% Encoding: UTF-8
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approximations are derived for the projected propagator and energy
dependent Green's function associated with a given irreducible representation
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to the usual trace formula, that determine the energy levels in a
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Address = {Exeter, UK},
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@Incollection{CBook:projects,
Title = {Appendix: {Projects}},
Author = {P. Cvitanovi{\'c}},
Booktitle = {{Chaos: Classical and Quantum}},
Publisher = {Niels Bohr Inst.},
Year = {2016},
Address = {Copenhagen},
URL = {http://ChaosBook.org/paper.shtml#projects}
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@Incollection{aver_dasbuch,
Title = {Averaging},
Author = {P. Cvitanovi{\'c}},
Booktitle = {{Chaos: Classical and Quantum}},
Publisher = {Niels Bohr Inst.},
Year = {2016},
Address = {Copenhagen},
URL = {http://ChaosBook.org/paper.shtml#average}
}
@Incollection{CBcount,
Title = {Counting},
Author = {P. Cvitanovi{\'c}},
Booktitle = {{Chaos: Classical and Quantum}},
Publisher = {Niels Bohr Inst.},
Year = {2016},
Address = {Copenhagen},
URL = {http://ChaosBook.org/paper.shtml#count}
}
@Incollection{koopmania,
Title = {Implementing evolution},
Author = {P. Cvitanovi{\'c}},
Booktitle = {{Chaos: Classical and Quantum}},
Publisher = {Niels Bohr Inst.},
Year = {2016},
Address = {Copenhagen},
URL = {http://ChaosBook.org/paper.shtml#appendMeasure}
}
@Incollection{CBcontinuous,
Title = {Relativity for cyclists},
Author = {P. Cvitanovi{\'c}},
Booktitle = {{Chaos: Classical and Quantum}},
Publisher = {Niels Bohr Inst.},
Year = {2016},
Address = {Copenhagen},
Chapter = {12},
URL = {http://ChaosBook.org/paper.shtml#continuous}
}
@Incollection{CBtrace,
author = {P. Cvitanovi{\'c}},
title = {Trace formulas},
booktitle = {Chaos: Classical and Quantum},
publisher = {Niels Bohr Inst.},
year = {2017},
editor = {P. Cvitanovi{\'c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
address = {Copenhagen},
url = {http://ChaosBook.org/paper.shtml#trace},
}
@Incollection{DasBuchMirror,
Title = {World in a mirror},
Author = {P. Cvitanovi{\'c}},
Booktitle = {{Chaos: Classical and Quantum}},
Publisher = {Niels Bohr Inst.},
Year = {2016},
Address = {Copenhagen},
URL = {http://ChaosBook.org/paper.shtml#discrete}
}
@Article{SCD07,
author = {Cvitanovi{\'c}, P. and Davidchack, R. L. and Siminos, E.},
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journal = {SIAM J. Appl. Dyn. Syst.},
year = {2010},
volume = {9},
pages = {1--33},
doi = {10.1137/070705623},
}
@InProceedings{etc12,
author = {Cvitanovi{\'c}, P. and Gibson, J. F.},
title = {Geometry of state space in plane {Couette} flow},
booktitle = {{Advances in Turbulence XII}},
year = {2009},
editor = {B. Eckhardt},
series = {Proc. 12\textsuperscript{th} EUROMECH Eur. Turb. Conf., Marburg},
pages = {75--78},
address = {Berlin},
publisher = {Springer},
doi = {10.1007/978-3-642-03085-7-17},
}
@Article{CviGib10,
Title = {Geometry of turbulence in wall-bounded shear flows: {Periodic} orbits},
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Journal = {Phys. Scr. T},
Year = {2010},
Pages = {014007},
Volume = {142},
DOI = {10.1088/0031-8949/2010/T142/014007}
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@Article{SumRules,
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Year = {1998},
Pages = {1209--1232},
Volume = {11},
DOI = {10.1088/0951-7715/11/5/003}
}
@Inproceedings{CvitLanCrete02,
author = {P. Cvitanovi{\'c} and Y. Lan},
title = {Turbulent fields and their recurrences},
booktitle = {{Correlations and Fluctuations in QCD : Proceedings of 10. International Workshop on Multiparticle Production}},
year = {2003},
editor = {N. Antoniou},
isbn = {978-981-238-455-3},
pages = {313--325},
address = {Singapore},
publisher = {World Scientific},
doi = {10.1142/9789812704641_0032},
}
@Incollection{CBook:appendApplic,
Title = {Transport of vector fields},
Author = {P. Cvitanovi{\'c} and G. Vattay},
Booktitle = {{Chaos: Classical and Quantum}},
Publisher = {Niels Bohr Inst.},
Year = {2016},
Address = {Copenhagen},
URL = {http://ChaosBook.org/paper.shtml#appendApplic}
}
@Incollection{CVW96,
author = {Cvitanovi{\'c}, P. and Vattay, G. and Wirzba, A.},
title = {Quantum fluids and classical determinants},
booktitle = {Classical, Semiclassical and Quantum Dynamics in Atoms},
publisher = {Springer},
year = {1997},
editor = {H. Friedrich and B. Eckhardt},
pages = {29--62},
address = {New York},
doi = {10.1007/bfb0105968},
}
@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},
doi = {10.1017/jfm.2013.75},
}
@Unpublished{Cvi07,
Title = {Continuous symmetry reduced trace formulas},
Author = {P. Cvitanovi\'{c}},
Note = {Unpublished},
Year = {2007},
URL = {http://ChaosBook.org/~predrag/papers/Cvi07.pdf}
}
@Incollection{symb_dyn,
author = {P. Cvitanovi\'{c}},
title = {Charting the state space},
booktitle = {{Chaos: Classical and Quantum}},
publisher = {Niels Bohr Inst.},
year = {2018},
editor = {P. Cvitanovi{\'c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
address = {Copenhagen},
url = {http://ChaosBook.org/paper.shtml#knead},
}
@Book{DasBuch,
Title = {Chaos: {Classical and Quantum}},
Author = {P. Cvitanovi\'{c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
Publisher = {Niels Bohr Inst.},
Year = {2019},
Address = {Copenhagen},
URL = {http://ChaosBook.org}
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@Article{CE91,
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Journal = {J. Phys. A},
Year = {1991},
Pages = {L237--L241},
Volume = {24},
DOI = {10.1088/0305-4470/24/5/005}
}
@Article{pre88top,
Title = {Topological and metric properties of {H\'{e}non}-type strange attractors},
Author = {P. Cvitanovi\'{c} and G. H. Gunaratne and I. Procaccia},
Journal = {Phys. Rev. A},
Year = {1988},
Pages = {1503},
Volume = {38}
}
@Article{FredDet,
author = {Cvitanovi\'{c}, P. and Rosenqvist, P. E. and G. Vattay and Rugh, H. H.},
title = {A {Fredholm} determinant for semiclassical quantization},
journal = {Chaos},
year = {1993},
volume = {3},
pages = {619--636},
doi = {10.1063/1.165992},
}
@Article{PCar,
Title = {Group theory for {Feynman} diagrams in non-{Abelian} gauge theories},
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Journal = {Phys. Rev. D},
Year = {1976},
Pages = {1536--1553},
Volume = {14}
}
@Article{PlanFieldThe,
Title = {Planar perturbation expansion},
Author = {Cvitanovi\'c, P.},
Journal = {Phys. Lett. B},
Year = {1981},
Pages = {49},
Volume = {99}
}
@Article{Cvi92chaos,
Title = {Periodic orbit theory in classical and quantum mechanics},
Author = {Cvitanovi\'c, P.},
Journal = {Chaos},
Year = {1992},
Pages = {1--4},
Volume = {2},
DOI = {10.1063/1.165921}
}
@Article{atlas12,
Title = {Cartography of high-dimensional flows: {A} visual guide to sections and slices},
Author = {Cvitanovi\'c, P. and Borrero-Echeverry, D. and Carroll, K. and Robbins, B. and Siminos, E.},
Journal = {Chaos},
Year = {2012},
Pages = {047506},
Volume = {22},
DOI = {10.1063/1.4758309}
}
@Article{CE89,
author = {Cvitanovi\'c, P. and Eckhardt, B.},
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journal = {Phys. Rev. Lett.},
year = {1989},
volume = {63},
pages = {823--826},
doi = {10.1103/physrevlett.63.823},
}
@Article{CvitaEckardt,
author = {Cvitanovi\'c, P. and Eckhardt, B.},
title = {Symmetry decomposition of chaotic dynamics},
journal = {Nonlinearity},
year = {1993},
volume = {6},
pages = {277--311},
doi = {10.1088/0951-7715/6/2/008},
}
@Article{LorentzDiff,
author = {Cvitanovi\'c, P. and Eckmann, J.-P. and Gaspard, P.},
title = {Transport properties of the {Lorentz} gas in terms of periodic orbits},
journal = {Chaos Solit. Fract.},
year = {1995},
volume = {6},
pages = {113--120},
doi = {10.1016/0960-0779(95)80018-C},
}
@Article{CGS92,
Title = {Investigation of the {Lorentz} gas in terms of periodic orbits},
Author = {Cvitanovi\'c, P. and Gaspard, P. and Schreiber, T.},
Journal = {Chaos},
Year = {1992},
Pages = {85--90},
Volume = {2},
DOI = {10.1063/1.165902}
}
@Unpublished{CvGr12,
Title = {Slicing a heart to keep it ticking: {Dreams Of Grand Schemes}},
Author = {Cvitanovi\'c, P. and Grigoriev, R. O.},
Note = {In preparation},
Year = {2012}
}
@Article{CGV,
Title = {On the mode-locking universality for critical circle maps},
Author = {P. Cvitanovi\'c and G. H. Gunaratne and M. J. Vinson},
Journal = {Nonlinearity},
Year = {1990},
Pages = {873},
Volume = {3},
Abstract = {The conjectured universality of the Hausdorff dimension of the fractal
set formed by the set of the irrational winding parameter values
for critical circle maps is shown to follow from the universal scalings
for quadratic irrational winding numbers.}
}
@Inproceedings{CviLip12,
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@Incollection{Gaspard92a,
author = {Gaspard, P.},
title = {From dynamical chaos to diffusion},
booktitle = {{From Phase Transitions to Chaos: Topics in Modern Statistical Physics}},
publisher = {World Scientific},
year = {1992},
editor = {Gy{\"o}rgyi, G. and Kondor, I. and Sasv{\'a}ri, L. and T{\'e}l, T.},
pages = {322--334},
address = {Singapore},
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Publisher = {Plenum}
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Title = {{Computer Algebra Methods for Equivariant Dynamical Systems}},
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Publisher = {Springer},
Year = {2000},
Address = {New York},
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Journal = {Nature},
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Abstract = {Laser-driven accelerators, in which particles are accelerated by the
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laser, have demonstrated accelerating electric fields of hundreds
of {GV} m-1 (refs 1?3). These fields are thousands of times greater
than those achievable in conventional radio-frequency accelerators,
spurring interest in laser accelerators as compact next-generation
sources of energetic electrons and radiation. To date, however, acceleration
distances have been severely limited by the lack of a controllable
method for extending the propagation distance of the focused laser
pulse. The ensuing short acceleration distance results in low-energy
beams with 100 per cent electron energy spread, which limits potential
applications. Here we demonstrate a laser accelerator that produces
electron beams with an energy spread of a few per cent, low emittance
and increased energy (more than 109 electrons above 80 {MeV).} Our
technique involves the use of a preformed plasma density channel
to guide a relativistically intense laser, resulting in a longer
propagation distance. The results open the way for compact and tunable
high-brightness sources of electrons and radiation.},
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}
@Misc{GiKuLeTa11,
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@Phdthesis{GibsonPhD,
Title = {Dynamical-systems Models of Wall-bounded, Shear-flow Turbulence},
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School = {Cornell Univ.},
Year = {2002}
}
@Techreport{channelflow,
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Institution = {U. New Hampshire},
Year = {2013},
Note = {{\tt {Channelflow.org}}},
Publisher = {J. F. Gibson},
URL = {http://Channelflow.org}
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@Techreport{GibsonMovies,
Title = {Movies of plane {Couette}},
Author = {Gibson, J. F. and Cvitanovi{\'c}, P.},
Institution = {Georgia Inst. of Technology},
Year = {2015},
URL = {http://ChaosBook.org/tutorials}
}
@Article{GHCW07,
Title = {Visualizing the geometry of state-space in plane {Couette} flow},
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Journal = {J. Fluid Mech.},
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@Article{HGC08,
Title = {Equilibrium and traveling-wave solutions of plane {Couette} flow},
