% This file was created with JabRef 2.7.
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@UNPUBLISHED{dovidio08,
author = {F. d'Ovidio and J. Isern-Fontanet and C. Lopez and E. Hernandez-Garcia
and E. Garcia-Ladona},
title = {Comparison between {Eulerian} diagnostics and finite-size {Lyapunov}
exponents computed from altimetry in the {Algerian} basin},
note = {{\tt arXiv:0807.3848}},
year = {2008},
abstract = {Transport and mixing properties of surface currents can be detected
from altimetric data by both Eulerian and Lagrangian diagnostics.
In contrast with Eulerian diagnostics, Lagrangian tools like the
local Lyapunov exponents have the advantage of exploiting both spatial
and temporal variability of the velocity field and are in principle
able to unveil subgrid filaments generated by chaotic stirring.}
}
@UNPUBLISHED{GrPo12,
author = {{beim Graben}, P. and {Potthast}, R.},
title = {Implementing {Turing} machines in dynamic field architectures},
note = {\arXiv{1204.5462}},
year = {2012},
keywords = {computer science - formal languages and automata theory}
}
@ARTICLE{AbBrKe91,
author = {H. D. I. Abarbanel and R. Brown and M. B. Kennel},
title = {Lyapunov exponents in chaotic systems: their importance and their
evaluation using observed data},
journal = {Int. J. Mod. Phys. B},
year = {1991},
volume = {5},
pages = {1347--1375}
}
@ARTICLE{AbBrKe91a,
author = {H. D. I. Abarbanel and R. Brown and M. B. Kennel},
title = {Variation of {Lyapunov} exponents on a strange attractor},
journal = {J. Nonlin. Sci.},
year = {1991},
volume = {1},
pages = {175--199}
}
@ARTICLE{ACFK09,
author = {H. D. I. Abarbanel and D. R. Creveling and R. Farsian and M. Kostuk},
title = {Dynamical state and parameter estimation},
journal = {SIAM J. Appl. Math.},
year = {2009},
volume = {8},
pages = {1341-1381}
}
@BOOK{ablowbook,
title = {Solitions, Nonlinear Evolution Equations and Inverse Scattering},
publisher = {Cambridge Univ. Press},
year = {1992},
author = {M. J. Ablowitz and P. A. Clarkson},
address = {Cambridge}
}
@ARTICLE{Abraham95,
author = {N. B. Abraham and U. A. Allen and E. Peterson and A. Vicens and R.
Vilaseca and V. Espinosa and G. L. Lippi},
title = {Structural similarities and differences among attractors and their
intensity maps in the laser-{Lorenz} model},
journal = {Optics Comm.},
year = {1995},
volume = {117},
pages = {367--384},
abstract = {Numerical studies of the laser-Lorenz model using parameters reasonably
accessible for recent experiments with a single mode homogeneously
broadened laser demonstrate that the form of the return map of successive
peak values of the intensity changes from a sharply cusped map in
resonance to a map with a smoothly rounded maximum as the laser is
detuned into the period doubling regime. This transformation appears
to be related to the disappearance (with detuning) of the heteroclinic
structural basis for the stable manifold which exists in resonance.
This is in contrast to the evidence reported by Tang and Weiss (Phys.
Rev. A 49 (1994) 1296) of a cusped map for both the period doubling
chaos found with detuning and the spiral chaos found in resonance
for seemingly Lorenz-like behavior of the far-infrared ammonia laser
from which it was concluded that there existed a ?unique chaotic
attractor for a single-mode laser?. }
}
@BOOK{AbrMars78,
title = {Foundations of Mechanics},
publisher = {Benjamin-Cummings},
year = {1978},
author = {R. Abraham and J. E. Marsden},
address = {Reading, Mass.}
}
@BOOK{mta,
title = {Manifolds, Tensor Analysis, and Applications},
publisher = {Springer},
year = {1988},
author = {R. Abraham and J. E. Marsden and T. S. Ratiu},
address = {New York},
edition = {2nd}
}
@ARTICLE{AbSm70,
author = {R. Abraham and S. Smale},
title = {Nongenericity of $\Omega$-stability},
journal = {Proc. Symp. Pure Math.},
year = {1970},
volume = {14},
pages = {5--8}
}
@BOOK{AbSh92,
title = {Dynamics - The Geometry of Behavior},
publisher = {Wesley},
year = {1992},
author = {R. H. Abraham and C. D. Shaw},
address = {Reading, MA}
}
@ARTICLE{Abramov09,
author = {R. V. Abramov},
title = {Short-time linear response with reduced-rank tangent map},
journal = {Chinese Ann. Mathematics B},
year = {2009},
volume = {30},
pages = {447-462}
}
@ARTICLE{AbrMaj08,
author = {Abramov, R. V. and Majda, A. J.},
title = {New approximations and tests of linear fluctuation-response for chaotic
nonlinear forced-dissipative dynamical systems},
journal = {J. Nonlinear Sci.},
year = {2008},
volume = {18},
pages = {303--341},
abstract = {We develop and test two novel computational approaches for predicting
the mean linear response of a chaotic dynamical system to small change
in external forcing via the fluctuation-dissipation theorem. Unlike
the earlier work in developing fluctuation-dissipation theorem-type
computational strategies for chaotic nonlinear systems with forcing
and dissipation, the new methods are based on the theory of Sinai-Ruelle-Bowen
probability measures, which commonly describe the equilibrium state
of such dynamical systems. The new methods take into account the
fact that the dynamics of chaotic nonlinear forced-dissipative systems
often reside on chaotic fractal attractors, where the classical quasi-Gaussian
formula of the fluctuation-dissipation theorem often fails to produce
satisfactory response prediction, especially in dynamical regimes
with weak and moderate degrees of chaos. A simple new low-dimensional
chaotic nonlinear forced-dissipative model is used to study the response
of both linear and nonlinear functions to small external forcing
in a range of dynamical regimes with an adjustable degree of chaos.
We demonstrate that the two new methods are remarkably superior to
the classical fluctuation-dissipation formula with quasi-Gaussian
approximation in weakly and moderately chaotic dynamical regimes,
for both linear and nonlinear response functions. One straightforward
algorithm gives excellent results for short-time response while the
other algorithm, based on systematic rational approximation, improves
the intermediate and long time response predictions.},
doi = {10.1007/s00332-007-9011-9},
keyword = {Mathematics and Statistics}
}
@ARTICLE{AS87,
author = {M. S. Acarlar and C. R. Smith},
title = {A study of hairpin vortices in a laminar boundary layer},
journal = {J. Fluid Mech.},
year = {1987},
volume = {175},
pages = {1--41 and 45--83}
}
@ARTICLE{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.}
}
@INPROCEEDINGS{sunil_ahuja_template-based_2006,
author = {S. Ahuja and {I.G.} Kevrekidis and {C.W.} Rowley},
title = {Template-based stabilization of relative equilibria},
booktitle = {2006 American Control Conference, 14-16 June 2006},
year = {2006},
pages = {6},
address = {Piscataway, NJ},
publisher = {{IEEE}},
abstract = {We present an approach to the design of feedback control laws that
stabilize the relative equilibria of general nonlinear systems with
continuous symmetry. Using a template-based method, we factor out
the dynamics associated with the symmetry variables and obtain evolution
equations in a reduced frame that evolves in the symmetry direction.
The relative equilibria of the original system are fixed points of
these reduced equations. Our controller design methodology is based
on the linearization of the reduced equations about such fixed points.
Assuming equivariant actuation, we derive feedback laws for the reduced
system that are optimal in the sense that they minimize a quadratic
cost function. We illustrate the method by stabilizing unstable traveling
waves of a dissipative {PDE} possessing translational invariance},
keywords = {control system synthesis,feedback,invariance, linearisation techniques,nonlinear
control systems, partial differential equations,stability}
}
@MISC{AllBar12,
author = {A. Allahem and T. Bartsch},
title = {Chaotic dynamics in multidimensional transition states},
year = {2012}
}
@ARTICLE{AllgGeorg88,
author = {E. L. Allgower and K. Georg},
title = {Numerically stable homotopy methods without an extra dimension},
journal = {Lect. Appl. Math.},
year = {1990},
volume = {26},
pages = {1--13},
note = {{\tt www.math.colostate.edu/emeriti/georg/homExtra.pdf}}
}
@ARTICLE{alvarez_monodromy_2005,
author = {M. Alvarez},
title = {Monodromy and stability for nilpotent critical points},
journal = {Int. J. Bifur. Chaos},
year = {2005},
volume = {15},
pages = {1253--1266},
issn = {0218-1274}
}
@ARTICLE{AmLeAg06,
author = {G. F. V. Amaral and C. Letellier and L. A. Aguirre},
title = {Piecewise affine models of chaotic attractors: {The R\"ossler} and
{Lorenz} systems},
journal = {Chaos},
year = {2006},
volume = {16},
pages = {013115},
doi = {10.1063/1.2149527},
numpages = {14}
}
@ARTICLE{AnBoAi07,
author = {Ando, H. and Boccaletti, S. and Aihara, K.},
title = {Automatic control and tracking of periodic orbits in chaotic systems},
journal = {Phys. Rev. E},
year = {2007},
volume = {75},
pages = {066211}
}
@ARTICLE{angen88,
author = {S. B. Angenent},
title = {The periodic orbits of an area preserving twist-map},
journal = {Commun. Math. Phys.},
year = {1988},
volume = {115},
pages = {353--374}
}
@ARTICLE{anosov67,
author = {D. V. Anosov},
title = {Geodesic flows on compact {Riemannian} manifolds of negative curvature},
journal = {Proc.\ Steklov.\ Inst.\ of Math.},
year = {1967},
volume = {90},
annote = {His famous paper about Anosov systems?}
}
@BOOK{AnAr88,
title = {Dynamical Systems {I}: {Ordinary} Differential Equations and Smooth
Dynamical Systems},
publisher = {Springer},
year = {1988},
author = {D. V. Anosov and V. I. Arnol'd},
address = {New York}
}
@ARTICLE{Arai07a,
author = {Arai, Z.},
title = {On hyperbolic plateaus of the {H\'enon} map},
journal = {Experimental Math.},
year = {2007},
volume = {16},
pages = {181--188},
abstract = { We propose a rigorous computational method to prove the uniform hyperbolicity
of discrete dynamical systems. Applying the method to the real {H\'enon}
family, we prove the existence of many regions of hyperbolic parameters
in the parameter plane of the family. },
doi = {10.1080/10586458.2007.10128992}
}
@UNPUBLISHED{Arai07b,
author = {Arai, Z.},
title = {On loops in the hyperbolic locus of the complex {H\'enon} map and
their monodromies},
note = {\arXiv{0704.2978}},
year = {2007}
}
@ARTICLE{ArMi06,
author = {Arai, Z. and Mischaikow, K.},
title = {Rigorous computations of homoclinic tangencies},
journal = {SIAM J. Appl. Dyn. Syst.},
year = {2006},
volume = {5},
pages = {280-292},
doi = {10.1137/050626429}
}
@UNPUBLISHED{BrCv12,
author = {S. Ortega Arango},
title = {Towards reducing continuous symmetry of baroclinic flows},
note = {ChaosBook.org project, \\ \wwwcb{/projects/Saldana/BrCv12.pdf}},
year = {2012}
}
@ARTICLE{AKcgl02,
author = {I. S. Aranson and L. Kramer},
title = {The world of the complex {Ginzburg-Landau} equation},
journal = {Rev. Mod. Phys.},
year = {2002},
volume = {74},
pages = {99--143}
}
@ARTICLE{cgl,
author = {I. S. Aranson and L. Kramer},
title = {The world of complex {Ginzburg-Landau} equation},
journal = {Rev. Mod. Phys.},
year = {2002},
volume = {74},
pages = {99--143}
}
@BOOK{ArWe05,
title = {Mathematical Methods for Physicists: A Comprehensive Guide},
publisher = {Academic Press},
year = {2005},
author = {G. B. Arfken and H. J. Weber},
address = {New York},
edition = {6}
}
@ARTICLE{ArCho91,
author = {D. Armbruster and P. Chossat},
title = {Heteroclinic orbits in a spherically invariant system},
journal = {Physica D},
year = {1991},
volume = {50},
pages = {155--176},
abstract = {The existence and stability of structurally stable heteroclinic cycles
are discussed in a codimension-2 bifurcation problem with O(3)-symmetry,
when the critical spherical modes l = 1 and l = 2 occur simultaneously.
Several types of heteroclinic cycles are found which may explain
aperiodic attractors found in numerical simulations for the onset
of convection in a self-gravitating fluid in a spherical shell.},
doi = {10.1016/0167-2789(91)90173-7}
}
@ARTICLE{AGHks89,
author = {D. Armbruster and J. Guckenheimer and P. Holmes},
title = {{Kuramoto-Sivashinsky} dynamics on the center-unstable manifold},
journal = {SIAM J. Appl. Math.},
year = {1989},
volume = {49},
pages = {676--691}
}
@ARTICLE{AGHO288,
author = {D. Armbruster and J. Guckenheimer and P. Holmes},
title = {Heteroclinic cycles and modulated travelling waves in systems with
{O(2)} symmetry},
journal = {Physica D},
year = {1988},
volume = {29},
pages = {257--282}
}
@UNPUBLISHED{ABBBBB08,
author = {A. Arneodo and R. Benzi and J. Berg and L. Biferale and E. Bodenschatz
and A. Busse and E. Calzavarini and B. Castaing and M. Cencini and
L. Chevillard and R. Fisher and R. Grauer and H. Homann and D. Lamb
and A.S. Lanotte and E. Leveque and B. Luethi and J. Mann and N.
Mordant and W.-C. Mueller and S. Ott and N.T. Ouellette and J.-F.
Pinton and S. B. Pope and S.G. Roux and F. Toschi and H. Xu and P.K.
Yeung},
title = {Universal intermittent properties of particle trajectories in highly
turbulent flows},
note = {{\tt arXiv:0802.3776}},
year = {1989}
}
@BOOK{arnold92,
title = {Ordinary Differential Equations},
publisher = {Springer},
year = {1992},
author = {V. I. Arnol'd},
address = {New York}
}
@ARTICLE{arnold91k,
author = {V. I. Arnol'd},
title = {Kolmogorov's hydrodynamic attractors},
journal = {Proc. R. Soc. Lond. A},
year = {1991},
volume = {434},
pages = {19--22},
number = {1890},
abstract = {A review of Kolmogorov's efforts relating the Navier-Stokes equation
to the theory of dynamical system. Several interesting questions
regarding the connection are exposed.}
}
@BOOK{arnold89,
title = {Mathematical Methods for Classical Mechanics},
publisher = {Springer},
year = {1989},
author = {V. I. Arnol'd},
address = {New York}
}
@BOOK{ArKoNe88,
title = {Mathematical Aspects of Classical and Celestial Mechanics},
publisher = {Springer},
year = {1988},
author = {V. I. Arnol'd and V. V. Kozlov and A. I. Neishtadt},
address = {New York}
}
@ARTICLE{AACI,
author = {R. Artuso and E. Aurell and P. Cvitanovi{\'{c}}},
title = {Recycling of strange sets: {I}. {Cycle} expansions},
journal = {Nonlinearity},
year = {1990},
volume = {3},
pages = {325--359}
}
@ARTICLE{AACII,
author = {R. Artuso and E. Aurell and P. Cvitanovi{\'{c}}},
title = {Recycling of strange sets: {II}. {Applications}},
journal = {Nonlinearity},
year = {1990},
volume = {3},
pages = {361}
}
@ARTICLE{ACK89,
author = {Artuso, R. and Cvitanovi\'{c}, P. and Kenny, B. G.},
title = {Phase transitions on strange irrational sets},
journal = {Phys. Rev. A},
year = {1989},
volume = {39},
pages = {268--281},
doi = {10.1103/PhysRevA.39.268}
}
@ARTICLE{art03int,
author = {R. Artuso and P. Cvitanovi\'{c} and G. Tanner},
title = {Cycle expansions for intermittent maps},
journal = {Proc. Theo. Phys. Supp.},
year = {2003},
volume = {150},
pages = {1--21}
}
@ARTICLE{Asami06,
author = {Asamizuya, T.},
title = {Statistical properties of periodic orbits in a 4-disk billiard system
---{The} pruning-proof property---},
journal = {Progr. Theor. Phys.},
year = {2006},
volume = {116},
pages = {247--271},
abstract = {We investigate the statistical properties of the actual periodic orbits
in a 4-disk billiard system in the context of pruning. For served
value of the system parameter, we numerically obtain approximately
1,000,000 periodic orbits that have no more than 20 collisions. We
also compute some statistical quantities of the resultant periodic
orbits. In these statistics, we observe a periodic peak structure
that survives pruning; that is, it possesses a "pruning-proof property".}
}
@ARTICLE{Ascher95,
author = {U. Ascher and S. Ruuth and B. Wetton},
title = {Implicit-explicit methods for time-dependent partial differential
equations},
journal = {SIAM J. Numer. Anal.},
year = {1995},
volume = {32},
pages = {797--823},
number = {3},
month = jun
}
@ARTICLE{AshBoMe96,
author = {Ashwin, P. and B\"ohmer, K. and Mei, Z.},
title = {Forced symmetry breaking of homoclinic cycles in a {PDE} with {O(2)}
symmetry},
journal = {J. Comput. Appl. Math.},
year = {1996},
volume = {70},
pages = {297--310},
number = {2},
url = {http://wrap.warwick.ac.uk/18622/}
}
@ARTICLE{AshDang05,
author = {Ashwin, P and Dangelmayr, G.},
title = {Reduced dynamics and symmetric solutions for globally coupled weakly
dissipative oscillators},
journal = {Dyn. Sys.},
year = {2005},
volume = {20},
pages = {333--367}
}
@ARTICLE{AstMelb06,
author = {P. Aston and I. Melbourne},
title = {Lyapunov exponents of symmetric attractors},
journal = {Nonlinearity},
year = {2006},
volume = {19},
pages = {2455}
}
@ARTICLE{Astor10,
author = {Astorino, M.},
title = {Kauffman knot invariant from $\mathrm{SO}(N)$ or $\mathrm{Sp}(N)$
{Chern-Simons} theory and the {Potts} model},
journal = {Phys. Rev. D},
year = {2010},
volume = {81},
pages = {125026},
doi = {10.1103/PhysRevD.81.125026}
}
@ARTICLE{Aubry88,
author = {N. Aubry and P. Holmes and J. L. Lumley and E. Stone},
title = {The dynamics of coherent structures in the wall region of turbulent
boundary layer},
journal = {J. Fluid Mech.},
year = {1988},
volume = {192},
pages = {115--173}
}
@ARTICLE{aub95ant,
author = {S. Aubry},
title = {Anti-integrability in dynamical and variational problems},
journal = {Physica D},
year = {1995},
volume = {86},
pages = {284--296}
}
@ARTICLE{AuAb90,
author = {Aubry, S. and Abramovici, G.},
title = {Chaotic trajectories in the standard map. {The} concept of anti-integrability},
journal = {Physica D},
year = {1990},
volume = {43},
pages = {199--219},
abstract = {A rigorous proof is given in the standard map for the existence of
chaotic trajectories with unbounded momenta for large enough coupling
constant k. The obtained chaotic trajectories correspond to stationary
configurations of the Frenkel-Kontorowa model with a finite (non-zero)
photon gap (called gap parameter in dimensionless units). This property
implies that the trajectory (or the configuration {ui}) can be uniquely
continued as a uniformly continuous function of the model parameter
k in some neighborhood of the initial configuration. A non-zero gap
parameter implies that the Lyapunov coefficient is strictly positive
(when it is defined). In addition, the existence of dilating and
contracting manifolds is proven for these chaotic trajectories. ``Exotic''�
trajectories such as ballistic trajectories are also proven to exist
as a consequence of these theorems. The concept of anti-integrability
emerges from these theorems. In the anti-integrable limit which can
be only defined for a discrete time dynamical system, the coordinates
of the trajectory at time i do not depend on the coordinates at time
i - 1. Thus, at this singular limit, the existence of chaotic trajectories
is trivial and the dynamical system reduces to a Bernoulli shift.
It appears that the chaotic trajectories of dynamical systems originate
by continuity from those which exists at the anti-integrable limits.},
doi = {10.1016/0167-2789(90)90133-A}
}
@ARTICLE{AuDae83,
author = {Aubry, S. and Le Daeron, P. Y.},
title = {The discrete {Frenkel-Kontorova} model and its extensions. {I}. {Exact}
results for the ground-states},
journal = {Phys. D},
year = {1983},
volume = {8},
pages = {381--422},
annote = {Original paper about Aubry-Mather sets}
}
@ARTICLE{pchaot,
author = {D. Auerbach and P. Cvitanovi\'{c} and J.-P. Eckmann and G. Gunaratne
and I. Procaccia},
title = {Exploring chaotic motion through periodic orbits},
journal = {Phys. Rev. Lett.},
year = {1987},
volume = {58},
pages = {2387--2389},
abstract = {Use near recurrence to extract periodic orbits and their stability
eigenvalues.}
}
@BOOK{ausloos_logistic_2005,
title = {The Logistic Map and the Route to Chaos: From the Beginnings to Modern
Applications},
publisher = {Springer},
year = {2005},
author = {M. Ausloos and M. Dirickx},
address = {New York}
}
@article{AEOGS10,
title = {Observation of periodic orbits on curved two-dimensional geometries},
author = {Avlund, M. and Ellegaard, C. and Oxborrow, M. and Guhr, T. and S\o{}ndergaard, N.},
journal = {Phys. Rev. Lett.},
volume = {104},
issue = {16},
pages = {164101},
year = {2010},
doi = {10.1103/PhysRevLett.104.164101},
}
@UNPUBLISHED{Axen11,
author = {Axenides, M.},
title = {Non {Hamiltonian} chaos from {Nambu} dynamics of surfaces},
note = {\arXiv{1109.0470}},
year = {2011}
}
@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{AxFl08,
author = {Axenides, M. and Floratos, E.},
title = {{Nambu-Lie} 3-algebras on fuzzy 3-manifolds},
journal = {JHEP},
year = {2009},
volume = {902},
pages = {39},
note = {\arXiv{0809.3493}},
doi = {10.1088/1126-6708/2009/02/039}
}
@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{EckHoPo89,
author = {B. Eckhardt, G. Hose and E. Pollak},
journal = {Phys. Rev. A},
year = {1989},
volume = {39},
pages = {3776}
}
@ARTICLE{BaCseGaHa08,
author = {B. B\'anhelyi and T. Csendes and B. M. Garay and L. Hatvani},
title = {A computer-assisted proof of $\Sigma_3$-chaos in the forced damped
pendulum equation},
journal = {SIAM J. Appl. Dyn. Syst.},
year = {2008},
pages = {843-867}
}
@ARTICLE{vanBaal91,
author = {van Baal, P.},
title = {More (thoughts on) {Gribov} copies},
journal = {Nucl. Phys.},
year = {1992},
volume = {B369},
pages = {259-275},
doi = {10.1016/0550-3213(92)90386-P}
}
@ARTICLE{BCFLR08,
author = {R. Bachelard and C. Chandre and D. Fanelli and X. Leoncini and S.
