% P. Cvitanovic Apr 25 2007
% extracted from ~lippolis/articles/lippolis/bibtex/lippolis.bib
% derived from ../y-lan/bibtex/nonlind.bib
% based on waleffebib.bib, Waleffe bibliography Sep 4 2005
% $Author: predrag $ $Date: 2007-05-05 15:27:24 -0400 (Sat, 05 May 2007) $
***************************************************************************
ARTICLE{,
author = {},
title = {},
journal = {},
volume = {},
number = {},
pages = {},
year = {},
abstract = {}
}
BOOK{,
author = {},
title = {},
publisher = {},
address = {},
year = {}
}
@article{R1883,
AUTHOR="Reynolds, Osborne",
TITLE=" An experimental investigation of the circumstances which determine
whether the motion of water shall be direct or sinuous, and the law of resistance
in parallel channels",
JOURNAL="Proc. Roy. Soc. Lond. Ser A",
YEAR="1883",
VOLUME="174",
PAGES="935-982"
}
@article{FE03,
AUTHOR="Faisst, H. and Eckhardt, B.",
TITLE="Traveling waves in Pipe Flow",
JOURNAL="Phys. Rev. Lett.",
YEAR="2003",
VOLUME="91",
PAGES="224502",
}
@article{WK04,
AUTHOR="Wedin, H. and Kerswell, R.R.",
TITLE="Exact coherent structures in Pipe flow",
JOURNAL="J.\ Fluid Mech.",
YEAR="2004",
VOLUME="508",
PAGES="333-371",
}
@article{TTRD93,
AUTHOR="Trefethen, N. and Trefethen, A.E. and Reddy, S.C. and
Driscoll, T.A.",
TITLE="Hydrodynamic Stability without Eigenvalues",
JOURNAL="Science",
YEAR="1993",
VOLUME="261",
PAGES="578-584",
}
@article{N90,
AUTHOR="Nagata, M.",
TITLE="Three-dimensional finite-amplitude solutions in plane {C}ouette flow: bifurcation from infinity. ",
JOURNAL="J.\ Fluid Mech.",
YEAR="1990",
VOLUME="217",
PAGES="519-527",
}
@article{EM99,
AUTHOR="Eckhardt, B. and Mersmann, A.",
TITLE="Transition to turbulence in a shear flow",
JOURNAL="Phys.\ Rev.\ E",
YEAR="1999",
VOLUME="60",
PAGES="509-517"}
@article{SE00,
AUTHOR="Schmiegel, A. and Eckhardt, B.",
TITLE="Persistent turbulence in annealed plane {C}ouette flow",
JOURNAL="Europhys.\ Lett.",
YEAR="2000",
VOLUME="51",
PAGES="395--400"}
@article{HJM03,
AUTHOR="Hof, B. and Juel, A. and Mullin, T.",
TITLE="Scaling of the Turbulence Transition Threshold in a Pipe",
JOURNAL="Phys.\ Rev.\ Lett.",
YEAR="2003",
VOLUME="91",
PAGES="244502"}
@article{science04,
author = {Hof, B. and van Doorne, C. W.H. and
Westerweel, J. and Nieuwstadt, F. T.M. and Faisst, H. and
Eckhardt, B. and Wedin, H. and Kerswell, R. R. and Waleffe, F.},
title = {Experimental Observation of Nonlinear Traveling Waves in Turbulent Pipe Flow},
journal = {Science},
volume = {305},
number = {5690},
pages = {1594-1598},
year = {2004},
abstract = {Transition to turbulence in pipe flow is one of the most fundamental and longest-standing problems in fluid dynamics. Stability theory suggests that the flow remains
laminar for all flow rates, but in practice pipe flow becomes turbulent even at moderate
speeds. This transition drastically affects the transport efficiency of mass, momentum, and heat. On the basis of the recent discovery of unstable traveling waves in computational studies of the Navier-Stokes equations and ideas from dynamical systems theory, a model for the transition process has been suggested. We report experimental observation of these traveling waves in pipe flow, confirming the proposed transition scenario and suggesting that the dynamics associated with these unstable states may indeed capture the nature
of fluid turbulence.