Author = {J. F. Gibson and J. Halcrow and P. Cvitanovi{\'c}},
Journal = {J. Fluid Mech.},
Year = {2009},
Pages = {243--266},
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DOI = {10.1017/S0022112009990863}
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Title = {{Stretch, Twist, Fold: the Fast Dynamo}},
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Year = {1995},
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Title = {Lie Groups, {L}ie Algebras, and some of their Applications},
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@Book{gilmore08,
Title = {Lie Groups, Physics, and Geometry},
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Publisher = {Cambridge Univ. Press},
Year = {2008},
Address = {Cambridge}
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@Article{GGJLF10,
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@Book{gilmore2003,
Title = {The Topology of Chaos},
Author = {R. Gilmore and M. Lefranc},
Publisher = {Wiley},
Year = {2003},
Address = {New York}
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@Book{GL-Gil07b,
title = {{The Symmetry of Chaos}},
publisher = {Oxford Univ. Press},
year = {2007},
author = {R. Gilmore and C. Letellier},
address = {Oxford},
isbn = {9780195310658},
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@Article{GiChLiPo12,
author = {Ginelli, F. and Chat\'{e}, H. and Livi, R. and Politi, A.},
title = {Covariant {Lyapunov} vectors},
journal = {J. Phys. A},
year = {2013},
volume = {46},
pages = {254005},
doi = {10.1088/1751-8113/46/25/254005},
}
@Article{ginelli-2007-99,
author = {Ginelli, F. and Poggi, P. and Turchi, A. and Chat\'e, H. and Livi, R. and Politi, A.},
title = {Characterizing dynamics with covariant {Lyapunov} vectors},
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year = {2007},
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pages = {130601},
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}
@Article{GiKaChPoAl11,
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journal = {Phys. Rev. E},
year = {2011},
volume = {84},
pages = {066211},
doi = {10.1103/physreve.84.066211},
}
@Article{Ginzburg04,
Title = {{Nobel Lecture}: On superconductivity and superfluidity (what I have and have not managed to do) as well as on the ``physical minimum'' at the beginning of the {XXI} century},
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Year = {2004},
Pages = {981--998},
Volume = {76},
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Author = {M. Giona and S. Cerbelli},
Journal = {Phys. Lett. A},
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Pages = {200--207},
Volume = {347},
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Year = {1991},
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Volume = {24},
Abstract = {A method to compute the curvature of the unstable manifold is introduced
and applied to H\'enon map and Duffing attractor, showing that it
allows the authors to locate the homoclinic tangencies and, in turn,
to construct a generating partition.}
}
@Article{GiPo92,
Title = {Generating partitions in {H\'enon}-type maps},
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Journal = {Phys. Lett. A},
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Pages = {332--336},
Volume = {161}
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@Article{GloKR85,
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Volume = {210},
Abstract = {A auxiliary infinite gradient-like flow is constructed for the area-preserving
map such that orbits of the original system are rest points of the
new system. In this way, existence of orbits of any rotation number
is proved.}
}
@Article{GoSuOr87,
author = {Goldhirsch, I. and Sulem, P. L. and Orszag, S. A.},
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pages = {311--337},
doi = {10.1016/0167-2789(87)90034-0},
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@Article{GoSi94,
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Volume = {7},
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@Misc{Goldobin12,
author = {Goldobin, D. S.},
title = {Limit distribution of averages over unstable periodic orbits forming chaotic attractor},
year = {2012},
url = {http://arXiv.org/abs/1208.1691},
}
@Misc{GoZa10,
author = {Goldobin, D. S. and Zaks, M. A.},
title = {Noise reduces disorder in chaotic dynamics},
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url = {http://arXiv.org/abs/1008.0073},
}
@Book{goldstein59,
Title = {Classical Mechanics},
Author = {Goldstein, H.},
Publisher = {Wesley},
Year = {1959},
Address = {Reading, MA}
}
@Book{goldstein80,
Title = {{Classical Mechanics}},
Author = {Goldstein, H.},
Publisher = {Wesley},
Year = {1980},
Address = {Reading, MA},
Edition = {2},
ISBN = {9788177582833}
}
@Book{goldstein01,
Title = {{Classical Mechanics}},
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Publisher = {Wesley},
Year = {2001},
Address = {Reading, MA},
Edition = {3\textsuperscript{rd}}
}
@Article{golubord,
Title = {Order and disorder in fluid motion},
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Volume = {92},
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}
@Book{GoVanLo96,
Title = {Matrix Computations},
Author = {Golub, G. H. and Van Loan, C. F.},
Publisher = {J. Hopkins Univ. Press},
Year = {1996},
Address = {Baltimore, MD}
}
@Book{golubI,
Title = {{Singularities and Groups in Bifurcation Theory}},
Author = {M. Golubitsky and D. G. Schaeffer},
Publisher = {Springer},
Year = {1984},
Address = {New York},
DOI = {10.1007/978-1-4612-5034-0},
ISBN = {9781461250340},
Volume = {1}
}
@Book{golubitsky2002sp,
Title = {{The Symmetry Perspective}},
Author = {Golubitsky, M. and Stewart, I.},
Publisher = {Birkh{\"a}user},
Year = {2002},
Address = {Boston},
ISBN = {9783034881678}
}
@Book{golubII,
Title = {{Singularities and Groups in Bifurcation Theory}},
Author = {M. Golubitsky and I. Stewart and D. G. Schaeffer},
Publisher = {Springer},
Year = {1988},
Address = {New York},
DOI = {10.1007/978-1-4612-4574-2},
ISBN = {978-0-387-96652-6},
Volume = {2}
}
@Article{golubsym84,
author = {M. Golubitsky and J. W. Swift and Knobloch, J.},
title = {Symmetries and pattern selection in {Rayleigh-B\'{e}nard} convection},
journal = {Physica D},
year = {1984},
volume = {10},
pages = {249},
doi = {10.1016/0167-2789(84)90179-9},
}
@Article{GoPa11,
Title = {Numerical simulation of asymptotic states of the damped {Kuramoto-Sivashinsky} equation},
Author = {Gomez, H. and Paris, J.},
Journal = {Phys. Rev. E},
Year = {2011},
Pages = {046702},
Volume = {83},
DOI = {10.1103/PhysRevE.83.046702}
}
@Article{Good94,
Title = {Stability of the {Kuramoto-Sivashinsky} and related systems},
Author = {J. Goodman},
Journal = {Commun. Pure Appl. Math.},
Year = {1994},
Pages = {293--306},
Volume = {47},
DOI = {10.1002/cpa.3160470304}
}
@Article{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},
doi = {10.1063/1.3080680},
}
@Book{Govaerts00,
Title = {Numerical Methods for Bifurcations of Dynamical Equilibria},
Author = {W. J. F. Govaerts},
Publisher = {SIAM},
Year = {2000},
Address = {Philadelphia}
}
@Article{GozReu94,
Title = {{Lyapunov} exponents, path-integrals and forms},
Author = {Gozzi, E. and Reuter, M.},
Journal = {Chaos Solit. Fract.},
Year = {1994},
Pages = {1117--1139},
Volume = {4},
DOI = {10.1016/0960-0779(94)90026-4}
}
@Book{GraRyz,
Title = {Tables of Integrals, Series and Products},
Author = {I. S. Gradshteyn and I. M. Ryzhik},
Publisher = {Academic},
Year = {1980},
Address = {New York}
}
@Incollection{GranatRBA,
author = {Granat, R. and Jonsson, I. and K{\aa}gstr\"{o}m, B.},
title = {Recursive blocked algorithms for solving periodic triangular {Sylvester}-type matrix equations},
booktitle = {{Applied Parallel Computing. State of the Art in Scientific Computing}},
publisher = {Springer},
year = {2007},
pages = {531--539},
address = {New York},
doi = {10.1007/978-3-540-75755-9_65},
}
@Article{GranatK06,
author = {R. Granat and B. K\r{a}gstr\"{o}m},
title = {Direct eigenvalue reordering in a product of matrices in periodic {Schur} form},
journal = {SIAM J. Matrix Anal. Appl.},
year = {2006},
volume = {28},
pages = {285--300},
doi = {10.1137/05062490x},
}
@Article{grant67,
Title = {{Fourier-{Hermit}e} solutions of the {Vlasov} equations in the linearized limit},
Author = {F. C. Grant and M. R. Feix},
Journal = {Phys. Fluids},
Year = {1967},
Pages = {696--702},
Volume = {10}
}
@Article{D_q,
Title = {Generalized dimensions of strange attractors},
Author = {P. Grassberger},
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Year = {1983},
Pages = {227--230},
Volume = {97},
DOI = {10.1016/0375-9601(83)90753-3}
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@Article{gras86,
Title = {Toward a quantitative theory of self-generated complexity},
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Year = {1986},
Pages = {907--938},
Volume = {25},
DOI = {10.1007/BF00668821},
ISSN = {1572-9575}
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@Article{grass88,
Title = {On symbolic dynamics of one-humped maps of the Iinterval},
Author = {Grassberger, P.},
Journal = {Z. Naturforsch. A},
Year = {1988},
Pages = {671--680},
Volume = {43},
DOI = {10.1515/zna-1988-0710}
}
@Article{Grassberger89,
Title = {Information content and predictability of lumped and distributed dynamical systems},
Author = {P. Grassberger},
Journal = {Phys. Scr.},
Year = {1989},
Pages = {346},
Volume = {40},
DOI = {10.1088/0031-8949/40/3/016}
}
@Article{GK85,
Title = {Generating partitions for the dissipative {H{\'e}non} map},
Author = {P. Grassberger and H. Kantz},
Journal = {Phys. Lett. A},
Year = {1985},
Pages = {235--238},
Volume = {113},
DOI = {10.1016/0375-9601(85)90016-7}
}
@Article{grass89,
author = {P. Grassberger and H. Kantz and U. Moenig},
title = {On the symbolic dynamics of {H{\'e}non} map},
journal = {J. Phys. A},
year = {1989},
volume = {22},
pages = {5217--5230},
doi = {10.1088/0305-4470/22/24/011},
}
@Article{GraPro83a,
Title = {Characterization of strange attractors},
Author = {Grassberger, P. and Procaccia, I.},
Journal = {Phys. Rev. Lett.},
Year = {1983},
Pages = {346--349},
Volume = {50},
DOI = {10.1103/PhysRevLett.50.346}
}
@Article{GraPro83,
Title = {Estimation of the {Kolmogorov} entropy from a chaotic signal},
Author = {Grassberger, P. and Procaccia, I.},
Journal = {Phys. Rev. A},
Year = {1983},
Pages = {2591--2593},
Volume = {28},
DOI = {10.1103/PhysRevA.28.2591}
}
@Article{GraPro83b,
Title = {Measuring the strangeness of strange attractors},
Author = {Grassberger, P. and Procaccia, I.},
Journal = {Physica D},
Year = {1983},
Pages = {189--208},
Volume = {9},
DOI = {10.1016/0167-2789(83)90298-1}
}
@Article{greb90shad,
author = {C. Grebogi and S. M. Hammel and J. A. Yorke and T. Sauer},
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year = {1990},
volume = {65},
pages = {1527--1530},
doi = {10.1103/physrevlett.65.1527},
}
@Article{yorke2,
Title = {Unstable periodic orbits and the dimensions of multifractal chaotic attractors},
Author = {C. Grebogi and E. Ott and J. A. Yorke},
Journal = {Phys. Rev. A},
Year = {1988},
Pages = {1711--1724},
Volume = {37},
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{gree98,
Title = {Two-dimensional measure-preserving mappings},
Author = {J. M. Greene},
Journal = {J. Math. Phys.},
Year = {1968},
Pages = {760},
Volume = {9}
}
@Article{gree79,
Title = {A method for determining a stochastic transition},
Author = {J. M. Greene},
Journal = {J. Math. Phys.},
Year = {1979},
Pages = {1183--1201},
Volume = {20}
}
@Article{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},
doi = {10.1016/0167-2789(87)90076-5},
}
@Article{ksgreene88,
author = {J. M. Greene and J.-S. Kim},
title = {The steady states of the {Kuramoto-Sivashinsky} equation},
journal = {Physica D},
year = {1988},
volume = {33},
pages = {99--120},
doi = {10.1016/S0167-2789(98)90013-6},
}
@Article{GrMaViFe81,
Title = {Universal behaviour in families of area-preserving maps},