Ruffo},
title = {Abundance of regular orbits and out-of-equilibrium phase transitions
in the thermodynamic limit for long-range systems},
journal = {Phys. Rev. Lett.},
year = {2008},
volume = {101},
pages = {260603}
}
@ARTICLE{BakasovAbraham93,
author = {Bakasov, A. A. and Abraham, N. B.},
title = {Laser second threshold: {Its} exact analytical dependence on detuning
and relaxation rates},
journal = {Phys. Rev. A},
year = {1993},
volume = {48},
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journal = {Physica D},
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volume = {160},
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author = {L. Brusch and M. G. Zimmermann and M. van Hecke and M. B{\"{a}}r
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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{Canuto88,
title = {Spectral Methods in Fluid Dynamics},
publisher = {Springer},
year = {1988},
author = {C. Canuto and M. Y. Hussaini and A. Quateroni and T. A. Zhang},
address = {New York}
}
@ARTICLE{car88,
author = {F. Cariello and M. Tabor},
title = {{P}ainlev\'{e} expansions for nonintegrable evolution equations},
journal = {Physica D},
year = {1989},
volume = {39},
pages = {77}
}
@ARTICLE{CaPe84,
author = {A. Carnegie and I. C. Percival},
journal = {J. Phys. A},
year = {1984},
volume = {17},
pages = {801}
}
@UNPUBLISHED{Carr12,
author = {K. M. Carroll},
title = {A review of return maps for {R\"ossler} and the complex {Lorenz}},
note = {ChaosBook.org project, \\ \wwwcb{/projects/Carroll/blog.pdf}},
year = {2012}
}
@BOOK{Cartan01,
title = {Riemannian geometry in an orthogonal frame: {From} lectures delivered
by \'Elie Cartan at the Sorbonne in 1926-1927},
publisher = {World Scientific},
year = {2001},
author = {Cartan, \'E.},
address = {River Edge, NJ},
isbn = {9810247478}
}
@BOOK{CartanMF,
title = {La m\'ethode du rep\`ere mobile, la th\'eorie des groupes continus,
et les espaces g\'en\'eralis\'es},
publisher = {Hermann},
year = {1935},
author = {Cartan, \'E.},
volume = {5},
series = {{Expos\'es} de {G\'eom\'etrie}},
address = {Paris}
}
@ARTICLE{Carvalho99,
author = {de Carvalho, A.},
title = {Pruning fronts and the formation of horseshoes},
journal = {Ergodic Theory Dynam. Systems},
year = {1999},
volume = {19},
pages = {851--894},
number = {4}
}
@ARTICLE{CaHa04a,
author = {de Carvalho, A. and Hall, T.},
title = {Braid forcing and star-shaped train tracks},
journal = {Topology},
year = {2004},
volume = {43},
pages = {247--287}
}
@ARTICLE{CaHa04b,
author = {de Carvalho, A. and Hall, T.},
title = {Unimodal generalized pseudo-{Anosov} maps},
journal = {Geometry and Topology},
year = {2004},
volume = {8},
pages = {1127--1188}
}
@ARTICLE{CaHa03,
author = {de Carvalho, A. and Hall, T.},
title = {Conjugacies between horseshoe braids},
journal = {Nonlinearity},
year = {2003},
volume = {16},
pages = {1329--1338}
}
@ARTICLE{CaHa01b,
author = {de Carvalho, A. and Hall, T.},
title = {The forcing relation for horseshoe braid types},
journal = {Experimental Math.},
year = {2002},
volume = {11},
pages = {271--288}
}
@ARTICLE{CaHa02,
author = {de Carvalho, A. and Hall, T.},
title = {How to prune a horseshoe},
journal = {Nonlinearity},
year = {2002},
volume = {15},
pages = {R19--R68}
}
@ARTICLE{CaHa01a,
author = {de Carvalho, A. and Hall, T.},
title = {Pruning theory and {Thurston}'s classification of surface homeomorphisms},
journal = {J. Eur. Math. Soc.},
year = {2001},
volume = {3},
pages = {287--333}
}
@BOOK{GiBo,
title = {Quantum chaos: Between order and disorder},
publisher = {Cambridge Univ. Press},
year = {1995},
author = {G. Casati and B. V. Chirikov},
address = {Cambridge}
}
@MISC{CaCaKi63,
author = {Cash, J. and Carter Cash, J. and Kilgore, M.},
title = {{The Ring of Fire}},
year = {1963},
address = {New York},
publisher = {Columbia Records}
}
@ARTICLE{Cassanas05b,
author = {R. Cassanas},
title = {Reduced {G}utzwiller formula with symmetry: {C}ase of a Lie group},
note = {\arXiv{math-ph/0509014}},
abstract = {We consider a classical {H}amiltonian $H$ on $\mathbb{R}^{2d}$, invariant
by a Lie group of symmetry $G$}
}
@ARTICLE{Cassanas05a,
author = {Roch Cassanas},
title = {Reduced {G}utzwiller formula with symmetry: {C}ase of a finite group},
journal = {J. Math. Phys.},
year = {2006},
volume = {47},
pages = {042102},
note = {\arXiv{math-ph/0506063}},
abstract = {assuming that periodic orbits are nondegenerate in SigmaE/G, we get
a reduced Gutzwiller trace formula which makes periodic orbits of
the reduced space}
}
@ARTICLE{castel01,
author = {Castelain, C. and Mokrani, A. and Le Guer, Y. and Peerhossaini, H.},
title = {Experimental study of chaotic advection regime in a twisted duct
flow},
journal = {European J. Mechanics B},
year = {2001},
volume = {20},
pages = {205--232},
doi = {10.1016/S0997-7546(00)01116-X},
keywords = {Lagrangian Chaos,Laser Doppler Velocimeter,Twisted Duct Flow}
}
@ARTICLE{Cattell2000,
author = {Kevin Cattell and Frank Ruskey and Joe Sawada and Micaela Serra and
C. R. Miers},
title = {Fast Algorithms to Generate Necklaces, Unlabeled Necklaces, and Irreducible
Polynomials over GF(2)},
journal = {J. Algorithms},
year = {2000},
volume = {37},
pages = {267--282}
}
@ARTICLE{CHHM98,
author = {Cendra, H. and Holm, D. D. and Hoyle, M. J. W. and Marsden, J. E.},
title = {The {Maxwell-Vlasov} equations in {Euler-Poincar\'e} form},
journal = {J. of Math. Phys.},
year = {1998},
volume = {39},
pages = {3138--3157}
}
@ARTICLE{detect,
author = {Champneys, A. R. and Kuznetsov, Y. A.},
title = {Numerical detection and continuation of codimension-two homoclinic
bifurcations},
journal = {Int. J. Bifur. Chaos},
year = {1994},
volume = {4},
pages = {785-822}
}
@ARTICLE{homcont,
author = {Champneys, A. R. and Kuznetsov, Y. A. and Sandstede, B.},
title = {A numerical toolbox for homoclinic bifurcation analysis},
journal = {Int. J. Bifur. Chaos},
year = {1996},
volume = {6},
pages = {867-887},
number = {5}
}
@MISC{ChaKer12,
author = {Chandler, G. J. and Kerswell, R. R.},
title = {Simple invariant solutions embedded in {2D Kolmogorov} turbulence},
year = {2012},
note = {{\tt arXiv:1207.4682}}
}
@ARTICLE{chand03tf,
author = {C. Chandre and S. Wiggins and T. Uzer},
title = {Time-frequency analysis of chaotic systems},
journal = {Physica D},
year = {2003},
volume = {181},
pages = {171},
abstract = {Use wavelets to extract local frequencies and use the major frequencies
to identify the characteristics of phase space motion.}
}
@ARTICLE{chang94,
author = {H. -C. Chang},
title = {Wave evolution on a falling film},
journal = {Ann. Rev. Fluid Mech.},
year = {1994},
volume = {26},
pages = {103--136}
}
@ARTICLE{kschang86,
author = {H.-C. Chang},
title = {Travelling waves on fluid interfaces: {Normal} form analysis of the
{Kuramoto-Sivashinsky} equation},
journal = {Phys. Fluids},
year = {1986},
volume = {29},
pages = {3142},
number = {10},
abstract = {The lowest order normal form analysis is give to the KSe and the approximate
analytical wavelength-amplitude and wavespeed-amplitude relation
are given. They agree quite well with the numerical calculations.}
}
@ARTICLE{C02,
author = {S. J. Chapman},
title = {Subcritical transition in channel flows},
journal = {J. Fluid Mech.},
year = {2002},
volume = {451},
pages = {35--97}
}
@ARTICLE{ChaDeVore79,
author = {J. G. Charney and J. G. DeVore},
title = {Multiple flow equilibria in the atmosphere and blocking},
journal = {J. Meteorology},
year = {1979},
volume = {36},
doi = {10.1175/1520-0469(1979)036<1205:MFEITA>2.0.CO;2}
}
@ARTICLE{chate94,
author = {H. Chat\'{e}},
title = {Spatiotemporal intermittency regimes of the one-dimensional complex
{Ginzburg-Landau} equation},
journal = {Nonlinearity},
year = {1994},
volume = {7},
pages = {185--204},
abstract = {In the Benjamin-Feir stable region, spatiotemporal intermittency regimes
are identified which consists of patches of linearly stale stable
plane waves separated by localized objects with a well defined dynamics.
In the transition reginon, asymptotic states with an irregular, frozen
spatial structure are shown to occur. The bistable region beyond
the Benjamin-Feir stable region is of the same spatiotemporal intermittency
type with phase turbulence as the ``laminar'' state.}
}
@ARTICLE{CM87tran,
author = {H. Chat\'{e} and Manneville, P.},
title = {Transition to Turbulence via spatiotemporal Intermittency},
journal = {Phys. Rev. Lett.},
year = {1987},
volume = {58},
pages = {112},
number = {2}
}
@BOOK{chat_mixing99,
title = {Mixing - chaos and turbulence},
publisher = {Kluwer},
year = {1999},
author = {H. Chat\'{e} and E. Villermaux and J.-M. Chomaz},
address = {New York}
}
@ARTICLE{Cha74,
author = {Chazarain, J.},
title = {Formule de {Poisson} pour les vari\'et\'es riemanniennes},
journal = {Invent. Math.},
year = {1974},
volume = {24},
pages = {65-82},
doi = {10.1007/BF01418788}
}
@ARTICLE{chemillier04,
author = {Marc Chemillier},
title = {Periodic musical sequences and {L}yndon words},
journal = {Soft Comput.},
volume = {8},
pages = {611--616}
}
@ARTICLE{Chen87,
author = {Q. Chen and Meiss, J. D., and I. C. Percival},
title = {Orbit extension methods for finding unstable orbits},
journal = {Physica D},
year = {1987},
volume = {29},
pages = {143--154}
}
@UNPUBLISHED{ChencinerLink,
author = {A. Chenciner},
title = {Three body problem},
note = {{\tt scholarpedia.org/article/Three\_body\_problem}},
year = {2007}
}
@ARTICLE{Chenc05,
author = {A. Chenciner},
title = {A note by {P}oincar\'e},
journal = {Regul. Chaotic Dyn.},
year = {2005},
volume = {10},
pages = {119-128}
}
@INPROCEEDINGS{CGMS02,
author = {A. Chenciner and J. Gerver and R. Montgomery and C. Sim\'o},
title = {Simple choreographic motions of $N$-bodies: A preliminary study},
booktitle = {Geometry, Mechanics and Dynamics},
year = {2002},
editor = {P. Newton and P. Holmes and A. Weinstein},
pages = {287-308},
address = {New York},
publisher = {Springer}
}
@ARTICLE{CheMon00,
author = {A. Chenciner and R. Montgomery},
title = {A remarkable solution of the 3-body problem in the case of equal
masses},
journal = {Ann. Math.},
year = {2000},
volume = {152},
pages = {881-901}
}
@ARTICLE{CheSu90,
author = {Cheng, C. Q. and Sun, Y. S.},
title = {Existence of invariant tori in three-dimensional measure-preserving
mappings},
journal = {Celestial Mech. Dynam. Astronom.},
year = {1990},
volume = {47},
pages = {275--92}
}
@ARTICLE{CheChe08,
author = {V. Y. Chernyak and M. Chertkov},
title = {Fermions and loops on graphs: {I. Loop} calculus for determinants},
journal = {J. Stat. Mech.},
year = {2008},
volume = {2008},
pages = {P12011},
abstract = {This paper is the first in a series devoted to evaluation of the partition
function in statistical models on graphs with loops in terms of the
Berezin/fermion integrals. The paper focuses on a representation
of the determinant of a square matrix in terms of a finite series,
where each term corresponds to a loop on the graph. The representation
is based on a fermion version of the loop calculus, previously introduced
by the authors for graphical models with finite alphabets. Our construction
contains two levels. First, we represent the determinant in terms
of an integral over anti-commuting Grassmann variables, with some
reparametrization/gauge freedom hidden in the formulation. Second,
we show that a special choice of the gauge, called the BP (Bethe-Peierls
or belief propagation) gauge, yields the desired loop representation.
The set of gauge fixing BP conditions is equivalent to the Gaussian
BP equations, discussed in the past as efficient (linear scaling)
heuristics for estimating the covariance of a sparse positive matrix.}
}
@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{CMRSY10,
author = {Chian, A. C. and Miranda, R. A. and Rempel, E. L. and Saiki, Y. and
Yamada, M.},
title = {Amplitude-phase synchronization at the onset of permanent spatiotemporal
chaos},
journal = {Phys. Rev. Lett.},
year = {2010},
volume = {104},
pages = {254102},
abstract = {Amplitude and phase synchronization due to multiscale interactions
in chaotic saddles at the onset of permanent spatiotemporal chaos
is analyzed using the Fourier-Lyapunov representation. By computing
the power-phase spectral entropy and the time-averaged power-phase
spectra, we show that the laminar (bursty) states in the on-off spatiotemporal
intermittency correspond, respectively, to the nonattracting coherent
structures with higher (lower) degrees of amplitude-phase synchronization
across spatial scales.}
}
@ARTICLE{CSRBHK07,
author = {Chian, A. C.-L. and Santana, W. M. and Rempel, E. L. and Borotto,
F. A. and Hada, T. and Kamide, Y.},
title = {Chaos in driven Alfv\'en systems: {Unstable} periodic orbits and
chaotic saddles},
journal = {Nonlinear Processes in Geophysics},
year = {2007},
volume = {14},
pages = {17--29},
abstract = { The chaotic dynamics of Alfv\'en waves in space plasmas governed
by the derivative nonlinear Schr\"odinger equation, in the low-dimensional
limit described by stationary spatial solutions, is studied. A bifurcation
diagram is constructed, by varying the driver amplitude, to identify
a number of nonlinear dynamical processes including saddle-node bifurcation,
boundary crisis, and interior crisis. The roles played by unstable
periodic orbits and chaotic saddles in these transitions are analyzed,
and the conversion from a chaotic saddle to a chaotic attractor in
these dynamical processes is demonstrated. In particular, the phenomenon
of gap-filling in the chaotic transition from weak chaos to strong
chaos via an interior crisis is investigated. A coupling unstable
periodic orbit created by an explosion, within the gaps of the chaotic
saddles embedded in a chaotic attractor following an interior crisis,
is found numerically. The gap-filling unstable periodic orbits are
responsible for coupling the banded chaotic saddle (BCS) to the surrounding
chaotic saddle (SCS), leading to crisis-induced intermittency. The
physical relevance of chaos for Alfv\'en intermittent turbulence
observed in the solar wind is discussed. },
doi = {10.5194/npg-14-17-2007}
}
@ARTICLE{CRMRF02,
author = {A. {C.-L.} Chian and E. L. Rempel and E. E. Macau and R. R. Rosa
and F. Christiansen},
title = {High-dimensional interior crisis in the {Kuramoto-Sivashinsky} equation},
journal = {Phys. Rev. E},
year = {2002},
volume = {65},
pages = {035203},
abstract = {An investigation of interior crisis of high dimensions in an extended
spatiotemporal system exemplified by the {Kuramoto-Sivashinsky} equation
is reported. It is shown that unstable periodic orbits and their
associated invariant manifolds in the Poincar\'e hyperplane can effectively
characterize the global bifurcation dynamics of high-dimensional
systems.}
}
@BOOK{Chicone2006,
title = {Ordinary Differential Equations with Applications},
publisher = {Springer},
year = {2006},
author = {C. Chicone},
address = {New York}
}
@ARTICLE{Chillingworth2000,
author = {D. Chillingworth},
title = {Generic multiparameter bifurcation from a manifold},
journal = {Dyn. Stab. Syst.},
year = {2000},
volume = {15},
pages = {101--137}
}
@ARTICLE{Chirikov79,
author = {B. V. Chirikov},
title = {A universal instability of many-dimensional oscillator system},
journal = {Phys. Rep.},
year = {1979},
volume = {263--379},
pages = {265}
}
@ARTICLE{ChoGuck99,
author = {W. G. Choe and J. Guckenheimer},
title = {Computing periodic orbits with high accuracy},
journal = {Computer Methods Appl. Mech. Engineering},
year = {1999},
volume = {170},
pages = {331--341},
abstract = {This paper introduces a new family of algorithms for computing periodic
orbits of vector fields. These global methods achieve high accuracy
with relatively coarse discretizations of periodic orbits through
the use of automatic differentiation. High degree Taylor series expansions
of trajectories are computed at mesh points. On a fixed mesh, we
construct closed curves that converge smoothly to periodic orbits
as the degree of the Taylor series expansions increase. The algorithms
have been implemented in Matlab together with the use of the automatic
differentiation code ADOL-C. Numerical tests of our codes are compared
with AUTO calculations using the Hodgkin-Huxley equations as a test
problem.},
doi = {10.1016/S0045-7825(98)00201-1}
}
@ARTICLE{ChPeCa90,
author = {{Chong}, M.~S. and {Perry}, A.~E. and {Cantwell}, B.~J.},
title = {A general classification of three-dimensional flow fields},
journal = {Phys. Fluids},
year = {1990},
volume = {2},
pages = {765-777}
}
@ARTICLE{Choss02,
author = {Chossat, P.},
title = {The reduction of equivariant dynamics to the orbit space for compact
group actions},
journal = {Acta Appl. Math.},
year = {2002},
volume = {70},
pages = {71--94}
}
@ARTICLE{Choss01,
author = {Chossat, P.},
title = {The bifurcation of heteroclinic cycles in systems of hydrodynamic
type},
journal = {Dyn. Contin. Discr. Impul. Syst., Ser. A.},
year = {2001},
volume = {8},
pages = {575--590}
}
@ARTICLE{Choss93,
author = {Chossat, P.},
title = {Forced reflectional symmetry breaking of an {O(2)}-symmetric homoclinic
tangle},
journal = {Nonlinearity},
year = {1993},
volume = {6},
pages = {723--731}
}
@BOOK{Chossat94,
title = {The {Taylor-Couette} Problem},
publisher = {Springer},
year = {1994},
author = {P. Chossat and G. Iooss},
address = {New York}
}
@BOOK{ChossLaut00,
title = {Methods in Equivariant Bifurcations and Dynamical Systems},
publisher = {World Scientific},
year = {2000},
author = {P. Chossat and R. Lauterbach},
address = {Singapore}
}
@ARTICLE{CCR,
author = {F. Christiansen and P. Cvitanovi\'c and H. H. Rugh},
title = {The spectrum of the period-doubling operator in terms of cycles},
journal = {J. Phys A},
year = {1990},
volume = {23},
pages = {L713}
}
@ARTICLE{Christiansen97,
author = {F. Christiansen and P. Cvitanovi\'{c} and V. Putkaradze},
title = {Spatiotemporal chaos in terms of unstable recurrent patterns},
journal = {Nonlinearity},
year = {1997},
volume = {10},
pages = {55--70},
note = {\arXiv{chao-dyn/9606016}}
}
@ARTICLE{Christiansen96,
author = {F. Christiansen and P. Cvitanovi\'{c} and V. Putkaradze},
title = {Hopf's last hope: spatiotemporal chaos in terms of unstable recurrent
patterns},
year = {1996},
note = {\arXiv{chao-dyn/9606016}}
}
@ARTICLE{ChPo97,
author = {F. Christiansen and A. Politi},
title = {Guidelines for the constuction of a generating partition in the standard
map},
journal = {Physica D},
year = {1997},
volume = {109},
pages = {32--41}
}
@ARTICLE{ChPo96,
author = {F. Christiansen and A. Politi},
title = {Symbolic encoding in symplectic maps},
journal = {Nonlinearity},
year = {1996},
volume = {9},
pages = {1623--1640}
}
@ARTICLE{chr95gen,
author = {F. Christiansen and A. Politi},
title = {Generating partition for the standard map},
journal = {Phys. Rev. E},
year = {1995},
volume = {51},
pages = {3811--3814}
}
@ARTICLE{ChRu97,
author = {Christiansen, F. and Rugh, H. H.},
title = {Computing {Lyapunov} spectra with continuous {Gram-Schmidt} orthonormalization},
journal = {Nonlinearity},
year = {1997},
volume = {10},
pages = {1063--1072},
abstract = {We present a continuous method for computing the Lyapunov spectrum
associated with a dynamical system specified by a set of differential
equations. We do this by introducing a stability parameter and augmenting
the dynamical system with an orthonormal k-dimensional frame and
a Lyapunov vector such that the frame is continuously Gram - Schmidt
orthonormalized and at most linear growth of the dynamical variables
is involved. We prove that the method is strongly stable when where
is the kth Lyapunov exponent in descending order and we show through
examples how the method is implemented. }
}
@ARTICLE{cima_number_2008,
author = {A. Cima},
title = {On the number of critical periods for planar polynomial systems},
journal = {Nonlinear Anal.},
year = {2008},
volume = {69},
pages = {1889--1903}
}
@ARTICLE{cima_dynamics_2006,
author = {A. Cima},
title = {Dynamics of some rational discrete dynamical systems via invariants},
journal = {Int. J. Bifur. Chaos},
year = {2006},
volume = {16},
pages = {631--646}
}
@ARTICLE{cima_relation_1998,
author = {A. Cima},
title = {On the relation between index and multiplicity},
journal = {J. London Mathematical Society},
year = {1998},
volume = {57},
pages = {757--768},
issn = {0024-6107}
}
@ARTICLE{cima_algebraic_1997,
author = {A. Cima},
title = {Algebraic properties of the {Lyapunov} and period constants},
journal = {Rocky Mountain J. of Math.},
year = {1997},
volume = {27},
pages = {471--502},
number = {2},
issn = {0035-7596}
}
@ARTICLE{cima_studying_2008,
author = {A. Cima and A. Gasull and V. Manosa},
title = {Studying discrete dynamical systems through differential equations},
journal = {J. Diff. Eqn.},
year = {2008},
volume = {244},
pages = {630--648},
issn = {0022-0396}
}
@ARTICLE{cima_periodic_2007,
author = {A. Cima and A. Gasull and F. Manosas},
title = {Periodic orbits in complex {Abel} equations},
journal = {J. Diff. Eqn.},
year = {2007},
volume = {232},
pages = {314--328},
number = {1},
issn = {0022-0396}
}
@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.}
}
@ARTICLE{CB92,
author = {R. M. Clever and F. H. Busse},
title = {Three-dimensional convection in a horizontal layer subjected to constant
shear},
journal = {J. Fluid Mech.},
year = {1992},
volume = {234},
pages = {511--527}
}
@ARTICLE{CoKrTeRo76,
author = {B. I. Cohen and J. A. Krommers and W. M. Tang and M. N. Rosenbluth},
title = {Non-linear saturation of the dissipative trapped ion mode by mode
coupling},
journal = {Nuclear Fusion},
year = {1976},
volume = {16},
pages = {971--992}
}
@ARTICLE{cohen72,
author = {B. I. Cohen and J. A. Krommes and W. M. Tang and M. N. Rosenbluth},
title = {Non-linear saturation of the dissipative trapped-ion mode by mode
coupling},
journal = {Nucl. Fusion},
year = {1972},
volume = {16},
pages = {971--991}
}
@BOOK{CoVanLo88,
title = {Handbook for Matrix Computations},
publisher = {SIAM},
year = {1988},
author = {Coleman, T. F. and Van Loan, C.},
address = {Philadelphia}
}
@ARTICLE{CEEks93,
author = {P. Collet and J. -P. Eckmann and H. Epstein and J. Stubbe},
title = {Analyticity for the {Kuramoto-Sivashinsky} equation},
journal = {Physica D},