},
URL = {http://www.sciencemag.org/cgi/content/abstract/305/5690/1594},
eprint = {http://www.sciencemag.org/cgi/reprint/305/5690/1594.pdf}
}
% = {Hof, Bjorn and van Doorne, Casimir W.H. and
Westerweel, Jerry and Nieuwstadt, Frans T.M. and Faisst, Holger and
Eckhardt, Bruno and Wedin, Hakan and Kerswell, Richard R. and Waleffe, Fabian},
@article{PhysToday04,
AUTHOR="Fitzgerald, Richard",
TITLE="New Experiments Set the Scale for the Onset of Turbulence in Pipe Flow",
JOURNAL="Physics Today",
YEAR="2004",
VOLUME="57",
NUMBER="2",
PAGES="21-23",}
@article{MFE04,
AUTHOR="Moehlis, J. and Faisst, H. and Eckhardt, B.",
TITLE="A low-dimensional model for turbulent shear flows",
JOURNAL="New Journal of Physics",
YEAR="2004",
VOLUME="6",
PAGES="56+17"}
@article{W90,
AUTHOR="Waleffe, F.",
TITLE="Proposal for a self-sustaining process in shear flows",
JOURNAL="Working paper, available at
\verb+www.math.wisc.edu/~waleffe/ECS/sspctr90.pdf+",
YEAR="1990",
VOLUME=" ",
PAGES=" "}
@article{W95a,
AUTHOR="Waleffe, F.",
TITLE="Hydrodynamic Stability and Turbulence:
Beyond transients to a self-sustaining process ",
JOURNAL="Stud.\ Applied Math.",
YEAR="1995",
VOLUME="95",
PAGES="319--343"}
@article{W95b,
AUTHOR="Waleffe, F.",
TITLE="Transition in shear flows:
Nonlinear normality versus non-normal linearity",
JOURNAL="Phys.\ Fluids",
YEAR="1995",
VOLUME="7",
PAGES="3060-3066"}
@article{W97,
AUTHOR="Waleffe, F.",
TITLE="On a self-sustaining process in shear flows",
JOURNAL="Phys.\ Fluids",
YEAR="1997",
VOLUME="9",
PAGES="883-900"}
@article{HaKiWa95,
AUTHOR="Hamilton, J. and Kim, J. and Waleffe, F.",
TITLE="Regeneration mechanisms of near-wall turbulence structures",
JOURNAL="J.\ Fluid Mech.",
YEAR="1995",
VOLUME="287",
PAGES="317-348"}
@inproceedings{WKH93,
AUTHOR="Waleffe, F. and Kim, J. and Hamilton, J.",
TITLE="On the origin of streaks in turbulent shear flows",
BOOKTITLE="Turbulent Shear Flows 8:
selected papers from the Eighth International
Symposium on Turbulent Shear Flows, Munich, Germany, Sept.\ 9-11, 1991",
EDITOR="Durst, F. and Friedrich, R. and Launder, B.E. and
Schmidt, F.W. and Schumann, U. and Whitelaw, J.H.",
YEAR="1993",
PUBLISHER="Springer-Verlag, Berlin",
PAGES="37-49"}
@inproceedings{W02,
AUTHOR="Waleffe, F. ",
TITLE=" Exact coherent structures and their instabilities: Toward a dynamical-system theory of shear turbulence",
BOOKTITLE="Proceedings of the International Symposium on ``Dynamics and Statistics of Coherent Structures in Turbulence: Roles of Elementary Vortices''",
EDITOR="Kida, Shigeo",
YEAR="2002",
PUBLISHER="National Center of Sciences, Tokyo, Japan",
PAGES="115-128"}
@article{W98,
AUTHOR="Waleffe, F.",
TITLE="Three-Dimensional Coherent States in Plane Shear Flows",
JOURNAL="Phys.\ Rev.\ Lett.",
YEAR="1998",
VOLUME="81",
PAGES="4140-4148"}
@article{W01,
AUTHOR="Waleffe, F. ",
TITLE="Exact Coherent Structures in Channel Flow",
JOURNAL="J.\ Fluid Mech.",
YEAR="2001",
VOLUME="435",
PAGES="93-102"}
@article{W03,
AUTHOR="Waleffe, F.",
TITLE="Homotopy of exact coherent structures in plane shear flows",
JOURNAL="Phys.\ Fluids",
YEAR="2003",
VOLUME="15",
PAGES="1517-1543"}
@inproceedings{WW05,
AUTHOR="Waleffe, F. and Wang, J.",
TITLE="Transition threshold and the Self-Sustaining Process",
BOOKTITLE="Non-uniqueness of Solutions to the Navier-Stokes Equations and their Connection with Laminar-Turbulent Transition",
EDITOR="Mullin, T. and Kerswell, R.R.",
YEAR="2005",
PUBLISHER="Kluwer",
PAGES="85-106"}
************** AAAAAAAAAAA ***************************
@ARTICLE{AGHks89,
author = {D. Armbuster and J. Guckenheimer and P. Holmes},
title = {{K}uramoto-{S}ivashinsky Dynamics on the Center-Unstable manifold},