Author = {J. M. Greene and R. S. MacKay and F. Vivaldi and M. J. Feigenbaum},
Journal = {Physica D},
Year = {1981},
Pages = {468--486},
Volume = {3},
DOI = {10.1016/0167-2789(81)90034-8}
}
@Book{Greensite11,
Title = {An introduction to the confinement problem},
Author = {Greensite, J.},
Publisher = {Springer},
Year = {2011},
Address = {New York},
Pages = {101--129}
}
@Book{greshosani,
Title = {Incompressible Flow and the Finite Element Method},
Author = {P. M. Gresho and R. L. Sani},
Publisher = {Wiley},
Year = {2000},
Address = {New York}
}
@Article{Gribov77,
Title = {Quantization of nonabelian gauge theories},
Author = {Gribov, V. N.},
Journal = {Nucl. Phys.},
Year = {1978},
Pages = {1--19},
Volume = {B139},
DOI = {10.1016/0550-3213(78)90175-X}
}
@Incollection{autod89,
Title = {On Automatic Differentiation},
Author = {A. Griewank},
Booktitle = {Mathematical Programming: Recent Developments and Applications},
Publisher = {Kluwer},
Year = {1989},
Address = {Dordrecht},
Editor = {M. Iri and K. Tanabe},
Pages = {83--108}
}
@Article{ksgrim91,
Title = {The non-existence of a certain class of travelling wave solutions of the {Kuramoto-Sivashinsky} equation},
Author = {R. Grimshaw and A. P. Hooper},
Journal = {Physica D},
Year = {1991},
Pages = {231--238},
Volume = {50},
Abstract = {If the effect of short-wave stability is small, there are no regular
shocks.},
DOI = {0.1016/0167-2789(91)90177-B}
}
@Article{Gritsun10,
Title = {Statistical characteristics of barotropic atmospeheric system and its unstable periodic solutions},
Author = {Gritsun, A.},
Journal = {Dokl. Earth Sciences},
Year = {2010},
Pages = {1688--1691},
Volume = {435},
Abstract = {This paper is devoted to the problem of approximating an invariant
measure and statistical characteristics of barotropic atmospheric
model with the help of its periodic trajectories. In this procedure
orbits are taken into account according to their weights defined
by the orbit instability characteristics. The method comes from the
dynamical systems theory where in several specific case (for hyperbolic
systems in particular) unstable periodic orbits define the system
invariant measure. In our study we show that the system PDF could
be reconstructed with the error less than 10\% provided that the
optimal orbit weight function is chosen.},
DOI = {10.1134/S1028334X10120287}
}
@Article{Gritsun11,
Title = {Connection of periodic orbits and variability patterns of circulation for the barotropic model of atmospheric dynamics},
Author = {Gritsun, A.},
Journal = {Dokl. Earth Sciences},
Year = {2011},
Pages = {636--640},
Volume = {438},
Abstract = {We have investigated the relationship between periodic trajectories
of barotropic atmospheric model and the modes of the model variability.
In particular, we have studied the nature of ``25 day'' mode of variability
(Branstator, 1987; Kushnir 1987). This mode arises as a first complex
empirical orthogonal function (or ``Hilbert EOF'' according to (H.
von Storch, Zwiers)) for a given system and is a dominant rotational
component of the system dynamics. It was shown that the mode structure
coincides with several least unstable periodic orbits of the system.
The phase portrait of the system in the plane of the first complex
EOF has regular shape with maximum of the probability density function
in the vicinity of these weakly unstable periodic orbits.},
DOI = {10.1134/S1028334X11050035}
}
@Article{Gritsun13,
Title = {Statistical characteristics, circulation regimes and unstable periodic orbits of a barotropic atmospheric model},
Author = {Gritsun, A.},
Journal = {Philos. Trans. Royal Soc. A},
Year = {2013},
Volume = {371},
DOI = {10.1098/rsta.2012.0336}
}
@Article{GrBrMa08,
Title = {Climate response of linear and quadratic functionals using the fluctuation-dissipation theorem},
Author = {A. Gritsun and G. Branstator and A. Majda},
Journal = {J. Atmos. Sci.},
Year = {2008},
Pages = {2824--2841},
Volume = {65},
DOI = {10.1175/2007JAS2496.1}
}
@Article{LucGri16,
author = {A. Gritsun and V. Lucarini},
title = {Fluctuations, response, and resonances in a simple atmospheric model},
journal = {Physica D},
year = {2017},
volume = {349},
doi = {10.1016/j.physd.2017.02.015},
}
@Article{Gritsun08,
Title = {Unstable periodic trajectories of a barotropic model of the atmosphere},
Author = {Gritsun, A. S.},
Journal = {Russian J. Numer. Analysis and Math. Modelling},
Year = {2008},
Pages = {345--367},
Volume = {23},
Abstract = {Unstable periodic trajectories of a chaotic dissipative system belong
to the attractor of the system and are its important characteristics.
Many chaotic systems have an infinite number of periodic solutions
forming the skeleton of the system attractor. This allows one to
approximate the system trajectories and statistical characteristics
by using periodic solutions. The least unstable orbits may generate
local maxima of the system state distribution functions on the attractor.
With respect to atmospheric systems this means that orbits may determine
dynamic circulation regimes and typical variability modes of the
system. In some cases, given a small number of periodic solutions,
one can describe the dynamics on the attractor of the system and
the basic statistics with sufficient precision. Thus, the information
concerning periodic trajectories of a particular dynamic system may
be very important for analysis of its behavior. A search for periodic
trajectories is reduced to the solution of a system of nonlinear
equations with respect to the initial condition of an orbit and its
period. The choice of a numerical solution method and an initial
guess is an important aspect here. In this paper we consider the
problem of the calculation of periodic trajectories for a barotropic
model of the atmosphere. Several methods for determination of periodic
orbits of the model are formulated and implemented, including the
Newton method with step suppression, the Newton method with a second
order tensor correction, the quasi-Newton method with step suppression,
the quasi-Newton method with minimization of the error functional
and approximate Hessian inversion by the LBFG scheme and the GMRES
method. A comparison of the efficiency of these numerical methods
and different choices of initial conditions is performed. Various
factors influencing the rate of convergence of the methods are considered.},
DOI = {10.1515/RJNAMM.2008.021}
}
@Article{G00,
author = {S. Grossmann},
title = {The onset of shear turbulence},
journal = {Rev. Mod. Phys.},
year = {2000},
volume = {72},
pages = {603--618},
}
@Article{GroFuj82,
Title = {Diffusion in discrete nonlinear dynamical systems},
Author = {Grossmann, S. and Fujisaka, H.},
Journal = {Phys. Rev. A},
Year = {1982},
Pages = {1779--1782},
Volume = {26},
DOI = {10.1103/PhysRevA.26.1779}
}
@Article{Adler98,
author = {Adler, R. L.},
title = {Symbolic dynamics and {Markov} partitions},
journal = {Bull. Amer. Math. Soc.},
year = {1998},
volume = {35},
pages = {1--56},
doi = {10.1090/S0273-0979-98-00737-X},
}
@Article{Guckenh77,
Title = {On the bifurcation of maps of the interval},
Author = {J. Guckenheimer},
Journal = {Inv. Math.},
Year = {1977},
Pages = {165--178},
Volume = {39},
DOI = {10.1007/BF01390107}
}
@Article{Guck79,
Title = {Sensitive dependence to initial conditions for one dimensional maps},
Author = {Guckenheimer, J.},
Journal = {Commun. Math. Phys.},
Year = {1979},
Pages = {133--160},
Volume = {70},
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},
doi = {10.1007/978-1-4612-1140-2},
}
@Article{GKOS15,
Title = {Invariant manifolds and global bifurcations},
Author = {J. Guckenheimer and B. Krauskopf and H. M. Osinga and B. Sandstede},
Journal = {Chaos},
Year = {2015},
Volume = {25},
DOI = {10.1063/1.4915528},
URL = {http://www.dam.brown.edu/people/sandsted/publications/gkos-survey.pdf}
}
@Article{GM00aut,
author = {J. Guckenheimer and B. Meloon},
title = {Computing periodic orbits and their bifurcations with automatic differentiation},
journal = {SIAM J. Sci. Comput.},
year = {2000},
volume = {22},
pages = {951--985},
doi = {10.1137/s1064827599359278},
}
@Incollection{GuIsLa08,
Title = {Ihara zeta functions for periodic simple graphs},
Author = {Guido, D. and Isola, T. and Lapidus, M. L.},
Booktitle = {{C*-algebras and Elliptic Theory II}},
Publisher = {Birkh{\"a}user},
Year = {2008},
Address = {Basel},
Editor = {Burghelea, D. and Melrose, R. and Mishchenko, A. S. and Troitsky, E. V.},
Pages = {103--121},
DOI = {10.1007/978-3-7643-8604-7_5},
ISBN = {978-3-7643-8604-7}
}
@Book{GuiSte90,
Title = {Symplectic Techniques in Physics},
Author = {V. Guillemin and S. Sternberg},
Publisher = {Cambridge Univ. Press},
Year = {1990},
Address = {Cambridge}
}
@Article{GuiUri87,
Title = {Reduction, the trace formula, and semiclassical asymptotics},
Author = {Guillemin, V. and Uribe, A.},
Journal = {Proc. Natl. Acad. Sci. USA},
Year = {1987},
Pages = {7799--7801},
Volume = {84},
DOI = {10.1073/pnas.84.22.7799}
}
@Article{GuiUri89,
Title = {Circular symmetry and the trace formula},
Author = {Guillemin, V. and Uribe, A.},
Journal = {Inv. Math.},
Year = {1989},
Pages = {385--423},
Volume = {96},
DOI = {10.1007/BF01393968}
}
@Article{GuiUri90,
Title = {Reduction and the trace formula},
Author = {Guillemin, V. and Uribe, A.},
Journal = {J. Diff. Geom.},
Year = {1990},
Pages = {315},
Volume = {32},
URL = {http://projecteuclid.org/euclid.jdg/1214445310}
}
@Book{MiraGum80,
Title = {Recurrances and Discrete Dynamical Systems},
Author = {I. Gumowski and C. Mira},
Publisher = {Springer},
Year = {1980},
Address = {Berlin}
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@Article{GunRan93I,
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author = {B. Gutkin and L. Han and R. Jafari and A. K. Saremi and P. Cvitanovi{\'c}},
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year = {2018},
note = {In preparation},
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Number = {44},
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Volume = {37},
Abstract = {An explicit algorithm to provide the pruning front for the area-preserving
H\'enon map is presented. The procedure terminates within finitely
many steps when the map has hyperbolic structure. The only information
required to specify the pruning front is a bifurcation diagram of
homoclinic orbits, and it is obtained by tracking orbits from the
anti-integrable limit. The pruned region thus determined is used
to construct the Markov partition of the map, and the topological
entropy is evaluated as an application.},
DOI = {10.1088/0305-4470/37/44/005}
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@Article{HaShu04b,
author = {Hagiwara, R. and Shudo, A.},
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@Book{Hairer06,
Title = {{Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations}},
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Address = {Berlin},
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Edition = {2}
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Year = {2008},
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URL = {http://ChaosBook.org/projects/theses.html}
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@Article{GHCV08,
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title = {{Oscillations in Nonlinear Systems}},
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Abstract = {We introduce a Lagrangian definition for the boundaries of coherent
structures in two-dimensional turbulence. The boundaries are defined
as material lines that are linearly stable or unstable for longer
times than any of their neighbors. Such material lines are responsible
for stretching and folding in the mixing of passive tracers. We derive
an analytic criterion that can be used to extract coherent structures
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criterion provides a rigorous link between the Lagrangian concept
of hyperbolicity, the {Okubo?Weiss} criterion, and vortex boundaries.