year = {1993},
volume = {67},
pages = {321--326},
abstract = {Proved the analyticity of the solution of the KSe on a periodic interval
near the real axis and the width of the region is estimated. Numerical
calculations show that this width should not depend on the size of
the interval.}
}
@ARTICLE{CEEksgl93,
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year = {1995},
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pages = {95--118},
abstract = {The author discusses semiclassical approximations that are adapted
to given symmetry classes in quantum mechanics. Arbitrary abelian
symmetries and also rotational symmetry are treated. Semiclassical
approximations are derived for the projected propagator and energy
dependent Green's function associated with a given irreducible representation
of the symmetry group. From these they derive trace formulae, analogous
to the usual trace formula, that determine the energy levels in a
given symmetry class in terms of classical orbits.}
}
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continuous symmetry. The usual trace formula must be modified in
such cases because periodic orbits occur in continuous families,
whereas the usual trace formula requires that the periodic orbits
be isolated at a given energy. These calculations extend the results
of a previous paper, in which they considered Abelian continuous
symmetries. The most important application of the results is to systems
with full three-dimensional rotational symmetry, and they give this
case special consideration.}
}
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is obtained; it involves a sum over classical periodic orbits on
a symmetry-reduced phase space, weighted by characters of the symmetry
group. These periodic orbits correspond to trajectories on the full
phase space which are not necessarily periodic, but whose end points
are related by symmetry. Examples: the stadium billiard, a particle
in a periodic potential, the Sinai billiard, the quartic oscillator,
and the rotational spectrum of SF6.},
numpages = {14}
}
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school = {Univ. of Leicester},
year = {2007},
address = {Leicester, UK},
note = {\arXiv{nlin.CD/0706.1940}}
}
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pages = {1569--1570},
abstract = {The phenomena of spatiotermporal chaos are introduced based on both
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chaotic state. The importance of the study of small subsystems is
emphasized.}
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publisher = {Princeton Univ. Press},
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author = {P. Cvitanovi\'c},
address = {Princeton, NJ},
note = {{\tt birdtracks.eu}}
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journal = {Chaos},
year = {1992},
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pages = {61}
}
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pages = {823--826}
}
@UNPUBLISHED{CvGr12,
author = {Cvitanovi\'c, P. and Grigoriev, R. O.},
title = {Slicing a heart to keep it ticking: {Dreams Of Grand Schemes}},
note = {In preparation},
year = {2012}
}
@ARTICLE{CGV,
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journal = {Nonlinearity},
year = {1990},
volume = {3},
pages = {873},
abstract = {The conjectured universality of the Hausdorff dimension of the fractal
set formed by the set of the irrational winding parameter values
for critical circle maps is shown to follow from the universal scalings
for quadratic irrational winding numbers.}
}
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journal = {Nonlinearity},
year = {1998},
volume = {11},
pages = {1209--1232},
note = {\arXiv{chao-dyn/9712002}}
}
@UNPUBLISHED{stadium95,
author = {Cvitanovi\'c, P. and Hansen, K. T.},
title = {Symbolic dynamics and {Markov} partitions for the stadium billiard},
note = {\arXiv{chao-dyn/9502005}; {\em J. Stat. Phys.}, accepted 1996, revised
version still not resubmitted}
}
@INPROCEEDINGS{CviLip12,
author = {P. Cvitanovi\'c and D. Lippolis},
title = {Knowing when to stop: {How} noise frees us from determinism},
booktitle = {Let's Face Chaos through Nonlinear Dynamics},
year = {2012},
editor = {M. Robnik and V. G. Romanovski},
pages = {82--126},
address = {Melville, New York},
publisher = {Am. Inst. of Phys.},
note = {\arXiv{1206.5506}},
doi = {10.1063/1.4745574}
}
@MISC{CBook:appendApplic,
author = {P. Cvitanovi\'c and G. Vattay},
title = {Appendix ``{Applications}''},
year = {2011},
note = {in \refref{DasBuch}}
}
@ARTICLE{CV93,
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thermodynamical spectra },
journal = {Phys. Rev. Lett.},
year = {1993},
volume = {71},
pages = {4138-4141}
}
@ARTICLE{Vattay,
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thermodynamical spectra},
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year = {1993},
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note = {\arXiv{chao-dyn/9307012}}
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author = {P. Cvitanovi\'{c}},
title = {Chapter ``{Relativity} for cyclists''},
year = {2011},
note = {in \refref{DasBuch}}
}
@UNPUBLISHED{Cvi07,
author = {P. Cvitanovi\'{c}},
title = {Continuous symmetry reduced trace formulas},
note = {\\ \wwwcb{/$\sim$predrag/papers/trace.pdf}},
year = {2007}
}
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year = {2000},
volume = {288},
pages = {61--80},
note = {\arXiv{nlin.CD/0001034}}
}
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journal = {Physica D},
year = {1991},
volume = {51},
pages = {138}
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pages = {2729}
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@BOOK{DasBuch,
title = {Chaos: Classical and Quantum},
publisher = {Niels Bohr Inst.},
year = {2012},
author = {P. Cvitanovi\'{c} and R. Artuso and R. Mainieri and G. Tanner and
G. Vattay},
address = {Copenhagen},
note = {{\wwwcb{}}}
}
@ARTICLE{CvitaEckardt,
author = {P. Cvitanovi\'{c} and B. Eckhardt},
title = {Symmetry decomposition of chaotic dynamics},
journal = {Nonlinearity},
year = {1993},
volume = {6},
pages = {277--311},
note = {\arXiv{chao-dyn/9303016}}
}
@ARTICLE{pexp,
author = {P. Cvitanovi\'{c} and B. Eckhardt},
title = {Periodic orbit expansions for classical smooth flows},
journal = {J. Phys. A},
year = {1991},
volume = {24},
pages = {L237}
}
@ARTICLE{pre88top,
author = {P. Cvitanovi\'{c} and G. H. Gunaratne and I. Procaccia},
title = {Topological and metric properties of {H\'{e}non}-type strange attractors},
journal = {Phys. Rev. A},
year = {1988},
volume = {38},
pages = {1503}
}
@ARTICLE{hansen1d,
author = {P. Cvitanovi\'{c} and Hansen, K. T.},
title = {Bifurcation structures in maps of {H\'enon} type},
journal = {Nonlinearity},
year = {1998},
volume = {11},
pages = {1233}
}
@MISC{DasBuchMirror,
author = {P. Cvitanovi{\'c}},
title = {Chapter ``{World} in a mirror''},
year = {2011},
note = {in \refref{DasBuch}}
}
@ARTICLE{SCD07,
author = {Cvitanovi{\'c}, P. and Davidchack, R. L. and Siminos, E.},
title = {On the state space geometry of the {Kuramoto-Sivashinsky} flow in
a periodic domain},
journal = {SIAM J. Appl. Dyn. Syst.},
year = {2010},
volume = {9},
pages = {1-33},
note = {\arXiv{0709.2944}}
}
@INPROCEEDINGS{etc12,
author = {Cvitanovi{\'c}, P. and Gibson, J. F.},
title = {Geometry of state space in plane {C}ouette flow},
booktitle = {Advances in Turbulence XII},
year = {2009},
editor = {B. Eckhardt},
series = {Proc. 12th EUROMECH Eur. Turb. Conf., Marburg},
pages = {75-78},
address = {Berlin},
publisher = {Springer}
}
@ARTICLE{CviGib10,
author = {Cvitanovi{\'c}, P. and Gibson, J. F.},
title = {Geometry of turbulence in wall-bounded shear flows: {Periodic} orbits},
journal = {Phys. Scr. T},
year = {2010},
volume = {142},
pages = {014007}
}
@INPROCEEDINGS{CvitLanCrete02,
author = {P. Cvitanovi{\'c} and Y. Lan},
title = {Turbulent fields and their recurrences},
booktitle = {Proceedings of 10th International Workshop on Multiparticle Production:
Correlations and Fluctuations in QCD},
year = {2003},
editor = {N. Antoniou},
pages = {313-325},
address = {Singapore},
publisher = {World Scientific},
note = {\arXiv{nlin.CD/0308006}}
}
@INPROCEEDINGS{CvWiAv12,
author = {Cvitanovi{\'c}, P. and Willis, A. P. and Avila, M.},
title = {Revealing the state space of turbulent pipe flow by symmetry reduction},
booktitle = {Proceed. ICTAM 2012 Intern. Congr. Theor. and Appl. Mech.},
year = {2012},
editor = {Jianxiang Wang}
}
@Unpublished{atlas12,
author = "Cvitanovi\'c, P. and Borrero-Echeverry, D. and Carroll, K. and
Robbins, B. and Siminos, E.",
title = {Cartography of high-dimensional flows:
{A} visual guide to sections and slices},
year = {2012},
note = "\arXiv{1209.4915}; Chaos J., to appear.",
doi = {10.1063/1.4758309}
}
@Unpublished{ CvGr12,
author = "Cvitanovi\'c, P. and Grigoriev, R. O.",
title = "Slicing a heart to keep it ticking: {Dreams Of Grand Schemes}",
year = "2012",
note = "In preparation"
}
@ARTICLE{AlIsPo91,
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year = {1991},
volume = {86},
pages = {1149--1157},
mrnumber = {93a:58052}
}
@ARTICLE{DAlesPol90,
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year = {1990},
volume = {64},
pages = {1609}
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author = {P. Dahlqvist and G. Russberg},
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year = {1991},
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pages = {4763}
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year = {1990},
volume = {65},
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@ARTICLE{DaMuPe88,
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@ARTICLE{DaVuDel00,
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{Ginzburg-Landau} equation},
year = {2000},
volume = {57},
pages = {459--472}
}
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author = {Dangelmayr, G.},
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year = {1986},
volume = {1},
pages = {159--185},
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}
@ARTICLE{DHBEks96,
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title = {Local models of spatio-temporally complex fields},
journal = {Physica D},
year = {1996},
volume = {90},
pages = {387--407}
}
@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 = {European Physical J. B},
year = {2000},
volume = {14},
pages = {377--381}
}
@ARTICLE{DLBconv92,
author = {F. Daviaud and J. Lega and P. Berg\'{e} and P. Coullet and M. Dubois},
title = {Spatio-temporal intermittency in a 1{D} convective pattern: theoretical
model and experiments},
journal = {Physica D},
year = {1992},
volume = {55},
pages = {287--308},
abstract = {Describe the occurrence of the spatio-temporal intermittency in a
1-d convective system that shows time-independent patterns based
on the amplitude equation approach}
}
@UNPUBLISHED{Davidchack_priv,
author = {R. L. Davidchack},
note = {private communication},
year = {2007}
}
@ARTICLE{mfind,
author = {R. L. Davidchack and Y-C Lai},
title = {Efficient algorithm for detecting unstable periodic orbits in chaotic
systems},
journal = {Phys. Rev. E},
year = {1999},
volume = {60},
pages = {6172}
}
@ARTICLE{dlbd00,
author = {R. L. Davidchack and Y-C Lai and E. M. Bollt and M. Dhamala},
title = {Estimating generating partitions of chaotic systems by unstable periodic
orbits},
journal = {Phys. Rev. E},
year = {2000},
volume = {61},
pages = {1353}
}
@ARTICLE{DaGrSaYo94,
author = {S. P. Dawson and C. Grebogi and T. Sauer and J. A. Yorke},
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journal = {Phys. Rev. Lett.},
year = {1994},
volume = {73},
pages = {1927}
}
@ARTICLE{dawson_collections_1997,
author = {S. P. Dawson and A. M. Mancho},
title = {Collections of heteroclinic cycles in the {Kuramoto-Sivashinsky}
equation},
journal = {Physica D},
year = {1997},
volume = {100},
pages = {231--256},
abstract = {We study the {Kuramoto-Sivashinky} equation with periodic boundary
conditions in the case of low-dimensional behavior. We analyze the
bifurcations that occur in a six-dimensional {(6D)} approximation
of its inertial manifold. We mainly focus on the attracting and structurally
stable heteroclinic connections that arise for these parameter values.
We reanalyze the ones that were previously described via a {4D} reduction
to the center-unstable manifold {(Ambruster} et al., 1988, 1989).
We also find a parameter region for which a manifold of structurally
stable heteroclinic cycles exist. The existence of such a manifold
is responsible for an intermittent behavior which has some features
of unpredictability.},
keywords = {Attracting heteroclinic cycles,Kuramoto-Sivashinsky}
}
@ARTICLE{DeArKo04,
author = {De Ara\'ujo, G. A. and Koiller, J.},
title = {Self-propulsion of {N}-hinged ``{Animats}'' at low {Reynolds} number},
journal = {Qualitative Theory of Dynamical Systems},
year = {2004},
volume = {4},
pages = {139--167}
}
@ARTICLE{DePSWP12,
author = {De Paula, A. S. and Savi, M. A. and Wiercigroch, M.N and Pavlovskaia,
E.},
title = {Bifurcation control of a parametric pendulum},
journal = {Int. J. Bifur. Chaos},
year = {2012},
volume = {22},
pages = {1250111},
doi = {10.1142/S0218127412501118}
}
@UNPUBLISHED{Dehaye05,
author = {P.-O. Dehaye},
title = {Averages over classical compact {L}ie groups and {W}eyl characters},
year = {2005}
}
@ARTICLE{DellAnZwan91,
author = {Dell'Antonio, G. and Zwanziger, D.},
title = {Every gauge orbit passes inside the {Gribov} horizon},
journal = {Commun. Math. Phys.},
year = {1991},
volume = {138},
pages = {291-299},
doi = {10.1007/BF02099494}
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@UNPUBLISHED{DemChaos,
author = {E. Demidov},
title = {Chaotic maps},
note = {{www.ibiblio.org/e-notes}},
year = {2009}
}
@BOOK{Demidovich1967,
title = {Lectures on Stability Theory},
publisher = {Nauka},
year = {1967},
author = {B. P. Demidovich},
address = {Moscow},
note = {in Russian}
}
@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}
}
@BOOK{Dennis96,
title = {Numerical Methods for Unconstrained Optimization and Nonlinear Equations},
publisher = {SIAM},
year = {1996},
author = {Dennis, J. E. and Schnabel, R. B.},
address = {Philadelphia}
}
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note = {{\arXiv{1001.4454}}},
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problems of nonlinear science in both numerical computations and
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detect the UPOs in a non-Lyapunov way. Firstly three special techniques
are added to quantum-behaved particle swarm optimization (QPSO),
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and boundaries restriction (NCB), then the new method NCB--QPSO is
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minimization through a proper translation in a non-Lyapunov way.
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different high order UPOs of 5 classic nonlinear maps are done by
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method in detecting the UPOs, and it has the advantages of fast convergence,
high precision and robustness.},
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year = {2012},
note = {\arXiv{1208.1691}}
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@BOOK{goldstein80,
title = {Classical Mechanics},
publisher = {Wesley},
year = {1980},
author = {Goldstein, H.},
address = {Reading, MA},
edition = {2nd}
}
@BOOK{goldstein59,
title = {Classical Mechanics},
publisher = {Wesley},
year = {1959},
author = {Goldstein, H.},
address = {Reading, MA}
}
@ARTICLE{goldstein01,
author = {Goldstein, H. and Poole, C. P. and Safko, J. L.},
title = {Classical Mechanics},
year = {2001},
address = {Reading, MA},
edition = {3rd},
publisher = {Wesley}
}
@ARTICLE{golubord,
author = {J. P. Gollub},
title = {Order and disorder in fluid motion},
journal = {Proc. Natl. Acad. Sci. USA},
year = {1995},
volume = {92},
pages = {6705},
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.}
}
@BOOK{GoVanLo96,
title = {Matrix Computations},
publisher = {J. Hopkins Univ. Press},
year = {1996},
author = {Golub, G. H. and Van Loan, C. F.},
address = {Baltimore, MD}
}
@BOOK{golubI,
title = {Singularities and Groups in Bifurcation Theory, vol. I},
publisher = {Springer},
year = {1984},
author = {M. Golubitsky and D. G. Schaeffer},
address = {New York}
}
@BOOK{golubitsky2002sp,
title = {The Symmetry Perspective},
publisher = {Birkh{\"a}user},
year = {2002},
author = {Golubitsky, M. and Stewart, I.},
address = {Boston}
}
@BOOK{golubII,
title = {Singularities and Groups in Bifurcation Theory, vol. II},
publisher = {Springer},
year = {1988},
author = {M. Golubitsky and I. Stewart and D. G. Schaeffer},
address = {New York}
}
@ARTICLE{golubsym84,
author = {M. Golubitsky and J. W. Swift and Knobloch, J.},
title = {Symmetries and pattern selection in {Rayleigh-Bernard} convection},
journal = {Physica D},
year = {1984},
volume = {10},
pages = {249}
}
@ARTICLE{GoPa11,
author = {Gomez, H. and Paris, J.},
title = {Numerical simulation of asymptotic states of the damped {Kuramoto-Sivashinsky}
equation},
journal = {Phys. Rev. E},
year = {2011},
volume = {83},
pages = {046702},
doi = {10.1103/PhysRevE.83.046702}
}
@ARTICLE{Good94,
author = {Jonathan Goodman},
title = {Stability of the {Kuramoto-Sivashinsky} and related systems},
journal = {Comm. Pure Appl. Math.},
year = {1994},
volume = {47},
pages = {293--306},
doi = {10.1002/cpa.3160470304}
}
@ARTICLE{GDTR08,
author = {E. Gouillart and O. Dauchot and {J.-L.} Thiffeault and S. Roux},
title = {Open-flow mixing: {Experimental} evidence for strange eigenmodes},
journal = {Phys. Fluids},
year = {2009},
volume = {21},
pages = {023603},
note = {\arXiv{0807.1723}},
abstract = {We investigate experimentally the mixing dynamics in a channel flow
with a finite stirring region undergoing chaotic advection. We study
the homogenization of dye in two variants of an eggbeater stirring
protocol that differ in the extent of their mixing region. In the
first case, the mixing region is separated from the side walls of
the channel, while in the second it extends to the walls. For the
first case, we observe the onset of a permanent concentration pattern
that repeats over time with decaying intensity. A quantitative analysis
of the concentration field of dye confirms the convergence to a self-similar
pattern, akin to the strange eigenmodes previously observed in closed
flows. We model this phenomenon using an idealized map, where an
analysis of the mixing dynamics explains the convergence to an eigenmode.
In contrast, for the second case the presence of no-slip walls and
separation points on the frontier of the mixing region leads to non-self-similar
mixing dynamics.}
}
@BOOK{Govaerts00,
title = {Numerical Methods for Bifurcations of Dynamical Equilibria},
publisher = {SIAM},
year = {2000},
author = {W. J. F. Govaerts},
address = {Philadelphia}
}
@ARTICLE{grant67,
author = {F. C. Grant and M. R. Feix},
title = {{Fourier-Hermite} solutions of the {Vlasov} equations in the linearized
limit},
journal = {Phys. Fluids},
year = {1967},
volume = {10},
pages = {696--702}
}
@ARTICLE{GrKa85,
author = {P. Grassberger and H. Kantz},
title = {Generating partitions for the dissipative {H\'enon} map},
journal = {Phys. Lett. A},
year = {1985},
volume = {113},
pages = {235--238},
abstract = {Discuss Biham-Wenzel method in {H\'enon} map and point out its defficiency:
1. It converges to a limit cycle sometimes. 2. Two symbol sequence
converges to the same cycle.}
}
@ARTICLE{grass89,
author = {P. Grassberger and H. Kantz and U. Moenig},
title = {On the symbolic dynamics of {H\'enon} map},
journal = {J. Phys. A},
year = {1989},
volume = {22},
pages = {5217--5230},
abstract = {Discuss Biham-Wenzel method in {H\'enon} map and point out its defficiency:
1. It converges to a limit cycle sometimes. 2. Two symbol sequence
converges to the same cycle.}
}
@ARTICLE{GraPro83,
author = {Grassberger, P. and Procaccia, I.},
title = {Estimation of the {Kolmogorov} entropy from a chaotic signal},
journal = {Phys. Rev. A},
year = {1983},
volume = {28},
pages = {2591--2593}
}
@ARTICLE{greb90shad,
author = {C. Grebogi and S. M. Hammel and J. A. Yorke and T. Sauer},
title = {Shadowing of physical trajectories in chaotic dynamics: {Containment}
and refinement},
journal = {Phys. Rev. Lett.},
year = {1990},
volume = {65},
pages = {1527--1530}
}
@ARTICLE{yorke2,
author = {C. Grebogi and E. Ott and J. A. Yorke},
title = {Unstable periodic orbits and the dimensions of multifractal chaotic
attractors},
journal = {Phys. Rev. A},
year = {1988},
volume = {37},
pages = {1711--1724},
abstract = {The probability measure generated by typical chaotic orbits of a dynamical
system can have an arbitrarily fine-scaled interwoven structure of
points with different singularity scalings. Recent work has characterized
such measures via a spectrum of fractal dimension values. In this
paper we pursue the idea that the infinite number of unstable periodic
orbits embedded in the support of the measure provides the key to
an understanding of the structure of the subsets with different singularity
scalings. In particular, a formulation relating the spectrum of dimensions
to unstable periodic orbits is presented for hyperbolic maps of arbitrary
dimensionality. Both chaotic attractors and chaotic repellers are
considered.},
doi = {10.1103/PhysRevA.37.1711}
}
@ARTICLE{gree79,
author = {J. M. Greene},
title = {A method for determining a stochastic transition},
journal = {J. Math. Phys.},
year = {1979},
volume = {20},
pages = {1183--1201}
}
@ARTICLE{gree98,
author = {J. M. Greene},
title = {Two-dimensional measure-preserving mappings},
journal = {J. Math. Phys.},
year = {1968},
volume = {9},
pages = {760}
}
@ARTICLE{ksgreene88,
author = {J. M. Greene and J.-S. Kim},
title = {The steady states of the {Kuramoto-Sivashinsky} equation},
journal = {Physica D},
year = {1988},
volume = {33},
pages = {99--120},
abtract = {Use Fourier modes to discuss the energy transport from long wavelength
modes to short ones. The generation of steady states with periodic
boundary condition is studied systematically with a bifurcation analysis.