journal = {SIAM J. Appl. Math.},
volume = {49},
number = {3},
pages = {676-691},
year = {1989}
}
@BOOK{arnold92,
author = {V. I. Arnol'd},
title = {Ordinary Differential Equations},
publisher = {Springer},
address = {New York},
year = {1992}
}
@ARTICLE{arnold91k,
author = {V. I. Arnold},
title = {Kolmogorov's hydrodynamic attractors},
journal = {Proc. R. Soc. Lond. A},
volume = {434},
number = {1890},
pages = {19-22},
year = {1991},
abstract = {A review of Kolmogorov's efforts relating the Navier-Stokes equation
to the theory of dynamical system. Several interesting questions regarding the
connection are exposed.}
}
@ARTICLE{AACI,
AUTHOR = {R. Artuso and E. Aurell and P. Cvitanovi{\'{c}}},
title = {Recycling of strange sets: I. Cycle expansions},
journal = {Nonlinearity},
volume = {3},
PAGES = {325-359},
year = {1990}
}
@ARTICLE{AACII,
AUTHOR = {R. Artuso and E. Aurell and P. Cvitanovi{\'{c}}},
title = {Recycling of strange sets: II. Applications},
journal = {Nonlinearity},
volume = {3},
pages = {361},
year = {1990}
}
@ARTICLE{AAC90,
AUTHOR = "Roberto Artuso and Erik Aurell and Predrag Cvitanovi\'c",
TITLE = "Recycling of strange sets: {I}. Cycle expansions",
JOURNAL = "Nonlinearity",
VOLUME = 3,
YEAR = 1990,
PAGES = "325--359"}
@ARTICLE{art03int,
author = {R. Artuso and P. Cvitanovi\'{c} and G. Tanner},
title = {Cycle expansions for intermittent maps},
journal = {Proc. Theo. Phys. Supp.},
volume = {150},
pages = {1-21},
year = {2003}
}
@ARTICLE{pchaot,
author = {D. Auerbach and P. Cvitanovi\'{c} and J.-P. Eckmann and
G. Gunaratne and I. Procaccia },
title = {Exploring Chaotic Motion Through Periodic Orbits},
journal = {Phys. Rev. Lett.},
volume = {58},
pages = {23},
year = {1987},
abstract = {Use near recurrence to extract periodic orbits and their stability
eigenvalues.}
}
************** BBBBBBBBBBB ***************************
@ARTICLE{BK90,
AUTHOR = {M. V. Berry and J. P. Keating},
TITLE = {A rule for quantizing chaos},
JOURNAL = {Journal of Physics A},
YEAR = 1990,
VOLUME = 23,
NUMBER = 21,
PAGES = {4839-4849}
}
@BOOK{gauge,
author = {T. S. Bir\'{o} and S. G. Matinyan and B. M\"{u}ller},
title = {Chaos and Gauge Field Theory},
publisher = {World Scientific},
address = {Singapore},
year = {1994}
}
@BOOK{BH86,
AUTHOR = {N. Bleistein and R. A. Handelsman},
TITLE = {Asymptotic Expansions of Integrals},
YEAR = 1986,
PUBLISHER = {Dover},
ADDRESS = {New York}
}
@BOOK{bowen,
AUTHOR = {R. Bowen},
TITLE = {Equilibrium states and the ergodic theory of {A}nosov-diffeomorphisms},
YEAR = 1975,
VOLUME = 470,
PUBLISHER = {Springer-Verlag},
SERIES = {Lecture Notes in Mathematics},
ADDRESS = {Berlin}
}
************** CCCCCCCCCCC ***************************
@Article{Christiansen:97,
author = {F. Christiansen and P. Cvitanovi\'{c} and V. Putkaradze},
title = {Spatiotemporal Chaos in Terms of Unstable Recurrent Patterns},
journal = {Nonlinearity},
volume = {10},
pages = {55},
year = {1997}
}
%cites Christiansen:97
@misc{misiurewicz-topological,
author = "Michal Misiurewicz and Piotr Zgliczynski",
title = "Topological Entropy for Multidimensional Perturbations of One-Dimensional
Maps",
url = "citeseer.ist.psu.edu/410408.html",
abstract = {Numerical experiments in [Christiansen:97] show that a suitably chosen
Poincar e map P is essentially one dimensional and can be modeled by a
one dimensional map p. Now we build a homotopy by F ( x; w) P (x; w) 1 ) p(x) 0)
This homotopy is compact. We cannot claim that we can use Theorem 1. 1 to estimate
rigorously topological entropy for the Poincar e map P , because even there Entropy
for multidimensional perturbations 3 is no rigorous proof that the Poincar e map P
studied numerically in [Christiansen:97] exists.