We apply the results to simulations of two-dimensional barotropic
turbulence.},
DOI = {10.1016/S0167-2789(00)00142-1}
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DOI = {10.1063/1.165900}
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Title = {Symbolic Dynamics in Chaotic systems},
Author = {Hansen, K. T.},
School = {Univ. of Oslo},
Year = {1993},
Address = {Oslo, Norway},
URL = {http://ChaosBook.org/projects/KTHansen/thesis}
}
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Title = {Symbolic dynamics. {I. Finite} dispersive billiards},
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Journal = {Nonlinearity},
Year = {1993},
Pages = {753--769},
Volume = {6},
DOI = {10.1088/0951-7715/6/5/005}
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Year = {1993},
Pages = {771--778},
Volume = {6},
DOI = {10.1088/0951-7715/6/5/006}
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doi = {10.1103/PhysRevE.52.2388},
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@Misc{stadium95,
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note = {{\em J. Stat. Phys.}, accepted 1996, revised version still not resubmitted},
url = {http://arXiv.org/abs/chao-dyn/9502005},
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Volume = {11},
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title = {{Applied Symbolic Dynamics and Chaos}},
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address = {Singapore},
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@Article{MvHhole01,
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Year = {2001},
Pages = {2018},
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DOI = {10.1142/S021812740701821X}
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@Incollection{Hell79,
author = {R. H. G. Helleman and T. Bountis},
title = {Periodic solutions of arbitrary period, variational methods},
booktitle = {{Stochastic Behavior in Classical and Quantum Hamiltonian Systems}},
publisher = {Springer},
year = {1979},
editor = {Casati, G. and Ford, J.},
pages = {353--375},
address = {Berlin},
doi = {10.1007/BFb0021758},
isbn = {978-3-540-09120-2},
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pages = {062922},
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year = {2018},
volume = {55},
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doi = {10.1016/j.cnsns.2017.06.025},
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doi = {10.21914/anziamj.v42i0.619},
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doi = {10.1137/090780055},
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author = {Hertsch, A. and Rudolph, G. and Schmidt, M.},
title = {{Gauge Orbit Types for Theories with Classical Compact Gauge Group O(n), SO(n) or Sp(n)}},
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year = {2011},
volume = {12},
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doi = {10.1007/s00023-011-0081-8},
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@Article{HeMuAlBrHa07,
author = {Heusler, S. and M\"uller, S. and Altland, A. and Braun, P. and Haake, F.},
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year = {2007},
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doi = {10.1103/physrevlett.98.044103},
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@Article{HHKR50yChaos,
Title = {Fifty years of chaos: {Applied} and theoretical},
Author = {T. Hikihara and P. Holmes and T. Kambe and G. Rega},
Journal = {Chaos},
Year = {2012},
Pages = {047501},
Volume = {22},
DOI = {10.1063/1.4769035}
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Title = {{\"U}ber die vollen {Invariantensysteme}},
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Year = {1893},
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@Book{hille,
Title = {Ordinary Differential Equations in the Complex Domain},
Author = {E. Hille},
Publisher = {Dover},
Year = {1997},
Address = {New York}
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@Incollection{hindmarsh1983,
Title = {{ODEPACK}, a systematized collection of {ODE} solvers},
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Booktitle = {{Scientific Computing}},
Publisher = {North-Holland},
Year = {1983},
Address = {Amsterdam},
Editor = {Stepleman, R. S.},
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year = {2016},
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Title = {Numerical justification of {Leonov} conjecture on {Lyapunov} dimension of {Rossler} attractor},
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@InProceedings{KuKuSa05,
Title = {Review and examples of non-{Feigenbaum} critical situations associated with period-doubling},
Author = {Kuznetsov, S. P. and Kuznetsov, A. P. and Sataev, I .R.},
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.},
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@Misc{lopezLink,
author = {Vanessa L{\'o}pez},
note = {private communication},
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Abstract = { We consider nonresonant and weakly resonant Hopf
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tori. Results are obtained for systems with compact and noncompact
symmetry groups. In the noncompact case, we distinguish between
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School = {School of Physics, Georgia Inst. of Technology},
Year = {2004},
Address = {Atlanta},
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to trajectories which are long relative to the predictability time
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quantities and given infinite computing resources. Alternatively,
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of initial conditions, because a finite time scale is required for
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Lorenz (1963) system, an intermediate time scale is found on which
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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},
url = {http://ChaosBook.org/projects/theses.html},
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Note = {In preparation},
Year = {2012}
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Series = {Advanced Courses in Mathematics - CRM Barcelona},
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of the axially symmetric and asymmetric thermal forcing, the equations
may possess one or two stable steady-state solutions, one or two
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be applied to the model, and checked for soundness by comparing the
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publisher = {Springer},
year = {2014},
editor = {Bock, G. H. and Hoang, P. X. and Rannacher, R. and Schl{\"o}der, P. J.},
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Volume = {85},
Abstract = {Ion acceleration driven by superintense laser pulses is attracting
an impressive and steadily increasing effort. Motivations can be
found in the applicative potential and in the perspective to investigate
novel regimes as available laser intensities will be increasing.
Experiments have demonstrated, over a wide range of laser and target
parameters, the generation of multi-{MeV} proton and ion beams with
unique properties such as ultrashort duration, high brilliance, and
low emittance. An overview is given of the state of the art of ion
acceleration by laser pulses as well as an outlook on its future
development and perspectives. The main features observed in the experiments,
the observed scaling with laser and plasma parameters, and the main
models used both to interpret experimental data and to suggest new
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DOI = {10.1103/RevModPhys.85.751}
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Journal = {Math. Comput. Simul.},
Year = {2007},
Pages = {266 --280},
Volume = {74},
DOI = {10.1016/j.matcom.2006.10.009}
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Year = {2007},
Pages = {281--291},
Volume = {74},
DOI = {10.1016/j.matcom.2006.10.022}
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@Article{ManCho09,
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Journal = {Nonlinear Dyn.},
Year = {2015},
Pages = {549--571},
Volume = {79},
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@Misc{ManKha13,
author = {S. C. Mancas and H. Khanal},
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Title = {Noether's theorem for smooth, difference and finite element schemes},
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Pages = {230--254},
Publisher = {Cambridge Univ. Press},
Series = {London Math. Soc. Lect. Notes},
Volume = {331},
URL = {https://www.kent.ac.uk/smsas/personal/elm2/liz/papers/focm.pdf}
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title = {A practical guide to the invariant calculus},
publisher = {Cambridge Univ. Press},
year = {2010},
author = {Mansfield, E. L.},
address = {Cambridge},
doi = {10.1017/cbo9780511844621},
}
@Article{Marion1989,
author = {Marion, M. and T{\'e}mam, R.},
title = {Nonlinear {Galerkin} methods},
journal = {SIAM J. Numer. Anal.},
year = {1989},
volume = {26},
pages = {1139--1157},
doi = {10.1137/0726063},
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@Article{Marques90,
Title = {On boundary conditions for velocity potentials in confined flows: {Application} to {Couette} flow},
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Journal = {Phys. Fluids A},
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@Article{Marquardt63,
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@Article{MarLopBla04,
Title = {Bifurcations in systems with {Z2} spatio-temporal and {O(2)} spatial symmetry},
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title = {Lectures on Mechanics},
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author = {Marsden, J. E.},
address = {Cambridge},
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}
@Book{marsdenbb,
title = {The {Hopf} Bifurcation and its Applications},
publisher = {Springer},
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@Book{marsden_hamiltonian_2007,
Title = {Hamiltonian Reduction by Stages},
Author = {Marsden, J. E. and G. Misiolek and {J.-P.} Ortega and M. Perlmutter and T. S. Ratiu},
Publisher = {Springer},
Year = {1964},
Address = {New York}
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@Book{MarRat99,
title = {{Introduction to Mechanics and Symmetry}},
publisher = {Springer},
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author = {Marsden, J. E. and Ratiu, T. S.},
address = {New York},
doi = {10.1007/978-0-387-21792-5},
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@Article{MaWe74,
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@Article{mart87lfa,
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@Article{nongal89,
author = {M. Martine and R. T{\'e}mam},
title = {Nonlinear {G{\"a}lerkin} methods},
journal = {SIAM J. Numer. Anal.},
year = {1989},
volume = {26},
number = {5},
pages = {1139--1157},
doi = {10.1137/0726063},
abstract = {nonlinear {G}{\"{a}}lerkin methods},
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@Article{MaSaTAS81,
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Annote = {Original paper abour Aubry-Mather sets},
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@Book{MathWalk73,
title = {Mathematical Methods of Physics},
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year = {1973},
author = {J. Mathews and R. L. Walker},
address = {Reading, MA},
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@Article{MHPRS07,
author = {Mathur, M. and Haller, G. and Peacock, T. and Ruppert-Felsot, J. E. and Swinney, H. L.},
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@Article{stabpcgl,
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@Article{maucher2013,
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Pages = {083055},
Volume = {15},
Abstract = {Quasiperiodic oscillations and shape-transformations of
higher-order bright solitons in nonlinear nonlocal media have been
frequently observed numerically in recent years, however, the origin of
these phenomena was never completely elucidated. In this paper, we
perform a linear stability analysis of these higher-order solitons by
solving the Bogoliubov-de Gennes equations. This enables us to understand
the emergence of a new oscillatory state as a growing unstable mode of a
higher-order soliton. Using dynamically important states as a basis, we
provide low-dimensional visualizations of the dynamics and identify
quasiperiodic and homoclinic orbits, linking the latter to
shape-transformations.},
DOI = {10.1088/1367-2630/15/8/083055}
}
@Article{MaRi83,
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@Book{Mayer80,
title = {{The Ruelle-Araki Transfer Operator in Classical Statistical Mechanics}},
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author = {Mayer, D. H.},
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Year = {2005},