Their stability is investigated and the scaling in the limit of large
system size is presented.}
}
@ARTICLE{GreeKim87,
author = {J. M. Greene and J.-S. Kim},
title = {The calculation of {Lyapunov} spectra},
journal = {Physica D},
year = {1987},
volume = {24},
pages = {213--225}
}
@ARTICLE{GrMaViFe81,
author = {J. M. Greene and R. S. MacKay and F. Vivaldi and M. J. Feigenbaum},
title = {Universal behaviour in families of area-preserving maps},
journal = {Physica D},
year = {1981},
volume = {3},
pages = {468}
}
@BOOK{Greensite11,
title = {An introduction to the confinement problem},
publisher = {Springer},
year = {2011},
author = {Greensite, J.},
pages = {101--129},
address = {New York}
}
@BOOK{greshosani,
title = {Incompressible Flow and the Finite Element Method},
publisher = {Wiley},
year = {2000},
author = {P. M. Gresho and R. L. Sani},
address = {New York}
}
@ARTICLE{Gribov77,
author = {Gribov, V. N.},
title = {Quantization of nonabelian gauge theories},
journal = {Nucl. Phys.},
year = {1978},
volume = {B139},
pages = {1},
doi = {10.1016/0550-3213(78)90175-X}
}
@INCOLLECTION{autod89,
author = {A. Griewank},
title = {On Automatic Differentiation},
booktitle = {Mathematical Programming: Recent Developments and Applications},
publisher = {Kluwer Academic Publishers},
year = {1989},
editor = {M. Iri and K. Tanabe},
pages = {83--108},
address = {Amsterdam}
}
@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},
abstract = {If the effect of short-wave stability is small, there are no regular
shocks.}
}
@ARTICLE{Gritsun11,
author = {Gritsun, A.},
title = {Connection of periodic orbits and variability patterns of circulation
for the barotropic model of atmospheric dynamics},
journal = {Doklady Earth Sciences},
year = {2011},
volume = {438},
pages = {636--640},
abstract = {We have investigated the relationship between periodic trajectories
of barotropic atmospheric model and the modes of the model variability.
In particular, we have studied the nature of ``25 day'' mode of variability
(Branstator, 1987; Kushnir 1987). This mode arises as a first complex
empirical orthogonal function (or ``Hilbert EOF'' according to (H.
von Storch, Zwiers)) for a given system and is a dominant rotational
component of the system dynamics. It was shown that the mode structure
coincides with several least unstable periodic orbits of the system.
The phase portrait of the system in the plane of the first complex
EOF has regular shape with maximum of the probability density function
in the vicinity of these weakly unstable periodic orbits.},
doi = {10.1134/S1028334X11050035}
}
@ARTICLE{Gritsun10,
author = {Gritsun, A.},
title = {Statistical characteristics of barotropic atmospeheric system and
its unstable periodic solutions},
journal = {Doklady Earth Sciences},
year = {2010},
volume = {435},
pages = {1688--1691},
abstract = {This paper is devoted to the problem of approximating an invariant
measure and statistical characteristics of barotropic atmospheric
model with the help of its periodic trajectories. In this procedure
orbits are taken into account according to their weights defined
by the orbit instability characteristics. The method comes from the
dynamical systems theory where in several specific case (for hyperbolic
systems in particular) unstable periodic orbits define the system
invariant measure. In our study we show that the system PDF could
be reconstructed with the error less than 10\% provided that the
optimal orbit weight function is chosen.},
doi = {10.1134/S1028334X10120287},
issue = {2}
}
@ARTICLE{GrBrMa08,
author = {A. Gritsun and G. Branstator and A. Majda},
title = {Climate response of linear and quadratic functionals using the fluctuation-dissipation
theorem},
journal = {J. Atmos. Sci.},
year = {2008},
volume = {65},
pages = {2824--2841},
doi = {10.1175/2007JAS2496.1}
}
@ARTICLE{Gritsun08,
author = {Gritsun, A. S.},
title = {Unstable periodic trajectories of a barotropic model of the atmosphere},
journal = {Russian J. Numer. Analysis and Math. Modelling},
year = {2008},
volume = {23},
pages = {345--367},
abstract = {Unstable periodic trajectories of a chaotic dissipative system belong
to the attractor of the system and are its important characteristics.
Many chaotic systems have an infinite number of periodic solutions
forming the skeleton of the system attractor. This allows one to
approximate the system trajectories and statistical characteristics
by using periodic solutions. The least unstable orbits may generate
local maxima of the system state distribution functions on the attractor.
With respect to atmospheric systems this means that orbits may determine
dynamic circulation regimes and typical variability modes of the
system. In some cases, given a small number of periodic solutions,
one can describe the dynamics on the attractor of the system and
the basic statistics with sufficient precision. Thus, the information
concerning periodic trajectories of a particular dynamic system may
be very important for analysis of its behavior. A search for periodic
trajectories is reduced to the solution of a system of nonlinear
equations with respect to the initial condition of an orbit and its
period. The choice of a numerical solution method and an initial
guess is an important aspect here. In this paper we consider the
problem of the calculation of periodic trajectories for a barotropic
model of the atmosphere. Several methods for determination of periodic
orbits of the model are formulated and implemented, including the
Newton method with step suppression, the Newton method with a second
order tensor correction, the quasi-Newton method with step suppression,
the quasi-Newton method with minimization of the error functional
and approximate Hessian inversion by the LBFG scheme and the GMRES
method. A comparison of the efficiency of these numerical methods
and different choices of initial conditions is performed. Various
factors influencing the rate of convergence of the methods are considered.},
doi = {10.1515/RJNAMM.2008.021}
}
@ARTICLE{G00,
author = {S. Grossmann},
title = {The onset of shear turbulence},
journal = {Rev. Mod. Phys.},
year = {2000},
volume = {72},
pages = {603--618}
}
@ARTICLE{Guck79,
author = {Guckenheimer, J.},
title = {Sensitive dependence to initial conditions for one dimensional maps},
journal = {Commun. Math. Phys.},
year = {1979},
volume = {70},
pages = {133-160},
abstract = {This paper studies the iteration of maps of the interval which have
negative Schwarzian derivative and one critical point. The maps in
this class are classified up to topological equivalence. The equivalence
classes of maps which display sensitivity to initial conditions for
large sets of initial conditions are characterized.},
doi = {10.1007/BF01982351}
}
@BOOK{guckb,
title = {Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector
Fields},
publisher = {Springer},
year = {1983},
author = {J. Guckenheimer and P. Holmes},
address = {New York}
}
@ARTICLE{GM00aut,
author = {J. Guckenheimer and B. Meloon},
title = {Computing periodic orbits and their bifurcations with automatic differentiation},
journal = {SIAM J. Sci. Comp.},
year = {2000},
volume = {22},
pages = {951--985}
}
@BOOK{GuiSte90,
title = {Symplectic Techniques in Physics},
publisher = {Cambridge Univ. Press},
year = {1990},
author = {V. Guillemin and S. Sternberg},
address = {Cambridge}
}
@ARTICLE{GuiUri90,
author = {Guillemin, V. and Uribe, A.},
title = {Reduction and the trace formula},
journal = {J. Diff. Geom.},
year = {1990},
volume = {32},
pages = {315}
}
@ARTICLE{GuiUri89,
author = {Guillemin, V. and Uribe, A.},
title = {Circular symmetry and the trace formula},
journal = {Invent. Math.},
year = {1989},
volume = {96},
pages = {385-423},
doi = {10.1007/BF01393968}
}
@ARTICLE{GuiUri87,
author = {Guillemin, V. and Uribe, A.},
title = {Reduction, the trace formula, and semiclassical asymptotics},
journal = {Proc. Natl. Acad. Sci. USA},
year = {1987},
volume = {84},
pages = {7799--7801}
}
@BOOK{MiraGum80,
title = {Recurrances and Discrete Dynamical Systems},
publisher = {Springer},
year = {1980},
author = {I. Gumowski and C. Mira},
address = {Berlin}
}
@BOOK{gutbook,
title = {Chaos in Classical and Quantum Mechanics},
publisher = {Springer},
year = {1990},
author = {M. C. Gutzwiller},
address = {New York}
}
@ARTICLE{gut82,
author = {M. C. Gutzwiller},
title = {The quantization of a classically ergodic system},
journal = {Physica D},
year = {1982},
volume = {5},
pages = {183},
abstract = {uses Ising model to improve summations, introduces symmetrizations
of the spectra, computes the spectrum}
}
@ARTICLE{gutzwiller71,
author = {M. C. Gutzwiller},
title = {Periodic orbits and classical quantization conditions},
journal = {J. Math. Phys.},
year = {1971},
volume = {12},
pages = {343--358}
}
@ARTICLE{gutzwiller70,
author = {M. C. Gutzwiller},
title = {Energy spectrum according to classical mechanics},
journal = {J. Math. Phys.},
year = {1970},
volume = {11},
pages = {1791--1806},
number = {6}
}
@ARTICLE{gutzwiller69,
author = {M. C. Gutzwiller},
title = {Phase-integral approximation in momentum space and the bound states
of an atom. II},
journal = {J. Math. Phys.},
year = {1969},
volume = {10},
pages = {1004--1021}
}
@ARTICLE{gutzwiller67,
author = {M. C. Gutzwiller},
title = {Phase-integral approximation in momentum space and the bound states
of an atom},
journal = {J. Math. Phys.},
year = {1967},
volume = {8},
pages = {1979--2000}
}
@ARTICLE{Guyard99,
author = {F. Guyard},
title = {Gr\"obner bases, invariant theory and equivariant dynamics},
journal = {J. Symbolic Comp.},
year = {1999},
volume = {28},
pages = {275--302},
doi = {10.1006/jsco.1998.0277}
}
@ARTICLE{henon,
author = {M. H\'enon},
title = {A two-dimensional mapping with a strange attractor},
journal = {Comm. Math. Phys.},
year = {1976},
volume = {50},
pages = {69}
}
@BOOK{henonrtb2,
title = {Generating Families in the Restricted Three-Body Problem {II}. Quantitative
Study of Bifurcations},
publisher = {Springer},
year = {2001},
author = {M. H\'{e}non},
address = {New York}
}
@ARTICLE{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}
}
@BOOK{HaCh08,
title = {Nonlinear Dynamical Systems and Control: A {Lyapunov}-Based Approach},
publisher = {Princeton U. Press},
year = {2008},
author = {Haddad, W. M. and Chellaboina, V.}
}
@ARTICLE{HaShu04a,
author = {Hagiwara, R. and Shudo, A.},
title = {An algorithm to prune the area-preserving {H\'enon} map},
journal = {J. Phys. A},
year = {2004},
volume = {37},
pages = {10521},
number = {44},
abstract = {An explicit algorithm to provide the pruning front for the area-preserving
H�non map is presented. The procedure terminates within finitely
many steps when the map has hyperbolic structure. The only information
required to specify the pruning front is a bifurcation diagram of
homoclinic orbits, and it is obtained by tracking orbits from the
anti-integrable limit. The pruned region thus determined is used
to construct the Markov partition of the map, and the topological
entropy is evaluated as an application.},
doi = {10.1088/0305-4470/37/44/005}
}
@ARTICLE{HaShu04b,
author = {Hagiwara, R. and Shudo, A.},
title = {Grammatical complexity for two-dimensional maps},
journal = {J. Phys. A},
year = {2004},
volume = {37},
pages = {10545},
abstract = {We calculate the grammatical complexity of the symbol sequences generated
from the {H\'enon} map and the Lozi map using the recently developed
methods to construct the pruning front. When the map is hyperbolic,
the language of symbol sequences is regular in the sense of the Chomsky
hierarchy and the corresponding grammatical complexity takes finite
values. It is found that the complexity exhibits a self-similar structure
as a function of the system parameter, and the similarity of the
pruning fronts is discussed as an origin of such self-similarity.
For non-hyperbolic cases, it is observed that the complexity monotonically
increases as we increase the resolution of the pruning front.}
}
@ARTICLE{HaMaj10,
author = {M. Hairer and A. J. Majda},
title = {A simple framework to justify linear response theory},
journal = {Nonlinearity},
year = {2010},
volume = {23},
pages = {909},
note = {\arXiv{0909.4313}},
abstract = {The use of linear response theory for forced dissipative stochastic
dynamical systems through the fluctuation-dissipation theorem is
an attractive way to study climate change systematically among other
applications. Here, a mathematically rigorous justification of linear
response theory for forced dissipative stochastic dynamical systems
is developed. The main results are formulated in an abstract setting
and apply to suitable systems, in finite and infinite dimensions,
that are of interest in climate change science and other applications.}
}
@ARTICLE{HakenZeroLyap83,
author = {H. Haken},
title = {At least one {Lyapunov} exponent vanishes if the trajectory of an
attractor does not contain a fixed point},
journal = {Phys. Lett. A},
year = {1983},
volume = {94},
pages = {71--72},
abstract = {We treat a set of coupled ordinary nonlinear differential equations
and show that for each trajectory which belongs to an attractor (or
to its basin) and which does not contain a fixed point, at least
one Lyapunov exponent vanishes.}
}
@PHDTHESIS{HalcrowThesis,
author = {Halcrow, J.},
title = {Geometry of turbulence: An exploration of the state-space of plane
{Couette} flow},
school = {School of Physics, Georgia Inst. of Technology},
year = {2008},
address = {Atlanta},
note = {\\\wwwcb{/projects/theses.html}}
}
@ARTICLE{GHCV08,
author = {Halcrow, J. and Gibson, J. F. and Cvitanovi{\'c}, P. and Viswanath,
D.},
title = {Heteroclinic connections in plane {Couette} flow},
journal = {J. Fluid Mech.},
year = {2009},
volume = {621},
pages = {365--376},
note = {\arXiv{0808.1865}}
}
@BOOK{haleob,
title = {Ordinary Differential Equations},
publisher = {Wiley},
year = {1969},
author = {J. Hale},
address = {New York}
}
@BOOK{haledb,
title = {Dynamics and Bifurcations},
publisher = {Springer},
year = {1991},
author = {J. Hale and H. Ko\c{c}ak},
address = {New York}
}
@BOOK{jhos,
title = {Oscillations in Nonlinear Systems},
publisher = {McGraw-Hill},
year = {1963},
author = {J. K. Hale},
address = {New York}
}
@BOOK{haleinf84,
title = {An Introduction to Infinite Dimensional Systems-Geometric Theory},
publisher = {Springer},
year = {1984},
author = {J. K. Hale and L. T. Magalhaes},
address = {New York}
}
@BOOK{Hall03,
title = {Lie Groups, {Lie} Algebras, and Representations},
publisher = {Springer},
year = {2003},
author = {Hall, B. C},
address = {New York}
}
@ARTICLE{Hall94,
author = {T. Hall},
title = {The creation of horseshoes},
journal = {Nonlinearity},
year = {1994},
volume = {7},
pages = {861--924}
}
@ARTICLE{HaMe98,
author = {G. Haller and I. Mezi\'c},
title = {Reduction of three-dimensional, volume-preserving flows with symmetry},
journal = {Nonlinearity},
year = {1998},
volume = {11},
pages = {319--339}
}
@BOOK{hamer,
title = {Group Theory and Its Application to Physical Problems},
publisher = {Dover},
year = {1962},
author = {Morton Hamermesh},
address = {New York}
}
@ARTICLE{HaKiWa95,
author = {J. Hamilton and J. Kim and F. Waleffe},
title = {Regeneration mechanisms of near-wall turbulence structures},
journal = {J. Fluid Mech.},
year = {1995},
volume = {287},
pages = {317--348}
}
@PHDTHESIS{hansen,
author = {Hansen, K. T.},
title = {Symbolic Dynamics in Chaotic systems},
school = {Univ. of Oslo},
year = {1993},
address = {Blindern, N-0316 Oslo, Norway},
note = {\\\wwwcb{/projects/KTHansen/thesis}}
}
@ARTICLE{Hansen92-1,
author = {Hansen, K. T.},
title = {Symbolic dynamics. 1.~{Finite} dispersive billiards },
journal = {Nonlinearity},
year = {1993},
volume = {6},
pages = {753 -- 769}
}
@ARTICLE{Hansen92-2,
author = {Hansen, K. T.},
title = {Symbolic dynamics. 2.~{Bifurcations} in billiards and smooth potentials
},
journal = {Nonlinearity},
year = {1993},
volume = {6},
pages = {771--778}
}
@ARTICLE{hansen92,
author = {Hansen, K. T.},
title = {Remarks on the symbolic dynamics for the {H\'enon} 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\'enon} map.}
}
@ARTICLE{Hansen92b,
author = {Hansen, K. T.},
title = {Pruning of orbits in four-disk and hyperbola billiards},
journal = {Chaos},
year = {1992},
volume = {2},
pages = {71--75},
mrnumber = {93a:58053}
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are transported away from the neighborhood of the attractor, leading
to deformations which can be one to two orders of magnitude larger
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maximal Lyapunov exponent can be used as a measure of the system's
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algorithm to calculate the homoclinic tangencies in the entire phase
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title = {Topics in applied mathematics},
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@ARTICLE{KaiTom80,
author = {T. Kai and K. Tomita},
title = {Statistical mechanics of deterministic chaos},
journal = {Progr. Theor. Phys.},
year = {1980},
volume = {64},
pages = {1532--1550},
doi = {10.1143/PTP.64.1532}
}
@INPROCEEDINGS{KaCoCa02,
author = {{Kalnay}, E. and {Corazza}, M. and {Cai}, M.},
title = {Are bred vectors the same as {Lyapunov} vectors?},
booktitle = {EGS XXVII General Assembly, Nice, 21-26 April 2002},
year = {2002}
}
@ARTICLE{KaMaPaa96,
author = {T. Kapitula and S. Maier-Paape},
title = {Spatial dynamics of time-periodic solutions for the {Ginzburg-Landau}
equation},
journal = {Z. Angew. Math. Phys.},
year = {1996},
volume = {47},
pages = {265--305}
}
@ARTICLE{Karma1994,
author = {A. Karma},
title = {Electrical alternans and spiral wave breakup in cardiac tissue},
journal = {Chaos},
year = {1994},
volume = {4},
pages = {461-472},
doi = {10.1063/1.166024}
}
@ARTICLE{ks04com,
author = {A.-K. Kassam and L. N. Trefethen},
title = {Fourth-order time stepping for stiff {PDE}s},
journal = {SIAM J. Sci. Comput.},
year = {2005},
volume = {26},
pages = {1214--1233},
number = {4},
abstract = {A contour integral method is presented to evaluate accurately the
matrix functions with removable singularities.}
}
@ARTICLE{ks05com,
author = {A.-K. Kassam and L. N. Trefethen},
title = {Fourth-order time stepping for stiff {PDE}s},
journal = {SIAM J. Sci. Comput.},
year = {2005},
volume = {26},
pages = {1214--1233},
number = {4},
abstract = {A contour integral method is presented to evaluate accurately the
matrix functions with removable singularities.}
}
@ARTICLE{Kato03,
author = {S. Kato and M. Yamada},
title = {Unstable periodic solutions embedded in a shell model turbulence},
journal = {Phys. Rev. E},
year = {2003},
volume = {68},
pages = {025302},
abstract = {A UPO is found in the chaotic region of one shell turbulence model
(GOY). The scaling exponents of the structure function are calculated
to verify that the statistics are well approximated by the UPO.}
}
@BOOK{Katok95,
title = {Introduction to the Modern Theory of Dynamical Systems},
publisher = {Cambridge Univ. Press},
year = {1995},
author = {A. Katok and B. Hasselblatt},
address = {Cambridge}
}
@ARTICLE{kavousanakis_projective_2007,
author = {{M.E.} Kavousanakis and R. Erban and {A.G.} Boudouvis and {C.W.}
Gear and {I.G.} Kevrekidis},
title = {Projective and coarse projective integration for problems with continuous
symmetries},
journal = {J. Computational Physics},
year = {2007},
volume = {225},
pages = {382--407},
abstract = {Temporal integration of equations possessing continuous symmetries
(e.g. systems with translational invariance associated with traveling
solutions and scale invariance associated with self-similar solutions)
in a "co-evolving" frame (i.e. a frame which is co-traveling, co-collapsing
or co-exploding with the evolving solution) leads to improved accuracy
because of the smaller time derivative in the new spatial frame.
The slower time behavior permits the use of projective and coarse
projective integration with longer projective steps in the computation
of the time evolution of partial differential equations and multiscale
systems, respectively. These methods are also demonstrated to be
effective for systems which only approximately or asymptotically
possess continuous symmetries. The ideas of projective integration
in a co-evolving frame are illustrated on the one-dimensional, translationally
invariant Nagumo partial differential equation {(PDE).} A corresponding
kinetic Monte Carlo model, motivated from the Nagumo kinetics, is
used to illustrate the coarse-grained method. A simple, one-dimensional
diffusion problem is used to illustrate the scale invariant case.
