpublished{YTGCR08,
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.},
Author = {Yang, H.-L. and Takeuchi, K. A. and Ginelli, F. and Chat\'e, H. and Radons, G.},
Note = {\arXiv{0807.5073}},
Title = {Hyperbolicity and the effective dimension of spatially-extended dissipative systems}}
@article{yu93bd,
Author = {S. V. Vladimirov and M. Y. Yu},
Journal = {Phys. Fluids B},
Pages = {2888},
Title = {Boundary Effects on the Nonlinear Interactions of Surface Waves},
Volume = {5},
Year = {1993}}
@article{yu94sur,
Author = {S. V. Vladimirov and M. Y. Yu},
Journal = {Phys. Lett. A},
Pages = {459},
Title = {Coupled Electron Surface Plasma Wave Solitons in Slabs},
Volume = {184},
Year = {1994}}
@article{yuan00id,
Abstract = {By using the natural measure to construct the canonical
conjugate of the observed signal, the authors can identify the
dynamical equivalence of two signals from one process.},
Author = {G.-C. Yuan and J. A. Yorke and T. L. Carroll and E. Ott and L. M. Pecora},
Journal = {Phys. Rev. Lett.},
Pages = {4265},
Title = {Testing whether two chaotic one dimensional processes are dynamically identical},
Volume = {85},
Year = {2000}}
@article{Zagarola98,
Author = {M. V. Zagarola and A. J. Smits},
Journal = {J. Fluid Mech.},
Pages = {33--79},
Title = {Mean flow scaling of turbulent pipe flow},
Volume = {373},
Year = {1998}}
@Article{Zang91,
author = {T. A. Zang},
title = {On the rotation and skew-symmetric forms for incompressible flow simulations},
journal = {Appl. Num. Math.},
year = {1991},
volume = {7},
pages = {27--40}
}
@Article{ZeMa85,
author = {Zeghlache, H. and {Mandel}, P.},
title = {Influence of detuning on the properties of laser equations},
journal = {J. Optical Soc. of America B},
year = {1985},
volume = {2},
pages = {18--22}
}
@Article{ZG96,
author = {S. M. Zoldi and H. S. Greenside},
title = {Spatially localized unstable periodic orbits of a high-dimensional chaotic system},
journal = {Phys. Rev. E},
year = {1998},
volume = {57},
pages = {R2511--R2514},
addendum = {\arXiv{chao-dyn/9704005}},
doi = {10.1103/PhysRevE.57.R2511}
}
@article{Zhou92,
Author = {X. Zhou and L. Sirovich},
Journal = {Phys. Fluids A},
Pages = {2855--2874},
Title = {Coherence and Chaos in a model of turbulent boundary layer},
Volume = {4},
Year = {1992}}
@Article{zielinska95,
author = {B. J. A. Zielinska and J. E. Wesfreid},
title = {On the spatial structure of global modes in wake flow},
journal = {Phys. Fluids},
year = {1995},
volume = {7},
pages = {1418--1424}
}