Address = {Singapore},
Editor = {Dumortier, F. and Broer, H. W. and Mawhin, J. and Vanderbauwhede, A. and Verduyn Lunel, S.},
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Publisher = {World Scientific},
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@Unpublished{McICvi15,
Title = {Periodic orbit theory of linear response},
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Note = {In preparation},
Year = {2015}
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@Article{McKean74,
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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 distribution in fully developed pipe flow},
journal = {J. Fluid Mech.},
year = {2004},
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pages = {135--147},
doi = {10.1017/S0022112003007304},
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@Article{McLPerlQui03,
Title = {Lie group foliations: dynamical systems and integrators},
Author = {R.I. McLachlan and M. Perlmutter and G.R.W. Quispel},
Journal = {Future Generation Computer Systems},
Year = {2003},
Pages = {1207--1219},
Volume = {19},
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}
}
@Article{McNMar01,
Title = {Origin of the hydrodynamic {Lyapunov modes}},
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Pages = {051103},
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DOI = {10.1103/PhysRevE.64.051103}
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@Misc{MSBF16,
author = {E. S. Medeiros and I. L. Caldas and M. S. Baptista and U. Feudel},
title = {Trapping phenomenon attenuates tipping points for limit cycles},
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url = {https://arxiv.org/abs/1506.08555},
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@Book{meer_hamiltonian_1994,
title = {Hamiltonian Structure of the Reversible Nonsemisimple 1:1 Resonance},
publisher = {Eindhoven Univ. of Technology, Dept. of Math. and Comp. Sci.},
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author = {J.C. van der Meer and J.A. Sanders and Vanderbauwhede, A.},
series = {Reports on applied and numerical analysis, RANA 94-02},
address = {Eindhoven},
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@Book{Mees81,
Title = {Dynamics of Feedback Systems},
Author = {Mees, A. I.},
Publisher = {Wiley},
Year = {1981},
Address = {New York}
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@Book{mehta,
Title = {Random Matrices},
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Publisher = {Academic},
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Address = {New York}
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@Article{meis92,
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@Book{Meisso7,
title = {Differential Dynamical Systems},
publisher = {SIAM},
year = {2007},
author = {J. D. Meiss},
address = {Philadelphia},
doi = {10.1137/1.9780898718232},
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@Article{Mei08,
author = {J. D. Meiss},
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year = {2008},
volume = {70},
pages = {965--988},
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year = {1999},
volume = {351},
pages = {1575--1603},
doi = {10.1090/S0002-9947-99-02147-9},
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@Article{mellibovsky11,
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Journal = {J. Fluid Mech.},
Year = {2011},
Pages = {96--129},
Volume = {670},
DOI = {10.1017/s0022112010005239}
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@Article{mellibovsky12,
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Year = {2012},
Pages = {149--190},
Volume = {709},
DOI = {10.1017/jfm.2012.326}
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@Article{umb94pat,
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@Article{Mendoza13,
author = {Mendoza, V.},
title = {Proof of the {Pruning Front Conjecture} for certain {H\'enon} parameters},
journal = {Nonlinearity},
year = {2013},
volume = {26},
pages = {679--690},
doi = {10.1088/0951-7715/26/3/679},
abstract = {The Pruning Front Conjecture is proved for an open set of H\'enon
parameters far from unimodal. More specifically, for an open subset
of H\'enon parameter space, consisting of two connected components
one of which intersects the area-preserving locus, it is shown that
the associated H\'enon maps are prunings of the horseshoe. In particular,
their dynamics is a subshift of the two-sided two-shift.},
}
@Article{MePrKn01,
author = {Mercader, I. and Prat, J. and Knobloch, E.},
title = {The {1:2} mode interaction in {Rayleigh B\'enard} convection with weakly broken midplane symmetry},
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year = {2001},
volume = {11},
pages = {27--41},
doi = {10.1142/s0218127401002006},
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@Article{MePrKn02,
Title = {Robust heteroclinic cycles in two-dimensional {Rayleigh-B\'enard} convection without {Boussinesq} symmetry},
Author = {Mercader, I. and Prat, J. and Knobloch, E.},
Journal = {Int. J. Bifur. Chaos},
Year = {2002},
Pages = {2501--2522},
Volume = {12},
DOI = {10.1142/S0218127402006047}
}
@Article{Mermin92,
Title = {The space groups of icosahedral quasicrystals and cubic, orthorhombic, monoclinic, and triclinic crystals},
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Year = {1992},
Pages = {3--49},
Volume = {64},
DOI = {10.1103/RevModPhys.64.3}
}
@Article{Mertz79,
Title = {Speckle imaging, photon by photon},
Author = {L. N. Mertz},
Journal = {Appl. Opt.},
Year = {1979},
Pages = {611--614},
Volume = {18},
Abstract = {A speckle processing prescription is described that
should yield diffraction-limited performance for large telescopes
in spite of atmospheric turbulence and at light levels as low as
100 photons/sec in the picture. The prescription involves
rearranging the spatial frequency components of running glimpses of
the scene according to the complex information (entropy) of those
components. The imaginary part (phase) of the information is
rendered unambiguous by maintaining track of the phase. A 2-D
photon counter furnishes the raw observations. Comparisons and
ramifications of the procedure are discussed.},
DOI = {10.1364/AO.18.000611}
}
@Article{Meseguer03,
Title = {Streak breakdown instability in pipe {Poiseuille} flow},
Author = {A. Meseguer},
Journal = {Phys. Fluids},
Year = {2003},
Pages = {1203--1213},
Volume = {15},
DOI = {10.1063/1.1564093}
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@Article{MeseguerANM07,
author = {A. Meseguer and F. Mellibovsky},
title = {On a solenoidal {Fourier-Chebyshev} spectral method for stability analysis of the {Hagen--Poiseuille} flow},
journal = {Appl. Numer. Math.},
year = {2007},
volume = {57},
pages = {920--938},
doi = {10.1016/j.apnum.2006.09.002},
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@Article{MesTre03,
author = {{\'{A}}. Meseguer and L. N. Trefethen},
title = {Linearized pipe flow to {Reynolds} number {$10^7$}},
journal = {J. Comput. Phys.},
year = {2003},
volume = {186},
pages = {178--197},
doi = {10.1016/s0021-9991(03)00029-9},
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@Article{varcyc,
Title = {{Newton} method for highly unstable orbits},
Author = {B. D. Mestel and I. Percival},
Journal = {Physica D},
Year = {1987},
Pages = {172},
Volume = {24},
DOI = {10.1016/0167-2789(87)90072-8}
}
@Book{Meyer00,
title = {{Matrix Analysis and Applied Linear Algebra}},
publisher = {SIAM},
year = {2000},
author = {Meyer, C.},
address = {Philadelphia},
doi = {10.1137/1.9780898719512},
}
@Book{MeyerHall92,
title = {{Introduction to {Hamiltonian} Dynamical Systems}},
publisher = {Springer},
year = {1992},
author = {K. R. Meyer and G. R. Hall},
address = {New York},
}
@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},
}
@Article{MeWi94,
author = {I. Mezi\'c and S. Wiggins},
title = {On the integrability and perturbation of three-dimensional fluid flows with symmetry},
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year = {1994},
volume = {4},
pages = {157--194},
doi = {10.1007/bf02430631},
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@Article{Mezzadri02,
Title = {On the multiplicativity of quantum cat maps},
Author = {F. Mezzadri},
Journal = {Nonlinearity},
Year = {2002},
Pages = {905--922},
Volume = {15},
DOI = {10.1088/0951-7715/15/3/323}
}
@Article{MicZhi01,
Title = {Symmetry, invariants, topology. Basic tools},
Author = {L. Michel and B.I. Zhilinski\'i},
Journal = {Phys. Lett.},
Year = {2001},
Pages = {11--84},
Volume = {341},
Abstract = {Elementary concepts of group actions: orbits and their
stabilizers, orbit types and their strata are introduced and illustrated
by simple examples. We give the unified description of these notions
which are often used in the different domains of physics under different
names. We also explain some basic facts about rings of invariant
functions and their module structure. This leads to a geometrical study
of the orbit space and of the level surfaces of invariant functions (e.g.
energy levels of Hamiltonians). Combining these tools with Morse theory
we study the extrema of invariant functions. Some physical applications
(not studied in other chapters) are sketched. },
DOI = {10.1016/S0370-1573(00)00088-0}
}
@Article{Mks86,
Title = {Steady solutions of the {Kuramoto-Sivashinsky} equation},
Author = {D. Michelson},
Journal = {Physica D},
Year = {1986},
Pages = {89--111},
Volume = {19},
DOI = {10.1016/0167-2789(86)90055-2}
}
@Article{Michel90,
author = {D. Michelson},
title = {Elementary particles as solutions of the {Sivashinsky} equation},
journal = {Physica D},
year = {1990},
volume = {44},
pages = {502--556},
doi = {10.1016/0167-2789(90)90160-q},
}
@Article{michsiv77,
Title = {Nonlinear analysis of hydrodynamic instability in laminar flames---{II}. {Numerical} experiments},
Author = {D. M. Michelson and G. I. Sivashinsky},
Journal = {Acta Astronaut.},
Year = {1977},
Pages = {1207--1221},
Volume = {4},
DOI = {10.1016/0094-5765(77)90097-2}
}
@Book{Mielke91,
Title = {{Hamiltonian and Lagrangian} Flows on Center Manifolds},
Author = {A. Mielke},
Publisher = {Springer},
Year = {1991},
Address = {New York}
}
@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},
address = {New York},
doi = {10.1016/S1874-575X(02)80036-4},
}
@Incollection{MilThu88,
Title = {Iterated maps of the interval},
Author = {Milnor, J. and Thurston, W.},
Booktitle = {{Dynamical {Systems} ({Maryland} 1986-87)}},
Publisher = {Springer},
Year = {1988},
Address = {New York},
Editor = {A. Dold and B. Eckmann},
Pages = {465--563},
DOI = {10.1007/BFb0082847}
}
@Misc{MiPlSt11,
Title = {On the steady state distributions for turbulence},
Author = {Mini\'c, Dj. and Pleimling, M. and Staples, A. E.},
Year = {2011},
URL = {http://arXiv.org/abs/1105.2941}
}
@Book{Mira87,
Title = {Chaotic dynamics -- {From} one dimensional endomorphism to two dimen\-sional diffeo\-morphism},
Author = {C. Mira},
Publisher = {World Scientific},
Year = {1987},
Address = {Singapore}
}
@Article{GL-Mir93,
Title = {The proto-{Lorenz} system},
Author = {Miranda, R. and Stone, E.},
Journal = {Phys. Lett. A},
Year = {1993},
Pages = {105--113},
Volume = {178},
DOI = {10.1016/0375-9601(93)90735-I}
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@Article{mislor1,
Title = {Chaos in the {Lorenz} equations: {A} computer assisted proof part {II}: {Details}},
Author = {K. Mischaikow and M. Mrozek},
Journal = {Math. Comput.},
Year = {1998},
Pages = {1023--1047},
Volume = {67},
DOI = {10.1090/s0025-5718-98-00945-4}
}
@Article{misdis,
Title = {Construction of Symbolic Dynamics from Experimental Time Series},
Author = {K. Mischaikow and M. Mrozek and J. Reiss and A. Szymczak},
Journal = {Phys. Rev. Lett.},
Year = {1999},
Number = {6},
Pages = {1144--1147},
Volume = {82},
DOI = {10.1103/physrevlett.82.1144}
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@Article{MiZg01,
Title = {Topological entropy for multidimensional perturbations of one dimensional maps},
Author = {M. Misiurewicz and P. Zgliczynski},
Journal = {Int. J. Bifur. Chaos},
Year = {2001},
Pages = {1443--1446},
Volume = {11},
DOI = {10.1142/s021812740100281x}
}
@Article{Mitchell12,
Title = {Partitioning two-dimensional mixed phase spaces},
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pages = {195101},
doi = {10.1088/1751-8113/45/19/195101},
}
@Article{RoCaRe12,
Title = {Dual pairing of symmetry and dynamical groups in physics},
Author = {Rowe, D. J. and Carvalho, M. J. and Repka, J.},
Journal = {Rev. Mod. Phys.},
Year = {2012},
Pages = {711--757},
Volume = {84},
DOI = {10.1103/RevModPhys.84.711}
}
@Article{RoSa00,
Title = {Nonlinear dimensionality reduction by locally linear embedding},
Author = {Roweis, S. T. and Saul, L. K.},
Journal = {Science},
Year = {2000},
Pages = {2323--2326},
Volume = {290},
Abstract = {Many areas of science depend on exploratory data analysis and visualization.