The efficiency of projective integration in the co-evolving frame
for both the macroscopic diffusion {PDE} and for a random-walker
particle based model is again demonstrated.},
keywords = {Coarse projective {integration,Continuous} {symmetry,Dynamic} {renormalization,Multiscale}
{computation,Projective} integration}
}
@ARTICLE{KawKida01,
author = {G. Kawahara and S. Kida},
title = {Periodic motion embedded in plane {Couette} turbulence: {Regeneration}
cycle and burst},
journal = {J. Fluid Mech.},
year = {2001},
volume = {449},
pages = {291--300},
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.}
}
@INPROCEEDINGS{KK05,
author = {G. Kawahara and S. Kida and M. Nagata},
title = {Unstable periodic motion in plane {Couette} system: The skeleton
of turbulence},
booktitle = {One Hundred Years of Boundary Layer Research},
year = {2005},
publisher = {Kluwer}
}
@ARTICLE{KaSa05,
author = {Kawasaki, M. and Sasa, S.},
title = {Statistics of unstable periodic orbits of a chaotic dynamical system
with a large number of degrees of freedom},
journal = {Phys. Rev. E},
year = {2005},
volume = {72},
pages = {037202},
note = {\arXiv{9801020}}
}
@ARTICLE{Kazantsev01,
author = {Kazantsev, E.},
title = {Sensitivity of the attractor of the barotropic ocean model to external
influences: {Approach} by unstable periodic orbits},
journal = {Nonlinear Processes in Geophysics},
year = {2001},
volume = {8},
pages = {281--300},
doi = {10.5194/npg-8-281-2001},
url = {http://www.nonlin-processes-geophys.net/8/281/2001/}
}
@ARTICLE{Kazantsev01a,
author = {Kazantsev, E.},
title = {Sensitivity of attractor to external influences: {Approach} by unstable
periodic orbits},
journal = {Chaos, Solitons \&\ Fractals},
year = {2001},
volume = {12},
pages = {1989--2005},
abstract = {A description of a deterministic chaotic system in terms of unstable
periodic orbits (UPOs) is used to develop a method of an a priori
estimate of the sensitivity of statistical averages of the solution
to small external influences. This method allows us to determine
the forcing perturbation which maximizes the norm of the perturbation
of a statistical moment of the solution on the attractor. The method
was applied to the Lorenz model. The estimates of perturbations of
two statistical moments were compared with directly calculated values.
The comparison shows that some 100 UPOs are sufficient to realize
this approach and get a good accuracy. The linear approach remains
valid up to rather high norms of the forcing perturbation.},
doi = {10.1016/S0960-0779(00)00154-5}
}
@ARTICLE{Kazantsev98,
author = {Kazantsev, E.},
title = {Unstable periodic orbits and attractor of the barotropic ocean model},
journal = {Nonlinear Processes in Geophysics},
year = {1998},
volume = {5},
pages = {193--208},
doi = {10.5194/npg-5-193-1998}
}
@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}
}
@INPROCEEDINGS{Keller77,
author = {H. B. Keller},
title = {Numerical solution of bifurcation and nonlinear eigenvalue problems},
booktitle = {Applications of Bifurcation Theory},
year = {1977},
editor = {P. H. Rabinowitz},
pages = {359--384},
address = {New York},
publisher = {Academic Press}
}
@BOOK{kellbv,
title = {Numerical Methods for Two-Point Boundary-Value Problems},
publisher = {Dover},
year = {1992},
author = {H. B. Keller},
address = {New York}
}
@ARTICLE{BuKe03,
author = {{Kennel}, M.~B. and {Buhl}, M.},
title = {Estimating good discrete partitions from observed data: {Symbolic}
false nearest neighbors},
journal = {Phys. Rev. Lett.},
year = {2003},
volume = {91},
pages = {084102},
note = {\arXiv{nlin/0304054}}
}
@ARTICLE{kskent92,
author = {P. Kent and J. Elgin},
title = {Travelling-waves of the {Kuramoto-Sivashinsky} equation: {Period-multiplying}
bifurcations},
journal = {Nonlinearity},
year = {1992},
volume = {5},
pages = {899--919},
abstract = {Some properties of the KSe are discussed. A unified argument about
the connections and periodic orbits is given. In particular, the
K-bifurcation of the system is conjectured to arises in the 1:n resonances
of a fixed point.}
}
@MISC{Kerswell12,
author = {R. R. Kerswell},
title = {Misunderstanding the turbulence in a pipe},
year = {2012},
note = {In preparation}
}
@ARTICLE{Kerswell05,
author = {R. R. Kerswell},
title = {Recent progress in understanding the transition to turbulence in
a pipe},
journal = {Nonlinearity},
year = {2005},
volume = {18},
pages = {R17--R44},
abstract = {The problem of understanding the nature of fluid flow through a circular
straight pipe remains one of the oldest problems in fluid mechanics.
So far no explanation has been substantiated to rationalize the transition
process by which the steady unidirectional laminar flow state gives
way to a temporally and spatially disordered three-dimensional (turbulent)
solution as the flow rate increases. Recently, new travelling wave
solutions have been discovered which are saddle points in phase space.
These plausibly represent the lowest level in a hierarchy of spatio-temporal
periodic flow solutions which may be used to construct a cycle expansion
theory of turbulent pipe flows. We summarize this success against
the backdrop of past work and discuss its implications for future
research.}
}
@ARTICLE{KeTu06,
author = {Kerswell, R. R. and Tutty, O.R.},
title = {Recurrence of travelling waves in transitional pipe flow},
journal = {J.\ Fluid Mech.},
year = {2007},
volume = {584},
pages = {69--102},
note = {\arXiv{physics/0611009}},
abstract = { We find that travelling waves with low wall shear stresses (lower
branch solutions) are on a surface which separates initial conditions
which uneventfully relaminarise and those which lead to a turbulent
evolution. Evidence for recurrent travelling wave visits is found
in both 5D and 10D long periodic pipes but only for those travelling
waves with low-to-intermediate wall shear stress and for less than
about 10\%\ of the time in turbulent flow. Dynamical structures such
as periodic orbits need to be isolated and included in any such expansion.}
}
@ARTICLE{KNSks90,
author = {I. G. Kevrekidis and B. Nicolaenko and J. C. Scovel},
title = {Back in the saddle again: a computer assisted study of the {Kuramoto-Sivashinsky}
equation},
journal = {SIAM J. Appl. Math.},
year = {1990},
volume = {50},
pages = {760--790},
number = {3},
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.}
}
@UNPUBLISHED{Khel10,
author = {Khellat, M.R. and Mirjalili, A.},
title = {Finding patterns within perturbative approximation in {QCD} and indirect
relations},
note = {\arXiv{1012.2531}},
year = {2010}
}
@ARTICLE{Kim87,
author = {J. Kim and P. Moin and R. Moser},
title = {Turbulence statistics in fully developed channel flow at low {Reynolds}
number},
journal = {J. Fluid Mech.},
year = {1987},
volume = {177},
pages = {133--166}
}
@ARTICLE{kirby_reconstructing_1992,
author = {M. Kirby and D. Armbruster},
title = {Reconstructing phase space from {PDE} simulations},
journal = {Zeitschr. Angewandte Mathematik und Physik},
year = {1992},
volume = {43},
pages = {999--1022},
abstract = {We propose the {Karhunen-Lo\'eve} {(K-L)} decomposition as a tool
to analyze complex spatio-temporal structures in {PDE} simulations
in terms of concepts from dynamical systems theory. Taking the {Kuramoto-Sivashinsky}
equation as a model problem we discuss the {K-L} decomposition for
4 different values of its bifurcation parameter a. We distinguish
two modes of using the {K-L} decomposition: As an analytic and synthetic
tool respectively. Using the analytic mode we find unstable fixed
points and stable and unstable manifolds in a parameter regime with
structurally stable homoclinic orbits (a=17.75). Choosing the data
for a {K-L} analysis carefully by restricting them to certain burst
events, we can analyze a more complicated intermittent regime at
a=68. We establish that the spatially localized oscillations around
a so called `strange' fixed point which are considered as fore-runners
of spatially concentrated zones of turbulence are in fact created
by a very specific limit cycle (a=83.75) which, for a=87, bifurcates
into a modulated traveling wave. Using the {K-L} decomposition synthetically
by determining an optimal Galerkin system, we present evidence that
the {K-L} decomposition systematically destroys dissipation and leads
to blow up solutions.}
}
@ARTICLE{Kirwan88,
author = {Kirwan, F.},
title = {The topology of reduced phase spaces of the motion of vortices on
a sphere},
journal = {Physica D},
year = {1988},
volume = {30},
pages = {99--123}
}
@ARTICLE{KlBo011,
author = {A. Klebanoff and E. Bollt},
title = {Convergence analysis of {Davidchack and Lai's} algorithm for finding
periodic orbits},
journal = {Chaos Solit. Fract.},
year = {2001},
volume = {12},
pages = {1305--1322},
abstract = {We rigorously study a recent algorithm due to Davidchack and Lai (DL)
[Davidchack RL, Lai Y-C. Phys Rev E 1999;60(5):6172-5] for efficiently
locating complete sets of hyperbolic periodic orbits for chaotic
maps. We give theorems concerning sufficient conditions on convergence
and also describing variable sized basins of attraction of initial
seeds, thus pointing out a particularly attractive feature of the
DL-algorithm. We also point out the true role of involutary matrices
which is different from that implied by Schmelcher and Diakonos [Schmelcher
P, Diakonos FK. Phys Rev E 1998;57(3):2739-46] and propagated by
Davidchack and Lai.},
doi = {10.1016/S0960-0779(00)00099-0}
}
@ARTICLE{KlPe92,
author = {Klein, P. and Pedlosky, J.},
title = {The role of dissipation mechanisms in the nonlinear dynamics of unstable
baroclinic waves},
journal = {J. Atmos. Sci.},
year = {1992},
volume = {49},
pages = {29--48},
doi = {10.1175/1520-0469(1992)049<0029:TRODMI>2.0.CO;2}
}
@ARTICLE{KlPe86,
author = {Klein, P. and Pedlosky, J.},
title = {A numerical study of baroclinic instability at large super-criticality},
journal = {J. Atmos. Sci.},
year = {1986},
volume = {43},
pages = {1243--1262},
doi = {10.1175/1520-0469(1986)043<1263:ANSOBI>2.0.CO;2}
}
@INPROCEEDINGS{Kleiser80,
author = {L. Kleiser and U. Schuman},
title = {Treatment of incompressibility and boundary conditions in 3-{D} numerical
spectral simulations of plane channel flows},
booktitle = {Proc. 3rd GAMM Conf. Numerical Methods in Fluid Mechanics},
year = {1980},
editor = {E. Hirschel},
pages = {165--173},
address = {Viewweg, Braunschweig},
organization = {GAMM}
}
@ARTICLE{KRSR,
author = {S. J. Kline and W. C. Reynolds and F. A. Schraub and P. W. Rundstadler},
title = {The structure of turbulent boundary layers},
journal = {J. Fluid Mech.},
year = {1967},
volume = {30},
pages = {741--773}
}
@ARTICLE{KRSR67,
author = {S. J. Kline and W. C. Reynolds and F. A. Schraub and P. W. Rundstadler},
title = {The structure of turbulent boundary layers},
journal = {J. Fluid Mech.},
year = {1967},
volume = {30},
pages = {741--773}
}
@INPROCEEDINGS{knobloch_hopf_1994,
author = {Knobloch, J. and Vanderbauwhede, A.},
title = {Hopf bifurcation at k-fold resonances in equivariant reversible systems},
booktitle = {Dynamics, Bifurcation and Symmetry, New Trends and New Tools},
year = {1994},
editor = {P. Chossat},
pages = {167},
address = {Doredrecht},
publisher = {Kluwer}
}
@ARTICLE{knobloch_general_1996,
author = {Knobloch, J. and Vanderbauwhede, A.},
title = {A general reduction method for periodic solutions in conservative
and reversible systems},
journal = {J. Diff. Eqn.},
year = {1996},
volume = {8},
pages = {71}
}
@ARTICLE{knobloch_hopf_1996,
author = {Knobloch, J. and Vanderbauwhede, A.},
title = {Hopf bifurcation at k-fold resonances in conservative systems},
journal = {Progress in Nonlinear Differential Equations and Their Applications},
year = {1996},
volume = {19},
pages = {155--170}
}
@UNPUBLISHED{KoSa11,
author = {Kobayashi, M. U. and Saiki, Y.},
title = {Manifold structures of unstable periodic orbits and the appearance
of periodic windows in chaotic systems},
note = {submitted},
year = {2011}
}
@ARTICLE{Koenig97,
author = {M. Koenig},
title = {Linearization of vector fields on the orbit space of the action of
a compact {Lie} group},
journal = {Math. Proc. Cambridge Philos. Soc.},
year = {1997},
volume = {121},
pages = {401--424},
doi = {10.1017/S0305004196001314}
}
@ARTICLE{KohTak07,
author = {Koh, Y. W. and Takatsuka, K.},
title = {Finding periodic orbits of higher-dimensional flows by including
tangential components of trajectory motion},
journal = {Phys. Rev. E},
year = {2007},
volume = {76},
pages = {066205},
doi = {10.1103/PhysRevE.76.066205}
}
@ARTICLE{KoEhlMo96,
author = {Koiller, J. and Ehlers, K. and Montgomery, R.},
title = {Problems and progress in microswimming},
journal = {J. Nonlin. Sci.},
year = {1996},
volume = {6},
pages = {507--541}
}
@ARTICLE{kolm91,
author = {A. N. Kolmogorov},
title = {The local structure of turbulence in incompressible viscous fluid
for very large Reynolds numbers},
journal = {Proc. R. Soc. Lond. A},
year = {1991},
volume = {434},
pages = {9--13},
number = {1890},
abstract = {Study the distribution of velocity difference of neighboring points
based on the locally homogeneous and locally isotropic hypothesized
poperties of turbulence motion. Two more similarity hypothesis suggested
the existence of a universal distribution for different viscosity
and energy dissipation rate. The large distance asymptotic behavior
of the distribution is also deduced.}
}
@ARTICLE{kook89,
author = {H-T. Kook and J. D. Meiss},
title = {Periodic orbits for reversible symplectic mappings},
journal = {Physica D},
year = {1989},
volume = {35},
pages = {65--86},
abstract = {By constructing a 2N-dimensional symplectic map for a Langrangian
system, the orbit structure of the phase space is discussed for a
time-reversible system. Periodic orbits are classfied according to
the rotation number, symmetry, and morse index. Unstable orbits repel
other periodic orbits to form resonances where a orbit can be trapped
for long with the same rotation number. The chain of resonances with
the same commensurability forms channel which provides the communication
bridge between different resonances. The connection of the orbit
structure with the continued fraction representation of real numbers
is discussed.}
}
@ARTICLE{kooknewt,
author = {H-T Kook and J. D. Meiss},
title = {Application of {N}ewton's method to {L}agrangian mappings},
journal = {Physica D},
year = {1989},
volume = {36},
pages = {317--326},
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.}
}
@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}
}
@BOOK{SpectralPDE,
title = {Implementing Spectral Methods for Partial Differential Equations},
publisher = {Springer},
year = {2009},
author = {D. A. Kopriva},
address = {New York}
}
@ARTICLE{kostelich97,
author = {E. J. Kostelich and I. Kan and C. Grebogi and E. Ott and J. A. Yorke},
title = {Unstable dimension variability: a source of nonhyperbolicity in chaotic
systems},
journal = {Physica D},
year = {1997},
volume = {109},
pages = {81--90}
}
@ARTICLE{Kowa97,
author = {Kowalski, K.},
title = {Nonlinear dynamical systems and classical orthogonal polynomials},
journal = {J. Math. Phys.},
year = {1997},
volume = {38},
pages = {2483--2505},
note = {\arXiv{solv-int/9801018}}
}
@ARTICLE{KoRe98,
author = {K. Kowalski and J. Rembielinski},
title = {Groups and nonlinear dynamical systems. {Chaotic} dynamics on the
{SU(2)xSU(2)} group},
journal = {Chaos Solit. Fract.},
year = {1998},
volume = {9},
pages = {437--448},
note = {\arXiv{chao-dyn/9801020}}
}
@ARTICLE{KoRe96,
author = {Kowalski, K. and Rembielinski, J.},
title = {Groups and nonlinear dynamical systems. {Dynamics} on the {SU(2)}
group},
journal = {Physica D},
year = {1996},
volume = {99},
pages = {237--251},
note = {\arXiv{chao-dyn/9801019}}
}
@ARTICLE{Kras04,
author = {I. Krasikov and G. J. Rodgers and C. E. Tripp},
title = {Growing random sequences},
journal = {J. Phys. A: Math. Gen.},
year = {2004},
volume = {37},
pages = {2365--2370},
abstract = {The random sequence has been generalized both in the form of addition
and in the probability distributions. Critical parameter values are
discovered around which the system has very different behavior.}
}
@ARTICLE{krauskopf_survey_2005,
author = {B. Krauskopf and H. M. Osinga and E. J. Doedel and M. E. Henderson
and J. Guckenheimer and A. Vladimirsky and M. Dellnitz and O. Junge},
title = {A survey of methods for computing (un)stable manifolds of vector
fields},
journal = {Int. J. Bifur. Chaos},
year = {2005},
volume = {15},
pages = {763--791},
abstract = {The computation of global invariant manifolds has seen renewed interest
in recent years. We survey different approaches for computing a global
stable or unstable manifold of a vector field, where we concentrate
on the case of a two-dimensional manifold. All methods are illustrated
with the same example of the two-dimensional stable manifold of the
origin in the Lorenz system.}
}
@UNPUBLISHED{KreEck12,
author = {T. Kreilos and B. Eckhardt},
title = {Periodic orbits in plane {Couette} flow},
note = {\arXiv{1205.0347}, Chaos J. to appear.},
year = {2012}
}
@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},
month = {jul}
}
@ARTICLE{KrupaRobHetCyc97,
author = {M. Krupa},
title = {Robust Heteroclinic Cycles},
journal = {J. Nonlin. Sci.},
year = {1997},
volume = {7},
pages = {129--176},
abstract = {Examines the theoretical and applied research of robust cycles. Formation
of heteroclinic cycles in higher codimension; Stability of robust
cycles; Concept of heteroclinic cycles.}
}
@ARTICLE{Krupa90,
author = {M. Krupa},
title = {Bifurcations of relative equilibria},
journal = {{SIAM} J. Math. Anal.},
year = {1990},
volume = {21},
pages = {1453--1486}
}
@ARTICLE{kruskal62,
author = {M. Kruskal},
title = {Asymptotic theory of {H}amiltonian and other systems with all solutons
nearly periodic},
journal = {J. Math. Phys.},
year = {1962},
volume = {3},
pages = {806},
number = {4},
abstract = {Asymptotic expansion to all orders. A systematic way to construct
the adiabatic invariants.}
}
@ARTICLE{Kudry08,
author = {Kudryashov, N. A.},
title = {Solitary and periodic solutions of the generalized {Kuramoto-Sivashinsky}
equation},
journal = {Regul. Chaotic Dyn.},
year = {2008},
volume = {13},
pages = {234-238},
note = {\arXiv{1112.5707}},
doi = {10.1134/S1560354708030088}
}
@ARTICLE{Kudry08a,
author = {N. A. Kudryashov and D. I. Sinelschikov and I. L. Chernyavsky },
title = {Nonlinear evolution equations for description of perturbation in
tube},
journal = {Nonlinear Dynam.},
year = {2008},
volume = {4},
pages = {69--86}
}
@BOOK{kundu08,
title = {Fluid Mechanics},
publisher = {Academic Press},
year = {2008},
author = {P. K. Kundu and I.M. Cohen},
address = {San Diego, CA}
}
@UNPUBLISHED{KuPa09,
author = {Kuptsov, P. V. and Parlitz, U.},
title = {Strict and fussy modes splitting in the tangent space of the {Ginzburg-Landau}
equation},
note = {\arXiv{0912.2261}},
year = {2009}
}
@BOOK{kura84tur,
title = {Chemical Oscillations, Waves and Turbulence},
publisher = {Springer},
year = {1984},
author = {Y. Kuramoto},
address = {New York}
}
@ARTICLE{kuturb78,
author = {Y. Kuramoto},
title = {Diffusion-induced chaos in reaction systems},
journal = {Suppl. Progr. Theor. Phys.},
year = {1978},
volume = {64},
pages = {346--367},
abstract = {Phase turbulence and amplitude turbulence are named and distinguished
from a dynamical systems point of view. The prototyped equations
are derived.}
}
@ARTICLE{ku,
author = {Y. Kuramoto and T. Tsuzuki},
title = {Persistent propagation of concentration waves in dissipative media
far from thermal equilibrium},
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year = {1976},
volume = {55},
pages = {365--369},
abstract = {The KSe is derived through the CGLe near the bifurcation point. Some
scaling argument is used to sort out the domiant terms.}
}
@INPROCEEDINGS{KuKuSa05,
author = {Kuznetsov, S.P. and Kuznetsov, A.P. and Sataev, I.R.},
title = {Review and examples of non-{Feigenbaum} critical situations associated
with period-doubling},
booktitle = {Physics and Control, 2005 International Conference Proceedings},
year = {2005},
pages = { 610 - 615},
abstract = {We review several critical situations, linked with period-doubling
transition to chaos, which require using at least two-dimensional
maps as models representing the universality classes. Each of them
corresponds to a saddle solution of the two-dimensional generalization
of Feigenbaum-Cvitanovic equation and is characterized by a set of
distinct universal constants analogous to Feigenbaum's alpha; and
delta;. We present a number of examples (driven self-oscillators,
coupled Henon-like maps, coupled driven oscillators, coupled chaotic
self-oscillators), which manifest these types of behavior.},
doi = {10.1109/PHYCON.2005.1514057}
}
@BOOK{Kuzn04,
title = {Elements of Applied Bifurcation Theory},
publisher = {Springer},
year = {2004},
author = {Y. A. Kuznetsov},
address = {New York}
}
@ARTICLE{LaMi99,
author = {Lahme, B. and Miranda, R.},
title = {{Karhunen-Loeve} decomposition in the presence of symmetry. {I}},
journal = {IEEE Trans. Image Processing},
year = {1999},
volume = {8},
pages = {1183 - 1190},
abtract = {The Karhunen-Loeve (KL) decomposition is widely used for data which
very often exhibit some symmetry, afforded by a group action. For
a finite group, we derive an algorithm using group representation
theory to reduce the cost of determining the KL basis. We demonstrate
the technique on a Lorenz-type ODE system. For a compact group such
as tori or SO(3,R) the method also applies, and we extend results
to these cases. As a short example, we consider the circle group
S1.}
}
@ARTICLE{Lai97,
author = {Y.-C. Lai},
title = {Characterization of the natural measure by unstable periodic orbits
in nonhyperbolic chaotic systems}
}
@ARTICLE{lamb_bifurcationperiodic_2003,
author = {Lamb, J. S. W. and Melbourne, I. and Wulff, C.},
title = {Bifurcation from periodic solutions with spatiotemporal symmetry,
including resonances and mode interactions},
journal = {J. Diff. Eqn.},
year = {2003},
volume = {191},
pages = {377--407},
number = {2}
}
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author = {J. S. W. Lamb and J. A. G. Roberts},
title = {Time reversal symmetry in dynamical systems: {A} survey},
journal = {Physica D},
year = {1998},
volume = {112},
pages = {1}
}
@ARTICLE{Lan10,
author = {Y. Lan},
title = {Cycle expansions: {From} maps to turbulence},
journal = {Comm. Nonlinear Sci. and Numerical Simulation},
year = {2010},
volume = {15},
pages = {502--526},
abstract = {We present a derivation, a physical explanation and applications of
cycle expansions in different dynamical systems, ranging from simple
one-dimensional maps to turbulence in fluids. Cycle expansion is
a newly devised powerful tool for computing averages of physical
observables in nonlinear chaotic systems which combines many innovative
ideas developed in dynamical systems, such as hyperbolicity, invariant
manifolds, symbolic dynamics, measure theory and thermodynamic formalism.