However, the ideas used in the proof of Theorem 1.1, continuation of topological
horseshoes and relation between topological horseshoes and entropy,
combined with recently developed rigorous numerics for KS equations
(see [zgliczynski-rigorous, 2000])
}
}
@misc{zgliczynski-rigorous,
author = "Piotr Zgliczynski and et al.",
title = "Rigorous Numerics for Partial Differential Equations: the Kuramoto-Sivashinsky
equation",
url = "citeseer.ist.psu.edu/399185.html" }
%cites Christiansen:97
@misc{zimmermann-global,
author = "MartÃn G. Zimmermann",
title = "Global Bifurcations and Chaotic Dynamics in Physical Applications",
url = "citeseer.ist.psu.edu/zimmermann97global.html",
abstract = {
The complex interactions between spatial and temporal components of the dynamics,
known as spatiotemporal chaos, only recently begun to be studied using ideas from
fundamental dynamical systems theory
(Temam, 1988; Goren et al. 1995; Christiansen et al. 1997).
The present dissertation is primarily interested in studying the appearance
of global bifurcations in two specific physical examples, a laser with an
injected signal, and a catalytic reaction on a platinum surface. Characteristic
of these bifurcations is the presence of a highly degenerate ....
One of the main motivations to study these equations is that some of them display
spatiotemporal chaos, which in many ways, resembles turbulence in fluid dynamics.
Recently, different efforts to uncover the dynamical origin for these complex states
have been developing (Goren et al. 1995; Christiansen et al. 1997).
}
}
@Article{ cross93,
author = "M. C. Cross and P. C. Hohenberg",
title = "Pattern formation outside of equilibrium",
journal = "Rev. Mod. Phys",
volume = "65",
number = "3",
pages = "851",
year = "1993"
}
@BOOK{DasBuch,
author = {P. Cvitanovi\'{c} and
R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
title = {Chaos: Classical and Quantum},
publisher = {Niels Bohr Institute},
address = {Copenhagen},
year = {2005},
note = {{\tt ChaosBook.org}}
}
@ARTICLE{chfield,
author = {Predrag Cvitanovi\'{c}},
title = {Chaotic Field Theory: a Sketch},
journal = {Physica A},
volume = {288},
pages = {61},
year = {2000},
note = {{\tt nlin.CD/0001034}}
}
@ARTICLE{CvitaEckardt,
author = {P. Cvitanovi\'{c} and B. Eckhardt},
title = {Symmetry decomposition of chaotic dynamics},
journal = {Nonlinearity},
volume = {6},
pages = {277},
year = {1993},
note = {{\tt chao-dyn/9303016}}
}
@ARTICLE{CE91,
AUTHOR = "Predrag Cvitanovi\'c and Bruno Eckhardt",
TITLE = "Periodic orbit expansions for classical smooth flows",
JOURNAL = "Journal of Physics A",
VOLUME = 24,
PAGES = "L237--L241",
YEAR = 1991}
@INPROCEEDINGS{Cvi92,
AUTHOR = "Predrag Cvitanovi\'c",
TITLE = "The Power of Chaos",
BOOKTITLE = "Applications of Chaos",
ORGANIZATION = "EPRI workshop",
EDITOR = "J. H. Kim and J. Stringer",
PUBLISHER = "John Wiley \& Sons",
ADDRESS = "New York",
YEAR = 1992}
@ARTICLE{skeleton,
author = {P. Cvitanovi\'{c}},
title = {Periodic Orbits as the skeleton of classical and quantum chaos},
journal = {Physica D},
volume = {51},
pages = {138},
year = {1991}
}
@ARTICLE{inv,
author = {Predrag Cvitanovi\'{c}},
title = {Invariant measurement of strange sets in terms of cycles},
journal = {Phys. Rev. Lett.},
volume = {61},
pages = {2729},
year = {1988}
}
@INPROCEEDINGS{CvitLanCrete02,
AUTHOR = {P. Cvitanovi\'{c} and Y. Lan},
ADDRESS = {Singapore},
BOOKTITLE = {Proceedings of 10th International Workshop
on Multiparticle Production:
Correlations and Fluctuations in QCD},