The need to analyze large amounts of multivariate data raises the
fundamental problem of dimensionality reduction: how to discover
compact representations of high-dimensional data. Here, we introduce
locally linear embedding {(LLE),} an unsupervised learning algorithm
that computes low-dimensional, neighborhood-preserving embeddings
of high-dimensional inputs. Unlike clustering methods for local dimensionality
reduction, {LLE} maps its inputs into a single global coordinate
system of lower dimensionality, and its optimizations do not involve
local minima. By exploiting the local symmetries of linear reconstructions,
{LLE} is able to learn the global structure of nonlinear manifolds,
such as those generated by images of faces or documents of text.},
DOI = {10.1126/science.290.5500.2323}
}
@Article{rowley_reduction_2003,
Title = {Reduction and reconstruction for self-similar dynamical systems},
Author = {Rowley, C. W. and Kevrekidis, I. G. and Marsden, J. E. and Lust, K.},
Journal = {Nonlinearity},
Year = {2003},
Pages = {1257--1275},
Volume = {16},
DOI = {10.1088/0951-7715/16/4/304}
}
@Article{rowley_reconstruction_2000,
Title = {Reconstruction equations and the {Karhunen-Lo{\'e}ve} expansion for systems with symmetry},
Author = {Rowley, C. W. and Marsden, J. E.},
Journal = {Physica D},
Year = {2000},
Pages = {1--19},
Volume = {142},
DOI = {10.1016/S0167-2789(00)00042-7}
}
@Article{RoMeBaSchHe09,
Title = {Spectral analysis of nonlinear flows},
Author = {Rowley, C. W. and Mezi\'c, I. and Bagheri, S. and Schlatter, P. and Henningson, D. S.},
Journal = {J. Fluid Mech.},
Year = {2009},
Pages = {115},
Volume = {641},
DOI = {10.1017/s0022112009992059}
}
@Article{rue68,
Title = {Statistical mechanics of a one-dimensional lattice gas},
Author = {D. Ruelle},
Journal = {Commun. Math. Phys.},
Year = {1968},
Pages = {267--288},
Volume = {9},
DOI = {10.1007/BF01654281}
}
@Article{ruell73,
Title = {Bifurcations in presence of a symmetry group},
Author = {D. Ruelle},
Journal = {Arch. Rational Mech. Anal.},
Year = {1973},
Pages = {136--152},
Volume = {51},
DOI = {Bifurcations in presence of a symmetry group}
}
@Article{Ruelle73NY,
Title = {Some comments on chemical oscillations},
Author = {Ruelle, D.},
Journal = {Trans. N. Y. Acad. Sci.},
Year = {1973},
Pages = {66--71},
Volume = {35},
DOI = {10.1111/j.2164-0947.1973.tb01505.x}
}
@Article{Rue76,
Title = {A measure associated with {Axiom-A} attractors},
Author = {D. Ruelle},
Journal = {Amer. J. Math.},
Year = {1976},
Pages = {619--654},
Volume = {98},
DOI = {10.2307/2373810}
}
@Article{Ruelle76a,
author = {Ruelle, D.},
title = {Generalized zeta-functions for {Axiom A} basic sets},
journal = {Bull. Amer. Math. Soc},
year = {1976},
volume = {82},
pages = {153--157},
doi = {10.1090/S0002-9904-1976-14003-7},
}
@Article{Ruelle76,
Title = {Zeta-functions for expanding maps and {Anosov} flows},
Author = {D. Ruelle},
Journal = {Inv. Math.},
Year = {1976},
Pages = {231--242},
Volume = {34},
DOI = {10.1007/BF01403069}
}
@Article{Ruelle78,
Title = {An inequality for the entropy of differentiable maps},
Author = {Ruelle, D.},
Journal = {Bol. Soc. Bras. Mat.},
Year = {1978},
Pages = {83--87},
Volume = {9},
DOI = {10.1007/BF02584795}
}
@Article{ruelle79,
Title = {Ergodic theory of differentiable dynamical systems},
Author = {D. Ruelle},
Journal = {Publ. Math. IHES},
Year = {1979},
Pages = {27--58},
Volume = {50},
DOI = {10.1007/BF02684768}
}
@Article{ruelext,
Title = {Large volume limit of the distribution of characteristic exponents in turbulence},
Author = {D. Ruelle},
Journal = {Commun. Math. Phys.},
Year = {1982},
Pages = {287--302},
Volume = {87},
DOI = {10.1007/BF01218566}
}
@Article{ruelle86,
Title = {Locating resonances for {Axiom A} dynamical systems},
Author = {D. Ruelle},
Journal = {J. Stat. Phys.},
Year = {1986},
Pages = {281--292},
Volume = {44},
DOI = {10.1007/bf01011300}
}
@Article{ruelle86a,
Title = {Resonances of chaotic dynamical systems},
Author = {D. Ruelle},
Journal = {Phys. Rev. Lett.},
Year = {1986},
Pages = {405--407},
Volume = {56},
DOI = {10.1103/physrevlett.56.405}
}
@Article{rue87b,
Title = {One-dimensional {Gibbs} states and {Axiom A} diffeomorphisms},
Author = {D. Ruelle},
Journal = {J. Diff. Geom.},
Year = {1987},
Pages = {117--137},
Volume = {25},
URL = {http://projecteuclid.org/euclid.jdg/1214440727}
}
@Article{rue87a,
Title = {Resonances for {Axiom A} flows},
Author = {D. Ruelle},
Journal = {J. Diff. Geom.},
Year = {1987},
Pages = {99--116},
Volume = {25},
URL = {http://projecteuclid.org/euclid.jdg/1214440726}
}
@Article{ruelle89,
Title = {The thermodynamic formalism for expanding maps},
Author = {D. Ruelle},
Journal = {Commun. Math. Phys.},
Year = {1989},
Pages = {239--262},
Volume = {125},
DOI = {10.1007/bf01217908},
Idoi = {10.1007/BF01217908}
}
@Article{Ruelle90,
Title = {An extension of the theory of {Fredholm} determinants},
Author = {D. Ruelle},
Journal = {Inst. Hautes \'Etudes Sci. Publ. Math.},
Year = {1990},
Pages = {175--193},
Volume = {72},
DOI = {10.1007/BF02699133}
}
@Article{Ruelle96,
Title = {Differentiation of {SRB} states},
Author = {Ruelle, D.},
Journal = {Commun. Math. Phys.},
Year = {1997},
Pages = {227--241},
Volume = {187},
DOI = {10.1007/s002200050134}
}
@Article{Ruelle98,
Title = {General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium},
Author = {D. Ruelle},
Journal = {Phys. Lett. A },
Year = {1998},
Pages = {220--224},
Volume = {245},
DOI = {10.1016/S0375-9601(98)00419-8}
}
@Article{Ruelle98a,
Title = {Nonequilibrium statistical mechanics near equilibrium: computing higher-order terms},
Author = {D. Ruelle},
Journal = {Nonlinearity},
Year = {1998},
Pages = {5},
Volume = {11-18},
DOI = {10.1088/0951-7715/11/1/002}
}
@Article{Ruelle03,
Title = {Differentiation of {SRB} states: {Correction} and complements},
Author = {D. Ruelle},
Journal = {Commun. Math. Phys.},
Year = {2003},
Pages = {185--190},
Volume = {234},
DOI = {10.1007/s00220-002-0779-z}
}
@Article{rue04ne,
Title = {Conversations on nonequilibrium physics with an extraterrestrial},
Author = {D. Ruelle},
Journal = {Phys. Today},
Year = {2004},
Number = {5},
Pages = {48--53},
Volume = {57},
Abstract = {New ideas of nonequilibrium physics are introduced. Emphasize the
SRB measure approach and energy fluctuation theorem.},
DOI = {10.1063/1.1768674}
}
@Book{ruelle,
Title = {Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics},
Author = {D. Ruelle},
Publisher = {Cambridge Univ. Press},
Year = {2004},
Address = {Cambridge},
Edition = {2\textsuperscript{nd}},
ISBN = {9780521546492}
}
@Article{Ruelle07,
Title = {Nonequilibrium statistical mechanics and entropy production in a classical infinite system of rotators},
Author = {Ruelle, D.},
Journal = {Commun. Math. Phys.},
Year = {2007},
Pages = {233--265},
Volume = {270},
DOI = {10.1007/s00220-006-0126-x}
}
@Article{Ruelle09,
author = {D. Ruelle},
title = {A review of linear response theory for general differentiable dynamical systems},
journal = {Nonlinearity},
year = {2009},
volume = {22},
pages = {855--870},
doi = {10.1088/0951-7715/22/4/009},
}
@Article{Ruelle12,
Title = {A mechanical model for {Fourier's} law of heat conduction},
Author = {Ruelle, D.},
Journal = {Commun. Math. Phys.},
Year = {2012},
Pages = {755--768},
Volume = {311},
DOI = {10.1007/s00220-011-1304-z}
}
@Article{RueSin86,
Title = {From dynamical systems to statistical mechanics and back},
Author = {D. Ruelle and Ya. G. Sinai},
Journal = {Physica A},
Year = {1986},
Pages = {1--8},
Volume = {140},
DOI = {10.1016/0378-4371(86)90199-8}
}
@Article{ruell71,
Title = {On the nature of turbulence},
Author = {D. Ruelle and F. Takens},
Journal = {Commun. Math. Phys.},
Year = {1971},
Pages = {167},
Volume = {20},
DOI = {10.1007/bf01646553}
}
@Article{hhrugh92,
Title = {The correlation spectrum for hyperbolic analytic maps},
Author = {H. H. Rugh},
Journal = {Nonlinearity},
Year = {1992},
Pages = {1237},
Volume = {5},
DOI = {10.1088/0951-7715/5/6/003}
}
@Phdthesis{RughThesis,
Title = {Time Evoluation and Correlations in Chaotic Dynamical Systems},
Author = {Rugh, H. H.},
School = {Univ. of Copenhagen},
Year = {1992},
Address = {Copenhagen}
}
@Article{hhrugh94,
Title = {On the asymptotic form and the reality of spectra of {Perron-Frobenius} operators},
Author = {H. H. Rugh},
Journal = {Nonlinearity},
Year = {1994},
Pages = {1055},
Volume = {7},
Abstract = {We study the eigenvalue spectrum of the generalized Perron-Frobenius
operator for 1-D maps having two expanding branches. We show that
if one branch 'dominates' the other, the dominating branch determines
the asymptotic form of the spectrum. In particular, we obtain sufficient
conditions for the reality of the spectrum of the usual Perron-Frobenius
operator.},
DOI = {10.1088/0951-7715/7/3/015}
}
@Article{Ruhe10,
author = {Ruhe, A.},
title = {Rational {Krylov} for real pencils with complex eigenvalues},
journal = {Taiwanese J. Math.},
year = {2010},
volume = {14},
pages = {795},
url = {http://www.jstor.org/stable/43834817},
}
@Article{Rumb00,
author = {M. Rumberger},
title = {{Lyapunov} exponents on the orbit space},
journal = {Discrete Continuous Dyn. Syst. Ser. A},
year = {2000},
volume = {7},
pages = {91--113},
doi = {10.3934/dcds.2001.7.91},
}
@Article{Rumb01,
Title = {On eigenvalues on the orbit space},
Author = {M. Rumberger},
Journal = {J. Pure Appl. Algebra},
Year = {2001},
Pages = {89--99},
Volume = {158},
DOI = {10.1016/S0022-4049(00)00023-2}
}
@Incollection{RumSch01,
Title = {The orbit space method: {Theory} and application},
Author = {M. Rumberger and J. Scheurle},
Booktitle = {{Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems}},
Publisher = {Springer},
Year = {2001},
Address = {New York},
Editor = {B. Fiedler},
URL = {http://dynamics.mi.fu-berlin.de/danse/}
}
@Article{RusHanOtt1980,
Title = {Dimension of strange attractors},
Author = {Russell, D. A. and Hanson, J. D. and Ott, E.},
Journal = {Phys. Rev. Lett.},
Year = {1980},
Pages = {1175--1178},
Volume = {45},
DOI = {10.1103/PhysRevLett.45.1175}
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@Book{haleinf84,
Title = {{An Introduction to Infinite Dimensional Dynamical Systems - Geometric Theory}},
Author = {Rybakowski, K.P. and Hale, J.K. and Magalhaes, L.T. and Oliva, W.M.},
Publisher = {Springer},
Year = {1984},
Address = {New York},
ISBN = {9781475744934}
}
@Article{SanNet10,
Title = {On the multiple shooting continuation of periodic orbits by {Newton-Krylov} methods},
Author = {S{\'a}nchez, J. and Net, M.},
Journal = {Int. J. Bifur. Chaos},
Year = {2010},
Pages = {43--61},
Volume = {20},
DOI = {10.1142/S0218127410025399}
}
@Article{SanNet13,
Title = {A parallel algorithm for the computation of invariant tori in large-scale dissipative systems},
Author = {S{\'a}nchez, J. and M. Net},
Journal = {Physica D},
Year = {2013},
Pages = {22--33},