The concept of cycle expansion has a deep root in theoretical physics,
bearing a close analogy to cumulant expansion in statistical physics
and effective action functional in quantum field theory, the essence
of which is to represent a physical system in a hierarchical way
by utilizing certain multiplicative structures such that the dominant
parts of physical observables are captured by compact, maneuverable
objects while minor detailed variations are described by objects
with a larger space and time scale. The technique has been successfully
applied to many low-dimensional dynamical systems and much effort
has recently been made to extend its use to spatially extended systems.
For one-dimensional systems such as the Kuramoto-Sivashinsky equation,
the method turns out to be very effective while for more complex
real-world systems including the Navier-Stokes equation, the method
is only starting to yield its first fruits and much more work is
needed to enable practical computations. However, the experience
and knowledge accumulated so far is already very useful to a large
set of research problems. Several such applications are briefly described
in what follows. As more research effort is devoted to the study
of complex dynamics of nonlinear systems, cycle expansion will undergo
a fast development and find wide applications.},
doi = {10.1016/j.cnsns.2009.04.022}
}
@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 = {\\\wwwcb{/projects/theses.html}}
}
@ARTICLE{LCC06,
author = {Y. Lan and C. Chandre and P. Cvitanovi{\'c}},
title = {Variational method for locating invariant tori},
journal = {Phys. Rev. E},
year = {2006},
volume = {74},
pages = {046206},
note = {\arXiv{nlin.CD/0508026}}
}
@ARTICLE{lanCvit07,
author = {Y. Lan and P. Cvitanovi{\'c}},
title = {Unstable recurrent patterns in {Kuramoto-Sivashinsky} dynamics},
journal = {Phys. Rev. E},
year = {2008},
volume = {78},
pages = {026208},
note = {\arXiv{0804.2474}}
}
@ARTICLE{lanVar1,
author = {Y. Lan and P. Cvitanovi{\'c}},
title = {Variational method for finding periodic orbits in a general flow},
journal = {Phys. Rev. E},
year = {2004},
volume = {69},
pages = {016217},
note = {\arXiv{nlin.CD/0308008}}
}
@ARTICLE{lanmaw03,
author = {Y. Lan and N. Garnier and P. Cvitanovi\'{c}},
title = {Stationary modulated-amplitude waves in the 1{D} complex {Ginzburg-Landau}
equation},
journal = {Physica D},
year = {2004},
volume = {188},
pages = {193--212}
}
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year = {1980},
author = {L.D. Landau and E.M. Lifshitz},
address = {Oxford}
}
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title = {Statistical Physics, Part 2},
publisher = {Pergamon Press},
year = {1980},
author = {L.D. Landau and E.M. Lifshitz},
address = {Oxford}
}
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title = {Electrodynamics of Continuous Media},
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year = {1960},
author = {L.D. Landau and E.M. Lifshitz},
address = {Oxford}
}
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title = {Fluid Mechanics},
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year = {1959},
author = {L.D. Landau and E.M. Lifshitz},
address = {Oxford}
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title = {Mechanics},
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year = {1959},
author = {L.D. Landau and E.M. Lifshitz},
address = {Oxford}
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year = {1959},
author = {L.D. Landau and E.M. Lifshitz},
address = {Oxford}
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pages = {294}
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author = {A. Lasota and M. MacKey},
address = {New York}
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author = {Laufer, M. and Orland, P.},
title = {The geometry of {Yang-Mills} orbit space on the lattice},
note = {\arXiv{1203.5134}},
year = {2012}
}
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author = {Frederic Laurent-Polz},
title = {Relative periodic orbits in point vortex systems},
note = {\arXiv{math/0401022}},
year = {2004}
}
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theory},
journal = {J. Dynam. Diff. Eq.},
year = {1997},
volume = {9},
pages = {535--560},
doi = {10.1007/BF02219397}
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title = {Physics of Continuous Matter: Exotic and Everyday Phenomena in the
Macroscopic World},
publisher = {{CRC Press}},
year = {2011},
author = {Lautrup, B.},
address = {{Boca Raton, FL}}
}
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booktitle = {Chaos},
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year = {1986},
editor = {V. Holden},
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address = {Princeton}
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publisher = {Springer},
year = {1989},
author = {D. F. Lawden},
address = {New York}
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a review},
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year = {2001},
volume = {269},
pages = {152--153}
}
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author = {F. Lenz and C. Petri and F. N. R. Koch and F. K. Diakonos and P.
Schmelcher},
title = {Evolutionary phase space in driven elliptical billiards},
journal = {New J. Physics},
year = {2009},
volume = {11},
pages = {083035},
note = {\arXiv{0904.3636}}
}
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journal = {Int. J. Bifur. Chaos},
year = {2003},
volume = {13},
pages = {1573--1577}
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title = {Covering dynamical systems: {T}wo-fold covers},
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year = {2001},
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pages = {016206}
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author = {Liao, S. J.},
title = {Chaos: a bridge from microscopic uncertainty to macroscopic randomness},
year = {2011},
note = {\arXiv{1108.4472}}
}
@MISC{Liao11a,
author = {Liao, S. J.},
title = {On the numerical simulation of propagation of micro-level uncertainty
for chaotic dynamic systems},
year = {2011},
note = {\arXiv{1109.0130}}
}
@ARTICLE{Liao09,
author = {Liao, S. J.},
title = {On the reliability of computed chaotic solutions of non-linear differential
equations},
journal = {Tellus-A},
year = {2009},
volume = {61},
pages = {550--564},
note = {\arXiv{0901.2986}}
}
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title = {Well-posedness, smooth dependence and center manifold reduction for
a semilinear hyperbolic system from laser dynamics},
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year = {2007},
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pages = {931--960}
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year = {1974},
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number = {Apr29}
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title = {Locating unstable periodic orbits: {When} adaptation integrates into
delayed feedback control},
journal = {Phys. Rev. E},
year = {2010},
volume = {82},
pages = {046214},
doi = {10.1103/PhysRevE.82.046214}
}
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title = {An introduction to symbolic dynamics and coding},
publisher = {Cambridge University Press},
year = {1995},
author = {D.A. Lind and B. Marcus},
address = {Cambridge}
}
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title = {Lie Groups for Pedestrians},
publisher = {North-Holland},
year = {1966},
author = {Lipkin , H. J.},
address = {Amsterdam}
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@PHDTHESIS{LippolisThesis,
author = {Lippolis, D.},
title = {How well can one resolve the state space of a chaotic map?},
school = {School of Physics, Georgia Inst. of Technology},
year = {2010},
address = {Atlanta},
note = {\\\wwwcb{/projects/theses.html}}
}
@UNPUBLISHED{LipCvi07,
author = {D. Lippolis and P. Cvitanovi\'c},
title = {Optimal resolution of the state space of a chaotic flow in presence
of noise},
note = {In preparation},
year = {2012}
}
@ARTICLE{LipCvi08,
author = {D. Lippolis and P. Cvitanovi\'c},
title = {How well can one resolve the state space of a chaotic map?},
journal = {Phys. Rev. Lett.},
year = {2010},
volume = {104},
pages = {014101},
note = {\arXiv{0902.4269}}
}
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in the $n$-body problem},
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year = {1997},
volume = {69},
pages = {213--275}
}
@UNPUBLISHED{liuliupego05,
author = {J.-G. Liu and J. Liu and R. L. Pego},
title = {Divorcing pressure from viscosity in incompressible {N}avier-{S}tokes
dynamics},
note = {\arXiv{math.AP/0502549}},
year = {2005}
}
@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}}
}
@ARTICLE{LoCaCoPeGo11,
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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}
}
@UNPUBLISHED{LoTh10,
author = {G. J. Lord and V. Th\"ummler},
title = {Freezing stochastic travelling waves},
note = {{\arXiv{1006.0428}}},
year = {2010}
}
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year = {1969},
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year = {1963},
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pages = {130--141}
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@ARTICLE{Lorenz63a,
author = {E. N. Lorenz},
title = {The Mechanics of vacillation},
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year = {1963},
volume = {20},
pages = {448--465},
doi = {10.1175/1520-0469(1963)020<0448:TMOV>2.0.CO;2}
}
@ARTICLE{Low58,
author = {F. E. Low},
title = {A {{Lagrangian}} formulation of the {{Boltzmann-Vlasov}} equation
for plasmas},
journal = {Proc. R. Soc. London A},
year = {1958},
volume = {248},
pages = {282--287},
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.}
}
@UNPUBLISHED{deLMeAvHo12,
author = {A. de Lozar and F. Mellibovsky and M. Avila and B. Hof},
title = {Experimental observation of the edge state in pipe flow},
note = {In preparation},
year = {2012}
}
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year = {1995},
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pages = {553--581}
}
@ARTICLE{LucDoe1993,
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pages = {2389-2410}
}
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author = {K. Lust and D. Roose},
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note = {Russian original Kharkow, 1892}
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@UNPUBLISHED{lopezLink,
author = {Vanessa L{\'o}pez},
note = {private communication}
}
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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}},
abstract = {Define relative periodic orbits and use Fourier modes to find them.},
doi = {10.1137/040618977}
}
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title = {Renormalisation in area-preserving maps},
publisher = {World Scientific},
year = {1993},
author = {MacKay, R. S.},
address = {Singapore}
}
@ARTICLE{mackmeiss87,
author = {MacKay, R. S. and Meiss, J. D. and I. C. Percival},
title = {Resonances in area preserving maps},
journal = {Physica D},
year = {1987},
volume = {27},
pages = {1}
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title = {Analysis of hyperchaotic complex {Lorenz} systems},
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year = {2008},
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pages = {1477--1494}
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author = {Mahmoud, G.l M. and Al-Kashif, M. A. and Farghaly, A. A.},
title = {Chaotic and hyperchaotic attractors of a complex nonlinear system},
journal = {J. Phys. A},
year = {2008},
volume = {41}
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author = {Mahmoud, G. M. and Bountis, T. and Al-Kashif, M. A. and Aly, S. A.},
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for detuned lasers},
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pages = {63--79}
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author = {R. Mainieri},
note = {Appendix {\em ``A brief history of chaos''}, in \refref{DasBuch}}
}
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author = {Ronnie Mainieri},
title = {Can averaged orbits be used to extract scaling functions?},
note = {\arXiv{chao-dyn/9302004}},
year = {1993}
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title = {Introduction to PDEs and Waves for the Atmosphere and Ocean},
publisher = {American Mathematical Society},
year = {2003},
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address = {New York}
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year = {2011},
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pages = {3053-3071},
doi = {10.1175/JAS-D-11-053.1}
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author = {Malige, F. and Robutel, P. and Laskar, J.},
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pages = {283--316}
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year = {1991},
volume = {24},
pages = {L1149--L1153},
abtract = {A {H}amiltonian equation is proposed for the generalized nonlinear
Schr{\"{o}}dinger equation. Soliton solutions are found and the stability
of the plane waves is investigated.}
}
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title = {The Fractal Geometry of Nature},
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year = {1982},
author = {B. B. Mandelbrot},
address = {New York}
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Dynamics And Complex Systems},
publisher = {Imperial College Press},
year = {2004},
author = {Manneville, P.}
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publisher = {Academic Press},
year = {1990},
author = {Manneville, P.},
address = {New York}
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equation},
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year = {1996},
volume = {96},
pages = {30--46},
abtract = {instability of plane waves}
}
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author = {P. {Manneville}},
title = {{Spots and turbulent domains in a model of transitional plane {Couette}
flow}},
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author = {{Manos}, T. and {Skokos}, C. and {Antonopoulos}, C.},
title = {Probing the local dynamics of periodic orbits by the generalized
alignment index {(GALI)} method},
note = {\arXiv{1103.0700}},
year = {2011}
}
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title = {Lectures on Mechanics},
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author = {Marsden, J. E.},
address = {Cambridge}
}
@BOOK{marsdenbb,
title = {The {Hopf} Bifurcation and its Applications},
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address = {New York}
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title = {Introduction to Mechanics and Symmetry},
publisher = {Springer},
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address = {New York, NY}
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@BOOK{MarsdRat94,
title = {Introduction to Mechanics and Symmetry},
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author = {J. E. Marsden and T. S. Ratiu},
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by the frequency ratio. The transport zones and partial barriers
are identified according to the ``noble ratio'' consideration.}
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author = {Martens, W.},
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journal = {JHEP},
year = {2011},
volume = {2011},
pages = {1-35},
abstract = {We calculate the two-loop matching corrections for the gauge couplings
at the Grand Unification scale in a general framework that aims at
making as few assumptions on the underlying Grand Unified Theory
(GUT) as possible. In this paper we present an intermediate result
that is general enough to be applied to the Georgi-Glashow SU(5)
as a ``toy model''. The numerical effects in this theory are found
to be larger than the current experimental uncertainty on $\alpha$s
. Furthermore, we give many technical details regarding renormalization
procedure, tadpole terms, gauge fixing and the treatment of group
theory factors, which is useful preparative work for the extension
of the calculation to supersymmetric GUTs.}
}
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author = {N. Marwan},
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author = {Mather, J. N.},
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annulus},
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year = {1982},
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pages = {457--467},
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E. and Swinney, H. L.},
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@ARTICLE{MaRi83,
author = {{Maxey}, M.~R. and {Riley}, J.~J.},
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@INPROCEEDINGS{McCordMontaldi,
author = {C. McCord AND J. Montaldi AND M. Roberts AND L. Sbano},
title = {Relative periodic orbits of symmetric {L}agrangian systems},
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year = {2004},
editor = {Freddy Dumortier and et.al.},
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@ARTICLE{mcord86,
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pages = {584--592}
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@ARTICLE{McKeon04,
author = {B. J. McKeon and J. Li and W. Jiang and J. F. Morrison and A. J.
Smits},
title = {Further observations on the mean velocity in fully-developed pipe
flow},
journal = {J. Fluid Mech.},
year = {2004},
volume = {501},
pages = {135--147}
}
@ARTICLE{McLPerlQui03,
author = {R.I. McLachlan and M. Perlmutter and G.R.W. Quispel},
title = {Lie group foliations: dynamical systems and integrators},
journal = {Future Generation Computer Systems},
year = {2003},
volume = {19},
pages = {1207 - 1219},
abstract = {Foliate systems are those which preserve some (possibly singular)
foliation of phase space, such as systems with integrals, systems
with continuous symmetries, and skew product systems. We study numerical
integrators which also preserve the foliation. The case in which
the foliation is given by the orbits of an action of a Lie group
has a particularly nice structure, which we study in detail, giving
conditions under which all foliate vector fields can be written as
the sum of a vector field tangent to the orbits and a vector field
invariant under the group action. This allows the application of
many techniques of geometric integration, including splitting methods
and Lie group integrators.},
doi = {10.1016/S0167-739X(03)00046-3}
}
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title = {Hamiltonian Structure of the Reversible Nonsemisimple 1:1 Resonance},
publisher = {Eindhoven Univ. of Technology, Dept. of Math. and Comp. Sci.},
year = {1994},
author = {J.C. van der Meer and J.A. Sanders and Vanderbauwhede, A.},
series = {Reports on applied and numerical analysis, RANA 94-02},
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title = {Dynamics of Feedback Systems},
publisher = {Wiley},
year = {1981},
author = {Mees, A. I.},
address = {New York}
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title = {Random Matrices},
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address = {New York}
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author = {J. D. Meiss},
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volume = {70},
pages = {965-988},
note = {\arXiv{0801.0883}}
}
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title = {Differential Dynamical Systems},
publisher = {SIAM},
year = {2007},
author = {J. D. Meiss},
address = {Philadelphia}
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@ARTICLE{mellibovsky11,
author = {Mellibovsky, F. and Eckhardt, B.},
title = {{Takens--Bogdanov} bifurcation of travelling-wave solutions in pipe
flow},
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year = {2011},
volume = {670},
pages = {96--129}
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convection without {Boussinesq} symmetry},
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year = {2002},
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@ARTICLE{MePrKn01,
author = {Mercader, I. and Prat, J. and Knobloch, E.},
title = {The {1:2} mode interaction in {Rayleigh B\'enard} Convection with
weakly broken midplane symmetry},
journal = {Int. J. Bifur. Chaos},
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pages = {27-41},
timestamp = {2012.03.28}
}
@BOOK{MeyerHall92,
title = {Introduction to {Hamiltonian} Dynamical Systems},
publisher = {Springer},
year = {1992},
author = {K. R. Meyer and G. R. Hall},
address = {New York}
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@BOOK{MeyerHall09,
title = {Introduction to Dynamical Systems and the {N}-body Problem},
publisher = {Springer},
year = {2009},
author = {K. R. Meyer and G. R. Hall and D. Offin},
address = {New York}
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@ARTICLE{MeWi94,
author = {I. Mezi\'c and S. Wiggins},
title = {On the integrability and perturbation of three-dimensional fluid
flows with symmetry},
journal = {J. Nonlin. Sci.},
year = {1994},
volume = {4},
pages = {157--194},
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.}
}
@ARTICLE{Michel90,
author = {D. Michelson},
title = {Elementary particles as solutions of the {Sivashinsky} equation},
journal = {Physica D},
year = {1990},
volume = {44},
pages = {502--556}
}
@ARTICLE{Mks86,
author = {D. Michelson},
title = {Steady solutions of the {Kuramoto-Sivashinsky} equation},
journal = {Physica D},
year = {1986},
volume = {19},
pages = {89--111},
abstract = {The variety of steady solution of the KSe is discussed. For large
c, there is only one odd front-like bounded solution. In decreasing
c, odd solutions with more zeros are born until finally a periodic
solution is born. Associated with the periodic solution, infinite
many tori will appear in the elliptic case and Cantor-type set of
chaotic solutions, with infinite many homoclinic odd solutions.}
}
@ARTICLE{michsiv77,
author = {D. M. Michelson and G. I. Sivashinsky},
title = {Nonlinear analysis of hydrodynamic instability in laminar flames
- {II}. Numerical experiments},
journal = {Acta Astronaut.},
year = {1977},
volume = {4},
pages = {1207--1221},
abstract = {Consider the effect of hydrodynamics instability and diffusion thermal
instability through the numerical calculation of the KSe.}
}
@INCOLLECTION{Mielke02,
author = {A. Mielke},
title = {The {Ginzburg-Landau} equation in its role as a modulation equation},
booktitle = {Handbook of Dynamical Systems, Vol. 2},
publisher = {Elsevier},
year = {2002},
editor = {B. Fiedler},
pages = {759--834}
}
@BOOK{Mielke91,
title = {{Hamiltonian and Lagrangian} Flows on Center Manifolds},
publisher = {Springer},
year = {1991},
author = {A. Mielke},
address = {New York}
}
@INCOLLECTION{MilThu88,
author = {Milnor, J. and Thurston, W.},
title = {Iterated maps of the interval},
booktitle = {Dynamical {S}ystems ({M}aryland 1986-87)},
publisher = {Springer},
year = {1988},
editor = {A. Dold and B. Eckmann},
volume = {1342},
series = {Lect. Notes Math.},
pages = {465--563},
address = {New York}
}
@UNPUBLISHED{MiPlSt11,
author = {Mini\'c, Dj. and Pleimling, M. and Staples, A. E.},
title = {On the steady state distributions for turbulence},
note = {\arXiv{1105.2941}},
year = {2011}
}
@BOOK{Mira87,
title = {Chaotic dynamics -- {From} one dimensional endomorphism to two dimen\-sional
diffeo\-morphism},
publisher = {World Scientific},
year = {1987},
author = {C. Mira},
address = {Singapore}
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title = {The proto-{Lorenz} system},
journal = {Phys. Letters A},
year = {1993},
volume = {178},
pages = {105}
}
@ARTICLE{mislor1,
author = {K. Mischaikow and M. Mrozek},
title = {Chaos in the {Lorenz} equations: {A} computer assisted proof part
{II}: {Details}},
journal = {Math. Comp.},
year = {1998},
volume = {67},
pages = {1023--1046}
}
@ARTICLE{misdis,
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title = {Construction of Symbolic Dynamics from Experimental Time Series},
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year = {1999},
volume = {82},
pages = {1144},
number = {6}
}
@ARTICLE{MiZg01,
author = {M. Misiurewicz and P. Zgliczynski},
title = {Topological entropy for multidimensional perturbations of one dimensional
maps},
journal = {Int. J. Bifur. Chaos},
year = {2001},
volume = {5},
pages = {1443--1446},
abstract = {Numerical experiments in [Christiansen97] show that a suitably chosen
Poincar e map P is essentially one dimensional and can be modeled
by a one dimensional map p. Now we build a homotopy by F ( x; w)
P (x; w) 1 ) p(x) 0) This homotopy is compact. We cannot claim that
we can use Theorem 1. 1 to estimate rigorously topological entropy
for the Poincar e map P , because even there Entropy for multidimensional
perturbations 3 is no rigorous proof that the Poincar e map P studied
numerically in [Christiansen97] exists. However, the ideas used in
the proof of Theorem 1.1, continuation of topological horseshoes
and relation between topological horseshoes and entropy, combined
with recently developed rigorous numerics for KS equations (see [ZgRi01])}
}
@ARTICLE{Mitchell12,
author = {K. A. Mitchell},
title = {Partitioning two-dimensional mixed phase spaces},
journal = {Physica D},
year = {2012},
pages = { - },
doi = {10.1016/j.physd.2012.07.004},
keywords = {Partitions}
}
@ARTICLE{MEF04,
author = {J. Moehlis and B. Eckhardt and H. Faisst},
title = {Fractal lifetimes in the transition to turbulence},
journal = {Chaos},
year = {2004},
volume = {14},
pages = {S11}
}
@ARTICLE{MFE04,
author = {J. Moehlis and H. Faisst and B. Eckhardt},
title = {A low-dimensional model for turbulent shear flows},
journal = {New J. Physics},
year = {2004},
volume = {6},
pages = {56}
}
@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. Syst.},
year = {2004},
volume = {4},
pages = {352--376}
}
@ARTICLE{MHMwind96,
author = {R. Montagne and E. Hern\'{a}ndez-Garc\'{i}a and M. S. Miguel},
title = {Winding number instability in the phase-turbulence regime of the
complex {Ginzburg-Landau} equation},
journal = {Phys. Rev. Lett.},
year = {1996},
volume = {77},
pages = {267},
number = {2},
abstract = {There exist a band of winding numbers associated with stable states
in the phase turbulent regime. The states with large winding numbers
decay to the ones with smaller winding numbers. Defect chaos happens
when the range of stable winding numbers vanishes.}
}
@ARTICLE{MoSiSo07,
author = {Montgomery, K. A. and Silber, M. and Solla, S. A.},
title = {Amplification in the auditory periphery: {The} effect of coupling
tuning mechanisms},
journal = {Phys. Rev. E},
year = {2007},
volume = {75},
pages = {051924},
abstract = {A mathematical model describing the coupling between two independent
amplification mechanisms in auditory hair cells is proposed and analyzed.