PUBLISHER = {World Scientific},
EDITOR = {N. Antoniou},
TITLE = {Turbulent fields and their recurrences},
YEAR = {2003},
NOTE ={{\tt nlin.CD/0308006}}
}
************** DDDDDDDDDDD ***************************
@ARTICLE{DR91,
author = {P. Dahlqvist and G. Russberg},
title = {Cycle expansions for intermittent diffusion },
journal = {J. Phys. A},
volume = {24},
pages = {4763},
year = {1991}
}
@ARTICLE{DM97,
author = {C. P. Dettmann and G. P. Morriss},
journal = {Phys. Rev. Lett.},
volume = {78},
pages = {4201},
year = {1997}
}
@ARTICLE{carl97int,
author = {C. P. Dettmann and P. Cvitanovi\'{c}},
title = {Cycle expansions for intermittent diffusion },
journal = {Phys. Rev. E},
volume = {56},
pages = {6687},
year = {1997}
}
@ARTICLE{doercgl88,
author = {C. R. Doering and J. D. Gibbon and D. D. Holm
and B. Nikolaenko},
title = {Low-dimensional behaviour in the complex {G}inzburg-{L}andau
equation},
journal = {Nonlinearity},
volume = {1},
pages = {279-309},
year = {1988},
abstract = {The stability of plane waves of the CGLe is analyzed in a
periodic domain. The mass and energy integrals are estimated to give an
upper bound of the magnitude of the field variable. After that, cone
condition is established for the CGLe which indicates the existence of
the inertial manifold and proves the existence of finite dimensionality of
the attractor in this infinite-dimensional system. The Lyapunov dimension
is estimated. Note the defition of Lyapunov exponents which is a little
different from usual.}
}
@ARTICLE{doercgl87,
author = {C. R. Doering and J. D. Gibbon and D. D. Holm
and B. Nikolaenko},
title = {Exact {L}yapunov Dimension of the Universal Attractor for the
Complex {G}inzburg-{L}andau Equation},
journal = {Phys. Rev. Lett.},
volume = {59},
number = {26},
pages = {2911-2914},
year = {1987},
}
************** EEEEEEEEEEE ***************************
@ARTICLE{eck81,
author = {J.-P. Eckmann},
title = {Roads to turbulence in dissipative dynamical systems},
journal = {Rev. Mod. Phys.},
volume = {53},
pages = {643},
year = {1981}
}
@ARTICLE{eckerg,
author = {J.-P. Eckmann and D. Ruelle},
title = {Ergodic theory of chaos and strange attractors},
journal = {Rev. Mod. Phys.},
volume = {57},
number = {3},
pages = {617},
year = {1985}
}
@BOOK{eckman80map,
author = {J.-P. Eckman and P. Collet},
title = {Iterated Maps on the Interval as Dynamical Systems},
series = {Progress in Physics},
number = {1},
publisher = {Birkh\"{a}user},
address = {Boston},
year = {1980}
}
@ARTICLE{ecklp88,
author = {J. -P. Eckmann and C. E. Wayne },
title = {Liapunov spectra for infinite chains of nonlinear oscillators },
journal = {J. Stat. Phys.},
volume = {50},
pages = {853-878},
year = {1988}
}
************** FFFFFFFFFFF ***************************
@ARTICLE{fl92,
author = {C. Falcolini and R. de la Llave},
title = {Numerical calculation of domains of analyticity for perturbation
theories in the presence of small divisors},
journal = {J. Stat. Phys.},
volume = {67},
pages = {645-666},
year = {1992},
abstract = {Study the complex domains of validity in generalized
standard mappings and compare Pad\'{e} methods and periodic orbit methods.}
}
@ARTICLE{Foiasks88,
author = {C. Foias and B. Nicolaenko and G.R. Sell and R. T\'{e}mam},
title = {{K}uramoto-{S}ivashinsky equation},
journal = {J. Math. Pures et Appl.},
volume = {67},
pages = {197},
year = {1988}
}
************** GGGGGGGGGGG ***************************
@BOOK{PG97,
AUTHOR = {P. Gaspard},
TITLE = {Chaos, Scattering and Statistical Mechanics},
YEAR = 1997,