Volume = {252},
Abstract = {A parallelizable algorithm to compute invariant tori
of high-dimensional dissipative systems, obtained upon discretization of
PDEs is presented. The size of the set of equations to be solved is
only a small multiple of the dimension of the original system. The
sequential and parallel implementations are compared with a previous
method (Sanchez et{~}al. (2010)){~}[11], showing that important savings in
wall-clock time can be achieved. In order to test it, a thermal
convection problem of a binary mixture of fluids has been used. The new
method can also be applied to problems with very low rotation numbers,
for which the previous is not suitable. This is tested in two examples of
two-dimensional maps.},
DOI = {10.1016/j.physd.2013.02.008}
}
@Article{SaNeSi10,
Title = {Computation of invariant tori by {Newton--Krylov} methods in large-scale dissipative systems},
Author = {S{\'a}nchez, J. and Net, M. and Sim{\'o}, C.},
Journal = {Physica D},
Year = {2010},
Pages = {123--133},
Volume = {239},
DOI = {10.1016/j.physd.2009.10.012}
}
@Article{sanchez2016,
author = {S{\'a}nchez-Arriaga, G. and Siminos, E.},
title = {Relativistic quasi-solitons and embedded solitons with circular polarization in cold plasmas},
journal = {J. Phys. A},
year = {2017},
volume = {50},
pages = {185501},
doi = {10.1088/1751-8121/aa65a0},
}
@Article{SSL10-1,
Title = {Relativistic solitary waves modulating long laser pulses in plasmas},
Author = {S{\'a}nchez-Arriaga, G. and Siminos, E. and Lefebvre, E.},
Journal = {Plasma Phys. Control. Fusion},
Year = {2011},
Pages = {045011},
Volume = {53},
Abstract = {This paper discusses the existence of solitary electromagnetic waves
trapped in a self-generated Langmuir wave and embedded in an infinitely
long circularly polarized electromagnetic wave propagating through
a plasma. From a mathematical point of view they are exact solutions
of the one-dimensional relativistic cold fluid plasma model with
nonvanishing boundary conditions. Under the assumption of travelling
wave solutions with velocity V and vector potential frequency ?,
the fluid model is reduced to a Hamiltonian system. The solitary
waves are homoclinic (grey solitons) or heteroclinic (dark solitons)
orbits to fixed points. Using a dynamical systems description of
the Hamiltonian system and a spectral method, we identify a large
variety of solitary waves, including asymmetric ones, discuss their
disappearance for certain parameter values and classify them according
to (i) grey or dark character, (ii) the number of humps of the vector
potential envelope and (iii) their symmetries. The solutions come
in continuous families in the parametric V?? plane and extend up
to velocities that approach the speed of light. The stability of
certain types of grey solitary waves is investigated with the aid
of particle-in-cell simulations that demonstrate their propagation
for a few tens of the inverse of the plasma frequency.},
DOI = {10.1088/0741-3335/53/4/045011},
Language = {en}
}
@Article{SSL10,
Title = {Relativistic solitary waves with phase modulation embedded in long laser pulses in plasmas},
Author = {S{\'a}nchez-Arriaga, G. and Siminos, E. and Lefebvre, E.},
Journal = {Phys. Plasmas},
Year = {2011},
Pages = {082304--082304-10},
Volume = {18},
Abstract = {We investigate the existence of nonlinear phase-modulated relativistic
solitary waves embedded in an infinitely long circularly polarized
electromagnetic wave propagating through a plasma. These states are
exact nonlinear solutions of the 1-dimensional Maxwell-fluid model
for a cold plasma composed of electrons and ions. The solitary wave,
which consists of an electromagnetic wave trapped in a self-generated
Langmuir wave, presents a phase modulation when the group velocity
V and the phase velocity Vph of the long circularly polarized electromagnetic
wave do not match the condition {VVph}?=?c2. The main properties
of the waves as a function of their group velocities, wavevectors,
and frequencies are studied, as well as bifurcations of the dynamical
system that describes the waves when the parameter controlling the
phase modulation changes from zero to a finite value. Such a transition
is illustrated in the limit of small amplitude waves where an analytical
solution for a grey solitary wave exists. The solutions are interpreted
as the stationary state after the collision of a long laser pulse
with an isolated solitary wave.},
DOI = {10.1063/1.3624498}
}
@Article{sanchez2015,
Title = {Relativistic breather-type solitary waves with linear polarization in cold plasmas},
Author = {S\'{a}nchez-Arriaga, G. and Siminos, E. and Saxena, V. and Kourakis, I.},
Journal = {Phys. Rev. E},
Year = {2015},
Pages = {033102},
Volume = {91},
Abstract = {Linearly polarized solitary waves, arising from the interaction of an intense laser pulse with a plasma, are investigated. Localized structures, in the form of exact numerical nonlinear solutions of the one-dimensional Maxwell-fluid model for a cold plasma with fixed ions, are presented. Unlike stationary circularly polarized solitary waves, the linear polarization gives rise to a breather-type behavior and a periodic exchange of electromagnetic energy and electron kinetic energy at twice the frequency of the wave. A numerical method based on a finite-differences scheme allows us to compute a branch of solutions within the frequency range Ωmin{\textless}Ω{\textless}ωpe, where ωpe and Ωmin are the electron plasma frequency and the frequency value for which the plasma density vanishes locally, respectively. A detailed description of the spatiotemporal structure of the waves and their main properties as a function of Ω is presented. Small-amplitude oscillations appearing in the tail of the solitary waves, a consequence of the linear polarization and harmonic excitation, are explained with the aid of the Akhiezer-Polovin system. Direct numerical simulations of the Maxwell-fluid model show that these solitary waves propagate without change for a long time.},
DOI = {10.1103/PhysRevE.91.033102}
}
@Article{Saad1986,
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@InProceedings{SadoEfst05,
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Booktitle = {Geometric Mechanics and Symmetry: the Peyresq Lectures},
Year = {2005},
Address = {Cambridge},
Editor = {J. Montaldi and T. Ratiu},
Pages = {211--302},
Publisher = {Cambridge Univ. Press}
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@Misc{sahai2015,
Title = {A chaotic dynamical system that paints},
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Year = {2015},
URL = {http://arXiv.org/abs/1504.02010}
}
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Title = {Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors},
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Year = {2007},
Pages = {615--620},
Volume = {14},
DOI = {10.5194/npg-14-615-2007}
}
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Year = {2008},
Pages = {675--680},
Volume = {15},
DOI = {10.5194/npg-15-675-2008}
}
@Article{SYCMR15,
Title = {Reconstruction of chaotic saddles by classification of unstable periodic orbits: {Kuramoto-Sivashinsky} equation},
Author = {Saiki, Y. and Yamada, M. and Chian, A. C.-L. and Miranda, R. A. and Rempel, E. L.},
Journal = {Chaos},
Year = {2015},
Pages = {103123},
Volume = {25},
DOI = {10.1063/1.4933267}
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DOI = {10.1016/s0377-0427(03)00565-x}
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Title = {Oseledets' splitting of standard-like maps},
Author = {Sala, M. and Artuso, R.},
Journal = {Chaos},
Year = {2015},
Volume = {25},
DOI = {10.1063/1.4909524}
}
@Incollection{Sald12,
author = {L. E. Saldana},
title = {A survey of spiral wave studies: {Dynamics} and symmetries},
booktitle = {{ChaosBook.org/projects}},
publisher = {Georgia Inst. of Technology},
year = {2012},
chapter = {A survey of spiral wave studies: {Dynamics} and symmetries},
url = {http://ChaosBook.org/projects/index.shtml#Saldana},
}
@Book{Salmon98,
Title = {Lectures on Geophysical Fluid Dynamics},
Author = {Salmon, R.},
Publisher = {Oxford Univ. Press},
Year = {1998},
Address = {Oxford}
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@Article{samelson01,
Title = {{Lyapunov}, {Floquet}, and singular vectors for baroclinic waves},
Author = {Samelson, R. M.},
Journal = {Nonlin. Proc. Geophys.},
Year = {2001},
Pages = {439--448},
Volume = {8},
DOI = {10.5194/npg-8-439-2001}
}
@Article{same01,
Title = {Periodic orbits and disturbance growth for baroclinic waves},
Author = {Samelson, R. M.},
Journal = {J. Atmos. Sci.},
Year = {2001},
Pages = {436--450},
Volume = {58},
DOI = {10.1175/1520-0469(2001)058<0436:poadgf>2.0.co;2}
}
@Article{samelson03,
Title = {A nonlinear baroclinic wave-mean oscillation with multiple normal-mode instabilities},
Author = {Samelson, R. M. and Wolfe, C. L.},
Journal = {J. Atmos. Sci.},
Year = {2003},
Pages = {1186--1199},
Volume = {60},
DOI = {10.1175/1520-0469(2003)060<1186:anbwow>2.0.co;2}
}
@Article{Samoilenko2005,
Title = {Conditions for synchronization of one oscillation system},
Author = {A. M. Samoilenko and L. Recke},
Journal = {Ukrain. Math. J.},
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Title = {Recurrent Spatio-temporal Structures in Presence of Continuous Symmetries},
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School = {School of Physics, Georgia Inst. of Technology},
Year = {2009},
Address = {Atlanta},
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@Misc{siminos2014c,
author = {Siminos, E.},
title = {Signatures of relativistic chaos in optical lattices, work in progress},
year = {2014},
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@Article{siminos11,
Title = {Stability of nonlinear {Vlasov-Poisson} equilibria through spectral deformation and {Fourier-{Hermit}e} expansion},
Author = {Siminos, Evangelos and B\'enisti, Didier and Gremillet, Laurent},
Journal = {Phys. Rev. E},
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Pages = {056402},
Volume = {83},
Abstract = {We study the stability of spatially periodic, nonlinear Vlasov-Poisson
equilibria as an eigenproblem in a Fourier-Hermite basis (in the
space and velocity variables, respectively) of finite dimension,
N. When the advection term in the Vlasov equation is dominant, the
convergence with N of the eigenvalues is rather slow, limiting the
applicability of the method. We use the method of spectral deformation
introduced by Crawford and Hislop Ann. Phys. ({NY)} 189 265 (1989)
to selectively damp the continuum of neutral modes associated with
the advection term, thus accelerating convergence. We validate and
benchmark the performance of our method by reproducing the kinetic
dispersion relation results for linear (spatially homogeneous) equilibria.