Hair cells are cells in the inner ear responsible for translating
sound-induced mechanical stimuli into an electrical signal that can
then be recorded by the auditory nerve. In nonmammals, two separate
mechanisms have been postulated to contribute to the amplification
and tuning properties of the hair cells. Models of each of these
mechanisms have been shown to be poised near a Hopf bifurcation.
Through a weakly nonlinear analysis that assumes weak periodic forcing,
weak damping, and weak coupling, the physiologically based models
of the two mechanisms are reduced to a system of two coupled amplitude
equations describing the resonant response. The predictions that
follow from an analysis of the reduced equations, as well as performance
benefits due to the coupling of the two mechanisms, are discussed
and compared with published experimental auditory nerve data.},
doi = {10.1103/PhysRevE.75.051924}
}
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author = {Moore, C.},
title = {Generalized shifts: unpredictability and undecidability in dynamical
systems},
journal = {Nonlinearity},
year = {1991},
volume = {4},
pages = {199--230}
}
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journal = {Phys. Rev. Lett.},
year = {1990},
volume = {64},
pages = {2354--2357}
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year = {1966},
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pages = {871}
}
@ARTICLE{minpack,
author = {Mor\'{e}, J. J. and Garbow, B. S. and Hillstrom, K. E. },
title = {{User Guide for MINPACK-1}},
journal = {ANL-80-74, Argonne National Laboratory},
year = {1980},
keywords = {algorithm, fitting, optimization},
url = {http://www.mcs.anl.gov/\~{}more/ANL8074a.pdf}
}
@ARTICLE{pimsimp,
author = {P. Moresco and S. P. Dawson},
title = {The {PIM}-simplex method: an extension of the {PIM}-triple method
to saddles with an arbitrary number of expanding directions},
journal = {Physica},
year = {1999},
volume = {126D},
pages = {38},
abstract = {They generilized the PIM method to treat the chaotic saddle with more
than one unstable direction by introducing a simplex to cross the
stable manifolds. Combining with one method for finding local extremum
in a simplex, the authors do essentially the same job as in the simple
PIM method.}
}
@ARTICLE{MFKM10,
author = {Morita, Y. and Fujiwara, N. and Kobayashi, M. U. and Mizuguchi, T.},
title = {Scytale decodes chaos: {A} method for estimating unstable symmetric
solutions},
journal = {Chaos},
year = {2010},
volume = {20},
pages = {013126},
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.}
}
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author = {M. V. Morkovin},
title = {On the many faces of transition},
booktitle = {Viscous Drag Reduction},
year = {1969},
editor = {C. S. Wells},
pages = {1--31},
address = {New York},
publisher = {Plenum}
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author = {J. F. Morrison and W. Jiang and B. J. McKeon and A. J. Smits},
title = {Reynolds number dependence of streamwise velocity spectra in turbulent
pipe flow},
journal = {Phys. Rev. Lett.},
year = {2002},
volume = {88},
pages = {214501}
}
@ARTICLE{Morrison04,
author = {J. F. Morrison and B. J. McKeon and W. Jiang and A. J. Smits},
title = {Scaling of the streamwise velocity component in turbulent pipe flow},
journal = {J. Fluid Mech.},
year = {2004},
volume = {508},
pages = {99--131}
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author = {Morrison, P. J.},
title = {Hamiltonian description of the ideal fluid},
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volume = {70},
pages = {467--521}
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author = {Morrison, P. J.},
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journal = {Phys. Lett. A},
year = {1980},
volume = {80},
pages = {383--386}
}
@ARTICLE{MorrGree80,
author = {Morrison, P. J. and J. M. Greene},
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Ideal Magnetohydrodynamics},
journal = {Phys. Rev. Lett.},
year = {1980},
volume = {45},
pages = {790--794},
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author = {Moser, J.},
title = {Recent developments in the theory of {Hamiltonian} systems},
journal = {SIAM Rev.},
year = {1986},
volume = {28},
pages = {459--485},
annote = {overview over KAM theory and Aubry-Mather; stability of fixed points}
}
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title = {Stable and Random Motions in Dynamical Systems},
publisher = {Princeton Univ. Press},
year = {1973},
author = {J. Moser},
number = {77},
series = {Ann. Math. Stud.},
address = {Princeton}
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@ARTICLE{moser96,
author = {J. Moser},
title = {A rapidly converging iteration method and nonlinear partial differential
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author = {D. Paz\'o and I. G. Szendro and J. M. L\'opez and M. A. Rodr\'iguez},
title = {Structure of characteristic {Lyapunov} vectors in spatiotemporal
chaos},
journal = {Phys. Rev. E},
year = {2008},
volume = {78},
pages = {016209},
abstract = {We study Lyapunov vectors (LVs) corresponding to the largest Lyapunov
exponents in systems with spatiotemporal chaos. We focus on characteristic
LVs and compare the results with backward LVs obtained via successive
Gram-Schmidt orthonormalizations. Systems of a very different nature
such as coupled-map lattices and the (continuous-time) Lorenz `96
model exhibit the same features in quantitative and qualitative terms.
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the results for chaotic systems. Our work supports the claims about
universality of our earlier results [I. G. Szendro et al., Phys.
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keywords = {chaos; Lyapunov methods; spatiotemporal phenomena; vectors}
}
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note = {\arXiv{1101.2779}}
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address = {New York}
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title = {Introduction to Dynamics},
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}
@BOOK{perkob,
title = {Differential Equations and Dynamical Systems},
publisher = {Springer},
year = {1991},
author = {L. Perko},
address = {New York}
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@ARTICLE{PePaVr08,
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mappings},
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year = {2008},
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pages = {285 -- 291}
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pages = {575--613}
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title = {Spectral Methods for Incompressible Flows},
publisher = {Springer},
year = {2002},
author = {R. Peyret},
address = {New York}
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year = {2001},
volume = {64},
pages = {026214},
number = {2},
abstract = {Change the stability of the fixed points on the Poincar\'{e} section
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note = {\arXiv{nlin.CD/0006011}}
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publisher = {Guthier-Villars},
year = {1899},
author = {H. Poincar\'e},
address = {Paris},
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}
@ARTICLE{Poinc1896,
author = {H. Poincar\'e},
title = {Sur les solutions p\'eriodiques et le principe de moindre action},
journal = {C. R. Acad. Sci. Paris},
year = {1896},
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pages = {915--918}
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title = {New Methods in Celestial Mechanics},
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year = {1992},
author = {H. Poincar\'{e}},
address = {New York}
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@ARTICLE{PoGiYaMa06,
author = {Politi, A. and Ginelli, F. and Yanchuk, S. and Maistrenko, Y.},
title = {From synchronization to {Lyapunov} exponents and back},
journal = {Physica D},
year = {2006},
volume = {224},
pages = {90},
note = {\arXiv{nlin/0605012}},
abstract = {The goal of this paper is twofold. In the first part we discuss a
general approach to determine Lyapunov exponents from ensemble- rather
than time-averages. The approach passes through the identification
of locally stable and unstable manifolds (the Lyapunov vectors),
thereby revealing an analogy with generalized synchronization. The
method is then applied to a periodically forced chaotic oscillator
to show that the modulus of the Lyapunov exponent associated to the
phase dynamics increases quadratically with the coupling strength
and it is therefore different from zero already below the onset of
phase-synchronization. The analytical calculations are carried out
for a model, the generalized special flow, that we construct as a
simplified version of the periodically forced Rossler oscillator.}
}
@ARTICLE{PoToLe98,
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title = {Lyapunov exponents from node-counting arguments},
journal = {J. Phys. IV},
year = {1998},
volume = {8},
pages = {263}
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title = {Turbulent Flows},
publisher = {Cambridge Univ. Press},
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address = {Cambridge}
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discrete Klein-Gordon type with double-quadratic on-site potential},
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of steady states in a class of lattices of nonlinear discrete Klein-Gordon
type with double-quadratic on-site potential. We derive by virtue
of the admissible condition the critical value of the coupling strength,
below which the steady states persist without bifurcations. If the
coupling coefficient passes through the critical value, some of the
steady states disappear. Meanwhile there are no new steady states
created as varies. We obtain bifurcation values of some lower-order
spatially periodic steady states by introducing the concept 'characteristic
polynomial' of periodic sequences.}
}
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pages = {847--875},
note = {\arXiv{1108.5990}}
}
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S. },
title = {Frequency locking by external forcing in systems with rotational
symmetry},
year = {2011},
note = {\arXiv{1108.5990}}
}
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author = {Rempel, E. L. and Chian, A. C.},
title = {Origin of transient and intermittent dynamics in spatiotemporal chaotic
systems},
journal = {Phys. Rev. Lett.},
year = {2007},
volume = {98},
pages = {14101}
}
@ARTICLE{ReCi07,
author = {Rempel, E. L. and Chian, A. C.},
title = {Origin of transient and intermittent dynamics in spatiotemporal chaotic
systems},
journal = {Phys. Rev. Lett.},
year = {2007},
volume = {98},
pages = {014101},
abstract = {Nonattracting chaotic sets (chaotic saddles) are shown to be responsible
for transient and intermittent dynamics in an extended system exemplified
by a nonlinear regularized long-wave equation, relevant to plasma
and fluid studies. As the driver amplitude is increased, the system
undergoes a transition from quasiperiodicity to temporal chaos, then
to spatiotemporal chaos. The resulting intermittent time series of
spatiotemporal chaos displays random switching between laminar and
bursty phases. We identify temporally and spatiotemporally chaotic
saddles which are responsible for the laminar and bursty phases,
respectively. Prior to the transition to spatiotemporal chaos, a
spatiotemporally chaotic saddle is responsible for chaotic transients
that mimic the dynamics of the post-transition attractor.}
}
@ARTICLE{ReCi05,
author = {Rempel, E. L. and Chian, A. C.},
title = {Intermittency induced by attractor-merging crisis in the {Kuramoto-Sivashinsky}
equation},
journal = {Phys. Rev. E},
year = {2005},
volume = {71},
pages = {016203},
abstract = {We characterize an attractor-merging crisis in a spatially extended
system exemplified by the Kuramoto-Sivashinsky equation. The simultaneous
collision of two coexisting chaotic attractors with an unstable periodic
orbit and its associated stable manifold occurs in the high-dimensional
phase space of the system, giving rise to a single merged chaotic
attractor. The time series of the post-crisis regime displays intermittent
behavior. The origin of this crisis-induced intermittency is elucidated
in terms of alternate switching between two chaotic saddles embedded
in the merged chaotic attractor.}
}
@ARTICLE{RCMR04,
author = {Rempel, E. L. and Chian, A. C. and Macau, E. E. and Rosa, R. R.},
title = {Analysis of chaotic saddles in high-dimensional dynamical systems:
the {Kuramoto-Sivashinsky} equation},
journal = {Chaos},
year = {2004},
volume = {14},
pages = {545--56},
abstract = {This paper presents a methodology to study the role played by nonattracting
chaotic sets called chaotic saddles in chaotic transitions of high-dimensional
dynamical systems. Our methodology is applied to the Kuramoto-Sivashinsky
equation. The paper describes a novel technique that uses the stable
manifold of a chaotic saddle to characterize the homoclinic tangency
responsible for an interior crisis, a chaotic transition that results
in the enlargement of a chaotic attractor. The numerical techniques
explained here are important to improve the understanding of the
connection between low-dimensional chaotic systems and spatiotemporal
systems which exhibit temporal chaos and spatial coherence.}
}
@ARTICLE{ReChMi07,
author = {Rempel, E. L. and Chian, A. C. and Miranda, R. A.},
title = {Chaotic saddles at the onset of intermittent spatiotemporal chaos},
journal = {Phys. Rev. E},
year = {2007},
volume = {76},
pages = {056217},
abstract = {In a recent study [Rempel and Chian, Phys. Rev. Lett. 98, 014101 (2007)],
it has been shown that nonattracting chaotic sets (chaotic saddles)
are responsible for intermittency in the regularized long-wave equation
that undergoes a transition to spatiotemporal chaos (STC) via quasiperiodicity
and temporal chaos. In the present paper, it is demonstrated that
a similar mechanism is present in the damped Kuramoto-Sivashinsky
equation. Prior to the onset of STC, a spatiotemporally chaotic saddle
coexists with a spatially regular attractor. After the transition
to STC, the chaotic saddle merges with the attractor, generating
intermittent bursts of STC that dominate the post-transition dynamics.}
}
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pages = {31 - 42},
abstract = {The Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions
and n particles admits a large group of discrete symmetries. The
fixed point sets of these symmetries naturally form invariant symplectic
manifolds that are investigated in this short note. For each k dividing
n we find k degree of freedom invariant manifolds. They represent
short wavelength solutions composed of k Fourier modes and can be
interpreted as embedded lattices with periodic boundary conditions
and only k particles. Inside these invariant manifolds other invariant
structures and exact solutions are found which represent for instance
periodic and quasi-periodic solutions and standing and travelling
waves. Similar invariant manifolds exist also in the Klein-Gordon
(KG) lattice and in the thermodynamic and continuum limits.}
}
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title = {The Fokker-Planck Equation},
publisher = {Springer},
year = {1996},
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publisher = {AMS},
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pages = {2128--2136},
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author = {Robinson, James C.},
title = {Inertial manifolds for the Kuramoto-Sivashinsky equation},
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pages = {190--193},
number = {2},
month = jan,
abstract = {A new theorem is applied to the Kuramoto-Sivashinsky equation with
L-periodic boundary conditions, proving the existence of an asymptotically
complete inertial manifold attracting all initial data. Combining
this result with a new estimate of the size of the globally absorbing
set yields an improved estimate of the dimension, $N \sim L^{2.46}$.},
file = {:Robinson-PhysLettA-184(1994)-190.pdf:PDF},
issn = {0375-9601},
keywords = {math, inertial manifold, Kuramoto-Sivashinsky equation},
timestamp = {2010.07.02},
url = {http://www.sciencedirect.com/science/article/B6TVM-46SNJY4-HX/2/985821e6b59f5c3b9f72db0ebfdc3de9}
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volume = {131},
pages = {1-31},
note = {\arXiv{0709.3143}},
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title = {Fundamentals of Group Theory: An Advanced Approach},
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year = {2012},
author = {Roman, S.},
address = {Boston},
isbn = {0817683003}
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title = {A practical method for calculating largest {Lyapunov} exponents from
small data sets},
journal = {Physica D},
year = {1993},
volume = {65},
pages = {117--134},
abstract = {Detecting the presence of chaos in a dynamical system is an important
problem that is solved by measuring the largest Lyapunov exponent.
Lyapunov exponents quantify the exponential divergence of initially
close state-space trajectories and estimate the amount of chaos in
a system. We present a new method for calculating the largest Lyapunov
exponent from an experimental time series. The method follows directly
from the definition of the largest Lyapunov exponent and is accurate
because it takes ...},
url = {http://citeseer.ist.psu.edu/328190.html}
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author = {D. Rothstein and E. Henry and J. P. Gollub},
title = {Persistent patterns in transient chaotic fluid mixing},
journal = {Nature},
year = {1999},
volume = {401},
pages = {770--772},
abstract = {Experimentally generate a 2-d periodic(or weakly turbulent) velocity
field using glycerol-water on magnets driven by electric current.
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}
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year = {2012},
volume = {45},
pages = {195101},
note = {\arXiv{1110.0766}}
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year = {2012},
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pages = {711--757},
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author = {Roweis, S. T. and Saul, L. K.},
title = {Nonlinear dimensionality reduction by locally linear embedding},
journal = {Science},
year = {2000},
volume = {290},
pages = {2323--2326},
doi = {10.1126/science.290.5500.2323},
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.}
}
@ARTICLE{rowley_reduction_2003,
author = {Rowley, C. W. and Kevrekidis, I. G. and Marsden, J. E. and Lust,
K.},
title = {Reduction and reconstruction for self-similar dynamical systems},
journal = {Nonlinearity},
year = {2003},
volume = {16},
pages = {1257--1275},
abstract = {We present a general method for analysing and numerically solving
partial differential equations with self-similar solutions. The method
employs ideas from symmetry reduction in geometric mechanics, and
involves separating the dynamics on the shape space (which determines
the overall shape of the solution) from those on the group space
(which determines the size and scale of the solution). The method
is computationally tractable as well, allowing one to compute self-similar
solutions by evolving a dynamical system to a steady state, in a
scaled reference frame where the self-similarity has been factored
out. More generally, bifurcation techniques can be used to find self-similar
solutions, and determine their behaviour as parameters in the equations
are varied. The method is given for an arbitrary Lie group, providing
equations for the dynamics on the reduced space, for reconstructing
the full dynamics and for determining the resulting scaling laws
for self-similar solutions. We illustrate the technique with a numerical
example, computing self-similar solutions of the Burgers equation}
}
@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},
journal = {Physica D},
year = {2000},
volume = {142},
pages = {1--19},
abstract = {We present a method for applying the {Karhunen-Lo\'eve} decomposition
to systems with continuous symmetry. The techniques in this paper
contribute to the general procedure of removing variables associated
with the symmetry of a problem, and related ideas have been used
in previous works both to identify coherent structures in solutions
of {PDEs,} and to derive low-order models via Galerkin projection.