PUBLISHER = {Cambridge University Press},
ADDRESS = {Cambridge}
}
@ARTICLE{heron87,
author = {J. M. Ghidaglia and B. H\'{e}ron},
title = {Dimension of the attractors associated to the {G}inzburg-{L}andau partial
differential equation},
journal = {Physica D},
volume = {28},
pages = {282-304},
year = {1987}
}
@PhdThesis{GibsonPhD,
author = {J.F. Gibson},
title = {Dynamical-systems models of wall-bounded, shear-flow turbulence},
school = {Cornell University},
year = {2002}
}
@article{Berkooz94,
author = {G. Berkooz and P. Holmes and J.L. Lumley and N. Aubry and E. Stone},
title = {Observations regarding
``{C}oherence and chaos in a model of turbulent boundary layer''
by {X}. {Z}hou and {L}. {S}irovich},
journal = {Phys. Fluids},
year = {1994},
month = {April},
volume = {6},
number = {4},
pages = {1574--1578}
}
@unpublished{GHCW06,
author = {J. F. Gibson and J. Halcrow and P. Cvitanovi\'c
and F. Waleffe},
title = {On the geometry of state space of a turbulent
plane {C}ouette flow: I {E}xact Coherent Solutions},
year = {2007},
NOTE ={In preparation}
}
@ARTICLE{greb90shad,
author = {C. Grebogi and S. M. Hammel and J. A. Yorke and T. Sauer},
title = {Shadowing of physical trajectories in chaotic dynamics:
Containment and refinement},
journal = {Phys. Rev. Lett.},
volume = {65},
pages = {1527-1530},
year = {1990}
}
@BOOK{MiraGum80,
AUTHOR = {I. Gumowski and C. Mira},
TITLE = {Recurrances and Discrete Dynamical Systems},
YEAR = 1980,
PUBLISHER = {Springer-Verlag},
ADDRESS = {Berlin}
}
@BOOK{gutbook,
author = {M.C. Gutzwiller},
title = {Chaos in Classical and Quantum Mechanics},
publisher = {Springer-Verlag},
address = {New York},
year = {1990}
}
************** HHHHHHHHHHH ***************************
@PHDTHESIS{hansen,
author = {K. T. Hansen},
title = {Symbolic Dynamics in Chaotic systems},
school = {University of Oslo},
address = {Box 1048, Blindern, N-0316, Oslo, Norway},
month = {September},
year = {1993}
}
@BOOK{hao98,
author = {B.-L. Hao and W.-M. Zheng},
title = {Applied Symbolic Dynamics and Chaos},
publisher = {World Scientific},
address = {Singapore},
year = {1998}
}
@BOOK{hao90b,
author = {B.-L. Hao},
title = {Chaos II},
publisher = {World Scientific},
address = {Singapore},
year = {1990}
}
@Book{Holmes96,
author = {P. Holmes and J.L. Lumley and G. Berkooz},
title = {Turbulence, Coherent Structures, Dynamical Systems and Symmetry},
publisher = {Cambridge University Press},
year = {1996},
address = {Cambridge},
}
@ARTICLE{Hopf42,
author = {E. Hopf},
title = {Abzweigung einer periodischen L\"osung},
journal = {Bereich. S\"achs. Acad. Wiss. Leipzig, Math. Phys. Kl.},
volume = {94},
pages ={19}
}
@ARTICLE{hopf48,
author = {E. Hopf},
title = {A mathematical example displaying features of turbulence},
journal = {Comm. Appl. Math.},
volume = {1},
pages ={303-322},
year = {1948}
}
@ARTICLE{hopf50,
author = {E. Hopf},
title = {The partial differential equation $u_t + uu_x = u_{xx}$},
journal = {Comm. Pure Appl. Math.},
volume = {3},
pages ={201-230},
year = {1950}
}
@ARTICLE{optp1,
author = {B. R. Hunt and E. Ott},
title = {Optimal Periodic Orbits of Chaotic Systems},
journal = {Phys. Rev. Lett.},
volume = {76},
pages = {2254},
year = {1996}
}
@ARTICLE{optp2,
author = {T.-H. Yang and B. R. Hunt and E. Ott},
title = {Optimal periodic orbits of continuous time chaotic systems},
journal = {Phys. Rev. E},
volume = {62},
number = {2},
pages = {1950},
year = {2000}
}
************** IIIIIIIIIII ***************************
************** JJJJJJJJJJJ ***************************
************** KKKKKKKKKKK ***************************
@ARTICLE{KT,