Finally, we study the stability of a periodic Bernstein-Greene-Kruskal
mode with multiple phase-space vortices, compare our results with
numerical simulations of the Vlasov-Poisson system, and show that
the initial unstable equilibrium may evolve to different asymptotic
states depending on the way it was perturbed.},
DOI = {10.1103/PhysRevE.83.056402}
}
@Unpublished{SCD09b,
Title = {Symmetry reduction: {Geometry} of a {Kuramoto-Sivashinsky} attractor revealed},
Author = {Siminos, E. and Budanur, N. B. and Cvitanovi{\'c}, P. and Davidchack, R. L.},
Note = {In preparation},
Year = {2015}
}
@Article{SiCvi10,
Title = {Continuous symmetry reduction and return maps for high-dimensional flows},
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Journal = {Physica D},
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@Misc{siminos2014p,
author = {Siminos, E. and Grech, M. and Skupin, S.},
title = {Onset of self-induced transparency in the hole boring regime in relativistic laser-plasma interaction, in preparation},
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@Article{siminos2012,
Title = {Effect of electron heating on self-induced transparency in relativistic-intensity laser-plasma interactions},
Author = {Siminos, E. and Grech, M. and Skupin, S. and Schlegel, T. and Tikhonchuk, V. T.},
Journal = {Phys. Rev. E},
Year = {2012},
Pages = {056404},
Volume = {86},
Abstract = {The effective increase of the critical density associated with the
interaction of relativistically intense laser pulses with overcritical
plasmas, known as self-induced transparency, is revisited for the
case of circular polarization. A comparison of particle-in-cell simulations
to the predictions of a relativistic cold-fluid model for the transparency
threshold demonstrates that kinetic effects, such as electron heating,
can lead to a substantial increase of the effective critical density
compared to cold-fluid theory. These results are interpreted by a
study of separatrices in the single-electron phase space corresponding
to dynamics in the stationary fields predicted by the cold-fluid
model. It is shown that perturbations due to electron heating exceeding
a certain finite threshold can force electrons to escape into the
vacuum, leading to laser pulse propagation. The modification of the
transparency threshold is linked to the temporal pulse profile, through
its effect on electron heating.},
DOI = {10.1103/PhysRevE.86.056404}
}
@Article{siminos2014,
Title = {Modeling relativistic soliton interactions in overdense plasmas: {A} perturbed nonlinear {Schr{\"o}dinger} equation framework},
Author = {E. Siminos and G. S{\'a}nchez-Arriaga and V. Saxena and I. Kourakis},
Journal = {Phys. Rev. E},
Year = {2014},
Pages = {063104},
Volume = {90},
Abstract = {We investigate the dynamics of localized solutions of the relativistic cold-fluid plasma model in the small but finite amplitude limit, for slightly overcritical plasma density. Adopting a multiple scale analysis, we derive a perturbed nonlinear Schrödinger equation that describes the evolution of the envelope of circularly polarized electromagnetic field. Retaining terms up to fifth order in the small perturbation parameter, we derive a self-consistent framework for the description of the plasma response in the presence of localized electromagnetic field. The formalism is applied to standing electromagnetic soliton interactions and the results are validated by simulations of the full cold-fluid model. To lowest order, a cubic nonlinear Schrödinger equation with a focusing nonlinearity is recovered. Classical quasiparticle theory is used to obtain analytical estimates for the collision time and minimum distance of approach between solitons. For larger soliton amplitudes the inclusion of the fifth-order terms is essential for a qualitatively correct description of soliton interactions. The defocusing quintic nonlinearity leads to inelastic soliton collisions, while bound states of solitons do not persist under perturbations in the initial phase or amplitude.},
DOI = {10.1103/PhysRevE.90.063104},
Shorttitle = {Modeling relativistic soliton interactions in overdense plasmas}
}
@Misc{siminos2014s,
author = {Siminos, E. and S\'anchez-Arriaga, G. and Saxena, V. and Kourakis, I.},
title = {Describing relativistic soliton interactions through envelope equations, in preparation},
year = {2014},
}
@Phdthesis{Simonis2006,
author = {Simonisk, J. P.},
title = {{Inexact {Newton} Methods Applied to Under-Determined Systems}},
school = {Polytechnic Inst.},
year = {2006},
address = {Worcester, MA},
url = {https://web.wpi.edu/Pubs/ETD/Available/etd-050406-103442/unrestricted/simonis.pdf},
}
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Title = {{Introduction to Ergodic Theory}},
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editor = {Skiadas, C.H. and Dimotikalis, I. and Skiadas, C.},
publisher = {World Scientific},
isbn = {9789814460477},
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Abstract = {Chaotic dynamics with sensitive dependence on initial conditions may
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is, Lyapunov exponents and exponential decay rates, are related.
More specifically, for piecewise linear expanding Markov maps observed
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operator. In addition, we comment on similar relations for general
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Abstract = {We create polynomial differential equations for a suspension of the
{H\'enon} map embedded in R 3 . By globalizing the local tangent
vectors to suspended periodic orbits of the {H\'enon} map, we are
able to find approximate autonomous differential equations for that
geometric suspension. Using as few as two suspended periodic orbits,
we can generate a robust three dimensional attractor whose Poincar\'e
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Title = {{Exact Solutions of {Einstein's} Field Equations}},
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abstract = {We present a method for constructing a quantum Markov partition. Its elements are obtained by quantizing the characteristic function of the classical rectangles. The result is a set of quantum operators which behave asymptotically as projectors over the classical rectangles apart from edge and corner effects. We investigate their spectral properties and different methods of construction. The quantum partition is shown to induce a symbolic decomposition of the quantum evolution operator. In particular, an exact expression for the traces of the propagator is obtained having the same structure as the Gutzwiller periodic orbit sum.},
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title = {{Mechanics}},
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@Book{Landau59c,
title = {{Quantum Mechanics: Non-Relativistic Theory}},
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@Book{Landau60,
title = {{Electrodynamics of Continuous Media}},
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address = {Oxford},
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url = {https://archive.org/search.php?query=creator%3A%22L.D.+Landau+%26+E.M.+Lifshitz%22},
}
@Book{Landau80a,
title = {{Statistical Physics, Part 1}},
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address = {Oxford},
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}
@Book{Landau80b,
title = {{Statistical Physics, Part 2: Theory of the Condensed State}},
publisher = {Pergamon Press},
year = {1980},
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address = {Oxford},
url = {https://archive.org/search.php?query=creator%3A%22L.D.+Landau+%26+E.M.+Lifshitz%22},
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@Book{LifPit,
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author = {K{\"u}bra Duru, H. and Bozkurt, D.},
title = {Integer powers of certain complex pentadiagonal 2-{Toeplitz} matrices},
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url = {https://arxiv.org/abs/1708.04121},
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author = {T.E. Tezduyar, and M. Behr, S.K. Aliabadi, S. Mittal and S.E. Ray},
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title = {{Convex Optimization}},
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url = {http://www.cis.pku.edu.cn/faculty/vision/zlin/1983-A%20Method%20of%20Solving%20a%20Convex%20Programming%20Problem%20with%20Convergence%20Rate%20O(k%5E(-2))_Nesterov.pdf},
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volume = {18},
pages = {436--443},
abstract = {A finite-difference Green's function method for solving time-harmonic wave guide scattering problems involving metallic obstacles of finite size by computer is described. The method is applied to the two-dimensional problem of a TE10 mode impinging on cylindrical metallic posts of arbitrary shape in a rectangular waveguide. The equivalent susceptance of a transverse semidiaphragm computed using a 50 point approximation for the induced current distribution is found to be 1.5 percent less than the exact value. The S matrix of a thin bent window versus wavelength is also presented.},
doi = {10.1109/TMTT.1970.1127265},
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author = {Serkh, K. and Rokhlin, V.},
title = {On the solution of the {Helmholtz} equation on regions with corners},
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year = {2016},
volume = {113},
pages = {9171--9176},
doi = {10.1073/pnas.1609578113},
abstract = {In this paper we solve several boundary value problems for the Helmholtz equation on polygonal domains. We observe that when the problems are formulated as the boundary integral equations of potential theory, the solutions are representable by series of appropriately chosen Bessel functions. In addition to being analytically perspicuous, the resulting expressions lend themselves to the construction of accurate and efficient numerical algorithms. The results are illustrated by a number of numerical examples.},
}
@Inproceedings{dlLlave00,
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address = {Singapore},
publisher = {World Scientific},
doi = {10.1142/4557},
url = {http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=00-56},
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organization = {Soc. Italiana di Fisica Conf. Proceed.},
publisher = {Ed. Compositori},
url = {https://arXiv.org/abs/chao-dyn/9304003},
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title = {{Theoretical Mechanics of Particles and Continua}},
publisher = {Dover},
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address = {New York},
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@TechReport{Crutchfield18,
author = {J. Crutchfield},
title = {{Roadmap for Natural Computation and Self-Organization}},
institution = {U. California, Davis},
year = {2017},
note = {Physics 256A course},
url = {http://csc.ucdavis.edu/~chaos/courses/ncaso/Lectures/},
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author = {M. Axenides and E. Floratos and S. Nicolis},
title = {The quantum cat map on the modular discretization of extremal black hole horizons},
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@Incollection{CBMarkov,
author = {P. Cvitanovi{\'c}},
title = {Walkabout: {Transition} graphs},
booktitle = {{Chaos: Classical and Quantum}},
publisher = {Niels Bohr Inst.},
year = {2018},
editor = {P. Cvitanovi{\'c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
address = {Copenhagen},
url = {http://ChaosBook.org/paper.shtml#Marko},
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@Article{AWGBG17,
author = {M. Akila and D. Waltner and B. Gutkin and P. Braun and T. Guhr},
title = {Collectivity and periodic orbits in a chain of interacting, kicked spins},
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author = {Aoki, K.},
title = {Symmetry, chaos and temperature in the one-dimensional lattice {$\phi^4$} theory},
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author = {P. Cvitanovi\'{c} and R. Artuso and L. Rondoni and E. A. Spiegel},
title = {Transporting densities},
booktitle = {{Chaos: Classical and Quantum}},
publisher = {Niels Bohr Inst.},
year = {2017},
editor = {P. Cvitanovi{\'c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
address = {Copenhagen},
url = {http://ChaosBook.org/paper.shtml#measure},
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@Comment{jabref-meta: databaseType:bibtex;}