The main result of this paper is to derive a simple and easily implementable
set of reconstruction equations which close the system of {ODEs}
produced by Galerkin projection. The geometric interpretation of
the method closely parallels techniques used in geometric phases
and reconstruction techniques in geometric mechanics. We apply the
method to the {Kuramoto-Sivashinsky} equation and are able to derive
accurate models of considerably lower dimension than are possible
with the traditional {Karhunen-Lo\'eve} expansion.}
}
@ARTICLE{RoMeBaSchHe09,
author = {Rowley, C. W. and Mezi\'c, I. and Bagheri, S. and Schlatter, P. and
Henningson, D. S.},
title = {Spectral analysis of nonlinear flows},
journal = {J. Fluid M.},
year = {2009},
volume = {641},
pages = {115}
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title = {On the gauge orbit space stratification: {A} review},
journal = {J. Phys.A},
year = {2002},
volume = {A35},
pages = {R1-5794},
note = {\arXiv{hep-th/0203027}},
doi = {10.1088/0305-4470/35/28/201}
}
@ARTICLE{rue04ne,
author = {D. Ruelle},
title = {Conversations on nonequilibrium physics with an extraterrestrial},
journal = {Physics Today},
year = {2004},
volume = {57},
pages = {48--53},
number = {5},
abstract = {New ideas of nonequilibrium physics are introduced. Emphasize the
SRB measure approach and energy fluctuation theorem.}
}
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author = {D. Ruelle},
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year = {1986},
volume = {44},
pages = {281--292},
abstract = {For a class of differentiable dynamical systems (called Axiom A systems)
it has been shown by Pollicott and the author that correlation functions
have Fourier transforms which are meromorphic in a strip. The poles
(or resonances) are, however, not easy to locate. This note reviews
the results which are known and discusses a simple model where the
position of resonances can be estimated. The effect of noise is also
discussed.}
}
@ARTICLE{ruelle86a,
author = {D. Ruelle},
title = {Resonances of chaotic dynamical systems},
journal = {Phys. Rev. Lett.},
year = {1986},
volume = {56},
pages = {405--407}
}
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year = {1982},
volume = {87},
pages = {287--302}
}
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author = {D. Ruelle},
title = {Ergodic theory of differentiable dynamical systems},
journal = {Publ. Math. IHES},
year = {1979},
volume = {50},
pages = {275}
}
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title = {Statistical Mechanics, Thermodynamic Formalism},
publisher = {Wesley},
year = {1978},
author = {D. Ruelle},
address = {Reading, MA}
}
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year = {1976},
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author = {D. Ruelle},
title = {Bifurcations in presence of a symmetry group},
journal = {Arch. Rational Mech. Anal.},
year = {1973},
volume = {51},
pages = {136-152}
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year = {1971},
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pages = {167}
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title = {The correlation spectrum for hyperbolic analytic maps},
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author = {M. Rumberger},
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pages = {89--99},
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title = {Lyapunov exponents on the orbit space},
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year = {2000},
volume = {7},
pages = {91--113},
doi = {10.3934/dcds.2001.7.91}
}
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author = {M. Rumberger and J. Scheurle},
title = {The orbit space method: theory and application},
booktitle = {Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems},
publisher = {Springer},
year = {2001},
editor = {B. Fiedler},
address = {New York},
note = {{\tt http://dynamics.mi.fu-berlin.de/danse/}}
}
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plasmas},
journal = {submitted},
year = {2010},
url = {cns.gatech.edu/~siminos/papers/SSL10.pdf}
}
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year = {1992},
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pages = {303--367}
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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
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@ARTICLE{sternb59,
author = {S. Sternberg},
journal = {Amer. J. Math.},
year = {1959},
volume = {81},
pages = {578}
}
@ARTICLE{sternb58,
author = {S. Sternberg},
journal = {Amer. J. Math.},
year = {1958},
volume = {80},
pages = {623}
}
@ARTICLE{sternb57,
author = {S. Sternberg},
journal = {Amer. J. Math.},
year = {1957},
volume = {79},
pages = {809}
}
@BOOK{bl,
title = {Introduction to Numerical Analysis},
publisher = {Springer},
year = {1983},
author = {J. Stoer and R. Bulirsch},
address = {New York}
}
@BOOK{StGo09,
title = {Mathematics for Physics: A Guided Tour for Graduate Students},
publisher = {Cambridge Univ. Press},
year = {2009},
author = {Stone, M. and Goldbart, P.},
address = {Cambridge}
}
@ARTICLE{SWG02,
author = {P. A. Stone and F. Waleffe and M. D. Graham},
title = {Toward a structural understanding of turbulent drag reduction: nonlinear
coherent states in viscoelastic shear flows},
journal = {Phys. Rev. Lett.},
year = {2002},
volume = {89},
pages = {20831}
}
@ARTICLE{HSGsize98,
author = {M. C. Strain and H. S. Greenside},
title = {Size-dependent transition to high-dimensional chaotic dynamics in
a two-dimensional excitable media},
journal = {Phys. Rev. Lett.},
year = {1998},
volume = {80},
pages = {2306--2309},
number = {11}
}
@INPROCEEDINGS{DDS90,
author = {D. D. Stretch},
title = {Automated pattern eduction from turbulent flow diagnostics},
booktitle = {Annual Research Briefs},
year = {1990},
pages = {145--157},
publisher = {Center for Turbulence Research, Stanford University}
}
@BOOK{strogb,
title = {Nonlinear Dynamics and Chaos},
publisher = {Perseus Publishing},
year = {2000},
author = {S. H. Strogatz},
series = {Studies in Nonlinearity},
address = {Cambridge, Mass}
}
@ARTICLE{StrMag76,
author = {Strutinskii, V. M. and Magner, A. G.},
title = {Quasiclassical theory of nuclear shell structure},
journal = {Sov. J. Particles Nucl.},
year = {1976},
volume = {7},
pages = {138--163}
}
@ARTICLE{pimstag,
author = {D. Sweet and H. E. Nusse. and J. A. Yorke},
title = {Stagger-and-Step Method: Detecting and Computing Chaotic Saddles
in Higher Dimensions},
journal = {Phys. Rev. Lett.},
year = {2001},
volume = {86},
pages = {2261},
number = {11},
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.}
}
@BOOK{rtb,
title = {Theory of orbits},
publisher = {Academic Press},
year = {1967},
author = {V. Szebehely},
address = {New York}
}
@UNPUBLISHED{szendro-2007,
author = {I. G. Szendro and D. Pazo and M. A. Rodriguez and J. M. Lopez},
title = {Spatiotemporal structure of {Lyapunov} vectors in chaotic coupled-map
lattices},
note = {\arXiv{0706.1706}},
year = {2007},
abstract = {useful list of references on Lyapunov vectors}
}
@ARTICLE{Tel2000,
author = {T. {T\'el} and G. K\'{a}rolyi and {\'{A}}. P\'{e}ntek and I. Scheuring
and Z. Toroczkai and C. Grebogi and J. Kadtke},
title = {Chaotic advection, diffusion, and reactions in open flows},
journal = {Chaos},
year = {2000},
volume = {10},
pages = {89-98},
abstract = {scatters tracers off a Von Karman vortex street, where they get stuck
behind the cylinder for a while in the mixing region}
}
@BOOK{Tabor89,
title = {Chaos and Integrability in Nonlinear Dynamics: An Introduction},
publisher = {Wiley},
year = {1989},
author = {M. Tabor},
address = {New York}
}
@ARTICLE{tajima02,
author = {S. Tajima and H. S. Greenside},
title = {Microextensive chaos of a spatially extended system},
journal = {Phys. Rev. E},
year = {2002},
volume = {66},
pages = {017205}
}
@ARTICLE{takac98,
author = {P. Tak\'{a}\v{c}},
title = {Bifurcations to invariant 2-tori for the complex {Ginzburg-Landau}
equation},
journal = {Appl. Math. Comput.},
year = {1998},
volume = {89},
pages = {241--257}
}
@ARTICLE{toricgl,
author = {P. Tak\'{a}\v{c}},
title = {Invariant 2-tori in the time-dependent {Ginzburg-Landau} equation},
journal = {Nonlinearity},
year = {1992},
volume = {5},
pages = {289--321}
}
@MISC{TaCh12,
author = {Takeuchi, K. A. and Chat\'{e}, H.},
title = {Collective {Lyapunov} modes},
year = {2012},
note = {\arXiv{1207.5571}}
}
@MISC{TaCh11,
author = {Takeuchi, K. A. and Chat\'{e}, H.},
title = {Can the inertial manifold be captured by unstable periodic orbits?},
year = {2011}
}
@ARTICLE{TaGiCh09,
author = {Takeuchi, K. A. and Ginelli, F. and Chat\'{e}, H.},
title = {Lyapunov analysis captures the collective dynamics of large chaotic
systems},
journal = {Phys. Rev. Lett.},
year = {2009},
volume = {103},
pages = {154103},
note = {\arXiv{0907.4298}},
abstract = {We show, using generic globally-coupled systems, that the collective
dynamics of large chaotic systems is encoded in their Lyapunov spectra:
most modes are typically localized on a few degrees of freedom, but
some are delocalized, acting collectively on the trajectory. For
globally-coupled maps, we show moreover a quantitative correspondence
between the collective modes and some of the so-called Perron-Frobenius
dynamics. Our results imply that the conventional definition of extensivity
must be changed as soon as collective dynamics sets in.}
}
@ARTICLE{TaGiCh11,
author = {Takeuchi, K. A. and Yang, {H.-l.} and Ginelli, F. and Radons, G.
and Chat\'{e}, H.},
title = {Hyperbolic decoupling of tangent space and effective dimension of
dissipative systems},
journal = {Phys. Rev. E},
year = {2011},
volume = {84},
pages = {046214}
}
@ARTICLE{TaSa07,
author = {Takeuchi, K. and Sano, M.},
title = {Role of unstable periodic orbits in phase transitions of coupled
map lattices},
journal = {Phys. Rev. E},
year = {2007},
volume = {75},
pages = {036201},
doi = {10.1103/PhysRevE.75.036201}
}
@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}
}
@ARTICLE{TaWi92,
author = {G. Tanner and D. Wintgen},
title = {Quantization of chaotic systems},
journal = {Chaos},
year = {1992},
volume = {2},
pages = {53}
}
@ARTICLE{temam91edd,
author = {R. Temam},
title = {Approximation of attractors, large eddy simulations and multiscale
methods},
journal = {Proc. R. Soc. Lond. A},
year = {1991},
volume = {434},
pages = {23--39},
number = {1890},
abstract = {Estimates on the dimension of the attractors and the decay rates of
Fourier mode of NSe are given. By approximation of the inertial manifold,
the interaction between the large and small eddies are derived. Such
laws open the way to numerically efficient algorithms via multiscale
methods.}
}
@BOOK{infdymnon,
title = {Infinite-Dimensional Dynamical Systems in Mechanics and Physics},
publisher = {Springer},
year = {1988},
author = {R. Temam},
address = {New York}
}
@BOOK{spfunc1,
title = {Special Functions: an Introduction to the classical Functions of
Mathematical Physics},
publisher = {Wiley},
year = {1996},
author = {N. M. Temme},
address = {New York}
}
@ARTICLE{TeHaHe10,
author = {Tempelmann, D. and Hanifi, A. and Henningson, D. S.},
title = {Spatial optimal growth in three-dimensional boundary layers},
journal = {J.\ Fluid Mech.},
year = {2010},
volume = {646},
pages = {5--37},
doi = {10.1017/S0022112009993260}
}
@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},
volume = {290},
pages = {2319-2323},
year = {2000},
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.},
}
@PHDTHESIS{Thum05,
author = {V. Th\"ummler},
title = {Numerical Analysis of the Method of Freezing Traveling Waves},
school = {Bielefeld Univ.},
year = {2005}
}
@ARTICLE{Thiffeault11,
author = {Thiffeault, J.-L. },
title = {Using multiscale norms to quantify mixing and transport},
journal = {Nonlinearity},
year = {2012},
volume = {25},
pages = {1--44},
note = {\arXiv{1105.1101}},
doi = {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},
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{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.}
}
@BOOK{ns,
title = {Numerical Partial Differential Equations},
publisher = {Springer},
year = {1995},
author = {J. W. Thomas},
address = {New York}
}
@BOOK{Tinkham,
title = {Group Theory and Quantum Mechanics},
publisher = {Dover},
year = {2003},
author = {Tinkham, M.},
address = {New York}
}
@ARTICLE{tompaid96,
author = {S. Tompaidis},
title = {Numerical Study of Invariant Sets of a Quasi-periodic Perturbation
of a Symplectic Map},
journal = {Experimental Math.},
year = {1996},
volume = {5},
pages = {211--230},
abstract = {{Newton} method is used to construct periodic orbits of longer and
longer period to approach a invariant torus with specific rotation
vector. Behavior after the break of a torus is described.}
}
@ARTICLE{ToKa78,
author = {J. Topper and T. Kawahara},
title = {Approximate equation for long nonlinear waves on a viscous fluid},
journal = {J. Phys. Soc. Japan},
year = {1978},
volume = {44},
pages = {663--666}
}
@ARTICLE{torc96,
author = {A. Torcini},
title = {Order Parameter for the Transition from Phase to Amplitude Turbulence},
journal = {Phys. Rev. Lett.},
year = {1996},
volume = {77},
pages = {1047},
number = {6},
abstract = {The maximal consersed phase gradient is introduced as an order parameter
to characterize the transition from phase to defect turbulence in
the CGLe. It has a finite value in the PT regime and decreases to
zero when the transition to defect turbulence is approached. A modified
KSe is able to reproduce the main feature of the stable waves and
to explain their origin.}
}
@ARTICLE{TFGphase97,
author = {A. Torcini and H. Frauenkron and P. Grassberger},
title = {Studies of phase turbulence in the one-dimensional complex {Ginzburg-Landau}
equation},
journal = {Phys. Rev. E},
year = {1997},
volume = {55},
pages = {5073},
abstract = {The phase gradient is used as an order parameter. Different states
are identified. In the PT region, a modified KSe rules the phase
dynamics of the CGLe.}
}
@ARTICLE{ToDe98,
author = {Toronov, V. Y. and Derbov, V. L.},
title = {Topological properties of laser phase},
journal = {J. Optical Soc. of America B},
year = {1998},
volume = {15},
pages = {1282--1290}
}
@ARTICLE{ToDe97,
author = {Toronov, V. Y. and Derbov, V. L.},
title = {Geometric phases in a ring laser},
journal = {Quantum Electronics},
year = {1997},
volume = {27},
pages = {644--648},
abstract = {An investigation is made of the geometric phases in a ring laser with
counterpropagating waves. It is shown that the frequency splitting
of the counterpropagating waves that appears in the case of radiation
pulsations or is induced by displacement of an external mirror can
be regarded as a manifestation of a geometric phase.}
}
@ARTICLE{ToDe97a,
author = {Toronov, V. Y. and Derbov, V. L.},
title = {Boundedness of attractors in the complex {Lorenz} model},
journal = {Phys. Rev. E},
year = {1997},
volume = {55},
pages = {3689--3692}
}
@ARTICLE{ToDe94,
author = {Toronov, V. Y. and Derbov, V. L.},
title = {Geometric phases in lasers and liquid flows},
journal = {Phys. Rev. E},
year = {1994},
volume = {49},
pages = {1392--1399},
abstract = {Pancharatnam's geometric phase is introduced for such nonlinear dissipative
systems as lasers and liquid flows. Two types of geometric; phases
are shown to arise in these systems: the phase induced by the inner
dynamics of the system and the one caused by the cyclic and adiabatic
variation of the system parameters. A possible generalization of
the geometric-effects theory in other dissipative systems is discussed.}
}
@ARTICLE{ToDe94a,
author = {Toronov, V. Y. and Derbov, V. L.},
title = {Geometric-phase effects in laser dynamics},
journal = {Phys. Rev. A},
year = {1994},
volume = {50},
pages = {878--881},
abstract = {We show that such phenomena of laser dynamics as mean-phase-slope
jumps and temporal phase jumps at resonance between the cavity and
spectral line frequencies are intrinsically connected with the topology
of attractors in the space of rays and can be interpreted as the
manifestations of the geometric-phase properties of the evolution
operator.},
keywords = {complex Lorenz, geometric phase, Laser, symmetry}
}
@BOOK{Townsend76,
title = {The Structure of Turbulent Shear Flows, 2nd Edn.},
publisher = {Cambridge U. Press},
year = {1976},
author = {A. A. Townsend},
address = {Cambridge}
}
@BOOK{Townsend56,
title = {The Structure of Turbulent Shear Flows},
publisher = {Cambridge Univ. Press},
year = {1956},
author = {A. A. Townsend},
address = {Cambridge}
}
@BOOK{Trefethen97,
title = {Numerical Linear Algebra},
publisher = {SIAM},
year = {1997},
author = {L. N. Trefethen and D. Bau},
address = {Philadelphia}
}
@ARTICLE{TTRD93,
author = {L. N. Trefethen and A. E. Trefethen and S. C. Reddy and T. A. Driscoll},
title = {Hydrodynamic Stability without Eigenvalues},
journal = {Science},
year = {1993},
volume = {261},
pages = {578--584}
}
@UNPUBLISHED{TrIsTa09,
author = {A. Trevisan and M. D'Isidoro and O. Talagrand},
title = {Four-dimensional variational assimilation in the unstable subspace
{(4DVar-AUS)} and the optimal subspace dimension},
note = {\arXiv{0902.2714}},
year = {2009},
abstract = { A key a priori information used in 4DVar is the knowledge of the
system's evolution equations. We propose a method for taking full
advantage of the knowledge of the system's dynamical instabilities
in order to improve the quality of the analysis. We present an algorithm,
four-dimensional variational assimilation in the unstable subspace
(4DVar-AUS), that consists in confining in this subspace the increment
of the control variable. The existence of an optimal subspace dimension
for this confinement is hypothesized. Theoretical arguments in favor
of the present approach are supported by numerical experiments in
a simple perfect non-linear model scenario. It is found that the
RMS analysis error is a function of the dimension N of the subspace
where the analysis is confined and is minimum for N approximately
equal to the dimension of the unstable and neutral manifold. For
all assimilation windows, from 1 to 5 days, 4DVar-AUS performs better
than standard 4DVar. In the presence of observational noise, the
4DVar solution, while being closer to the observations, if farther
away from the truth. The implementation of 4DVar-AUS does not require
the adjoint integration. }
}
@ARTICLE{TrePan98,
author = {Trevisan, A. and Pancotti, F.},
title = {Periodic orbits, {Lyapunov} vectors, and singular vectors in the
{L}orenz system},
journal = {J. Atmos. Sci.},
year = {1998},
volume = {55},
pages = {390},
abstract = {A periodic orbit analysis in the Lorenz system and the study of the
properties of the associated tangent linear equations are performed.
A set of vectors are found that satisfy the Oseledec (1968) theorem
and reduce to Floquet eigenvectors in the particular case of a periodic
orbit. These vectors, called Lyapunov vectors, can be considered
the generalization to aperiodic orbits of the normal modes of the
instability problem and are not necessarily mutually orthogonal.
The relation between singular vectors and Lyapunov vectors is clarified.
The mechanism responsible for super-Lyapunov growth is shown to be
related to the nonorthogonality of Lyapunov vectors. The leading
Lyapunov vectors, as defined here, as well as the asymptotic final
singular vectors, are tangent to the attractor, while the leading
initial singular vectors, in general, point away from it. Perturbations
that are on the attractor can be found in the subspace of the leading
Lyapunov vectors.}
}
@ARTICLE{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},
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{tucker1-2,
author = {W. Tucker},
title = {The {Lorentz} attractor exists},
journal = {C. R. Acad. Sci. Paris S\'er. I Math.},
year = {1999},
volume = {328},
pages = {1197--1202}
}
@ARTICLE{tuckerman89,
author = {L.S. Tuckerman},
title = {Divergence-free velocity fields in nonperiodic geometries},
journal = {J. Comput. Phys.},
year = {1989},
volume = {80},
pages = {403--441},
note = {see also \weblink{www.pmmh.espci.fr/~laurette/highlights/polar/polar.html}}
}
@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{TyZh10,
author = {Tye, S.-H. Henry and Zhang, Y.},
title = {Dual identities inside the gluon and the graviton scattering amplitudes},
journal = {JHEP},
year = {2010},
volume = {2010},
pages = {1-45}
}
@ARTICLE{Bandelow1998,
author = {U. Bandelow, L. Recke and B. Sandstede},
title = {Frequency regions for forced locking of self-pulsating multi-section
{DFB} lasers},
journal = {Opt. Commun.},
year = {1998},
volume = {147},
pages = {212--218}
}
@BOOK{Vallis06,
title = {Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale
Circulation},
publisher = {Cambridge Univ. Press},
year = {2006},
author = {G. K. Vallis},
address = {Cambridge}
}
@INPROCEEDINGS{Vanderbauwhede1989,
author = {A. Vanderbauwhede},
title = {Centre manifolds, normal forms and elementary bifurcations},
booktitle = {Dynamics Reported, Volume 2},
year = {1989},
editor = {U. Kirchgraber and H.O. Walther},
pages = {89--169},
address = {New York},
publisher = {Wiley}
}
@UNPUBLISHED{Vanderb,
author = {Vanderbauwhede, A.},
title = {On the continuation of relative periodic orbits in reversible systems},
note = {{\tt cage.ugent.be/~avdb/eng/frameseng.html}}
}
@ARTICLE{vanderbauwhede_example_2000,
author = {Vanderbauwhede, A.},
title = {An example of symmetry reduction in {H}amiltonian systems},
journal = {International Conference on Differential Equations: Berlin, Germany,
1-7 August 1999},
year = {2000}
}
@UNPUBLISHED{vanderbauwhede_short_1997,
author = {Vanderbauwhede, A.},
title = {A short tutorial on {Hamiltonian} systems and their reduction near
a periodic orbit},
year = {1997}
}
@ARTICLE{vanderbauwhede_normal_1995,
author = {Vanderbauwhede, A.},
title = {Normal Forms and Versal Unfoldings of Symplectic Linear Mappings},
journal = {World Scientific Series in Applicable Analysis},
year = {1995},
volume = {4},
pages = {685--700}
}
@ARTICLE{vanderbauwhede_topics_1994,
author = {Vanderbauwhede, A.},
title = {Topics in Bifurcation Theory and Applications {(Gerard Iooss and
Moritz Adelmeyer)}},
journal = {SIAM review.},
year = {1994},
volume = {36},
pages = {323}
}
@BOOK{vanderbauwhede_heteroclinic_1992,
title = {Heteroclinic Cycles and Periodic Orbits in Reversible Systems},
publisher = {Pitman},
year = {1992},
author = {Vanderbauwhede, A.},
pages = {250},
address = {Boston}
}
@BOOK{vanderbauwhede_local_1982,
title = {Local Bifurcation and Symmetry},
publisher = {Pitman},
year = {1982},
author = {Vanderbauwhede, A.},
address = {Boston}
}
@PHDTHESIS{vanderbauwhede_lokale_1980,
author = {Vanderbauwhede, A.},
title = {Lokale Bifurcatietheorie en Symmetrie},
school = {Rijksuniv. Gent, Fac. van de Wetenschappen},
year = {1980}
}
@ARTICLE{VaZw12,
author = {Vandersickel, N. and Zwanziger, D.},
title = {The {Gribov} problem and {QCD} dynamics},
journal = {Phys. Rep.},
year = {2012},
note = {\arXiv{1202.1491}; To appear.}
}
@ARTICLE{vp98,
author = {A. Varga and S. Pieters},
title = {Gradient-Based Approach to Solve Optimal Periodic Output Feedback
Control Problems},
journal = {Automatica},
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author = {D. Viswanath},
title = {An Extension of the {L}indstedt-{P}oincar{\'e} Algorithm for Computing
Periodic Orbits}
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author = {F. Waleffe},
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note = {Working paper, available at \verb+www.math.wisc.edu/~waleffe/ECS/sspctr90.pdf+},
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Tsingou, whose pioneer works were not mentioned in Gleick's book.}
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author = {Wang, Q.},
title = {Forward and adjoint sensitivity computation of chaotic dynamical
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note = {\arXiv{1202.5229}},
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author = {R. Wilczak},
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abstract = {Symmetry reduction by the method of slices is applied to pipe flow
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symmetries of turbulent flow states. Within the symmetry-reduced
state space, all travelling wave solutions reduce to equilibria,
and all relative periodic orbits reduce to periodic orbits. Projections
of these solutions and their unstable manifolds from their \${\textbackslash}infty\$-dimensional
symmetry-reduced state space onto suitably chosen 2- or 3-dimensional
subspaces reveal their interrelations and the role they play in organising
turbulence in wall-bounded shear flows. Visualisations of the flow
within the slice and its linearisation at equilibria enable us to
trace out the unstable manifolds, determine close recurrences, identify
connections between different travelling wave solutions, and find,
for the first time for pipe flows, relative periodic orbits that
are embedded within the chaotic attractor, which capture turbulent
dynamics at transitional Reynolds numbers.}
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note = {\arXiv{0807.5073}},
abstract = {We show, using covariant Lyapunov vectors, that the chaotic solutions
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isolated from each other. In the context of dissipative partial differential
equations, our results imply that a faithful numerical integration
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numurical results support the anlytical calculation.}
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@ARTICLE{YaRa11,
author = {Yang, {H.-l.} and Radons, G.},
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volume = {108},
pages = {154101},
abstract = {A method for determining the dimension and state space geometry of
inertial manifolds of dissipative extended dynamical systems is presented.
It works by projecting vector differences between reference states
and recurrent states onto local linear subspaces spanned by the Lyapunov
vectors. A sharp characteristic transition of the projection error
occurs as soon as the number of basis vectors is increased beyond
the inertial manifold dimension. Since the method can be applied
using standard orthogonal Lyapunov vectors, it provides a possible
way to also determine experimentally inertial manifolds and their
geometric characteristics.}
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@UNPUBLISHED{YaRa10,
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@UNPUBLISHED{YeChe11,
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abstract = {Ishii and Sands (1998 Commun. Math. Phys. 198 379-406) show the monotonicity
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the a-b parameter space. We show the monotonicity of the entropy
in the vertical direction around a = 2 and in some other directions
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for the parameters at which the Lozi family has zero entropy.}
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note = {Copyright UMI - Dissertations Publishing 2006; 104 p.; 2006; 305276304;
3231362; Zarringhalam, Kourosh; 1221684071; 9780542871733; 66569;
35020461; English; M1: Ph.D.; M3: 3231362}
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note = {{\tt citeseer.ist.psu.edu/zimmermann97global.html}},
abstract = {The complex interactions between spatial and temporal components of
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dissertation is primarily interested in studying the appearance of
global bifurcations in two specific physical examples, a laser with
an injected signal, and a catalytic reaction on a platinum surface.
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degenerate .... One of the main motivations to study these equations
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ways, resembles turbulence in fluid dynamics. Recently, different
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}
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note = {\arXiv{chao-dyn/9903020}},
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@MISC{CNS,
note = {Center for Nonlinear Science,\weblink{www.cns.gatech.edu}}
}