AUTHOR = {L. Kadanoff and C. Tang},
TITLE = {Escape rate from strange repellers},
JOURNAL = {Proceedings of the National Academy of Science, USA},
YEAR = 1984,
VOLUME = 81,
PAGES = 1276
}
@BOOK{Katok95,
AUTHOR = {A. Katok and B. Hasselblatt},
TITLE = {Introduction to the Modern Theory of Dynamical Systems},
PUBLISHER = {Cambridge University Press},
ADDRESS = {Cambridge},
YEAR = 1995
}
@ARTICLE{KawKida01,
author = {G. Kawahara and S. Kida},
title = {Periodic motion embedded in {P}lane {C}ouette turbulence:
regeneration cycle and burst},
journal = {J. Fluid Mech.},
volume = {449},
pages = {291-300},
year = {2001},
abstract = {Two time-periodic solutions are found in a 3-d constrained
plane Couette flow, multishooting method being used. The turbulent state
mainly follows the periodic orbit with strong variations. The gentle one
is related to the bursting behavior of the system. Heteroclinic orbits
between these two periodic orbits are found.}
}
@inproceedings{KK05,
author = { G. Kawahara and S. Kida and M. Nagata},
title = {Unstable periodic motion in plane {C}ouette system: The
skeleton of turbulence},
booktitle = {One hundred years of boundary layer research},
year = {2005},
publisher = {Kluwer},
pages = {},
}
@ARTICLE{Kato03,
author={S. Kato and M. Yamada},
title={Unstable periodic solutions embedded in a shell model turbulence},
journal={Phys. Rev. E},
volume={68},
pages={025302},
year={2003},
abstract={A UPO is found in the chaotic region of one shell turbulence
model (GOY). The scaling exponents of the structure function are calculated
to verify that the statistics are well approximated by the UPO.}
}
@BOOK{kura84tur,
author = {Y. Kuramoto},
title = {Chemical Oscillations, Waves and Turbulence},
publisher = {Springer},
address = {Berlin},
year = {1984},
}
@ARTICLE{ku,
author = {Y. Kuramoto and T. Tsuzuki},
title = {Persistent propagation of concentration waves
in dissipative media far from thermal equilibrium},
journal = {Progr. Theor. Phys.},
volume = {55},
pages = {365},
year = {1976},
abstract = {The KSe is derived through the CGLe near the bifurcation point. Some
scaling argument is used to sort out the domiant terms.}
}
@ARTICLE{kuturb78,
author = {Y. Kuramoto},
title = {Diffusion-Induced Chaos in Reaction Systems},
journal = {Suppl. Progr. Theor. Phys.},
volume = {64},
pages = {346-367},
year = {1978},
abstract = {Phase turbulence and amplitude turbulence are named and distinguished
from a dynamical systems point of view. The prototyped equations are derived.}
}
************** LLLLLLLLLLL ***************************
@PhdThesis{Lan:Thesis,
author = {Y. Lan},
title = {Dynamical systems approach to 1-$d$ spatiotemporal chaos -- {A} cyclist's view},
school = {School of Physics, Geogia Institute of Technology},
year = {2004},
address = {Atlanta},
}
@ARTICLE{lanVar1,
author = {Y. Lan and P. Cvitanovi\'{c}},
title = {Variational method for finding periodic orbits in a
general flow},
journal = {Phys. Rev. E},
volume = {69},
pages = {016217},
year = {2004},
NOTE ={{\tt nlin.CD/0308008}}
}
@ARTICLE{LCC06,
author = {Y. Lan and C. Chandre and P. Cvitanovi\'{c}},
title = {Variational method for locating invariant tori},
journal = {Phys. Rev. E},
volume = {74},
pages = {046206},
year = {2006},
NOTE ={{\tt nlin.CD/0508026}}
}
@unpublished{LanCvi07,
author = {Y. Lan and P. Cvitanovi\'{c}},
title = {Unstable recurrent patterns in {K}uramoto-{S}ivashinsky dynamics},
year = {2007},
NOTE ={In preparation}
}
@ARTICLE{lanmaw03,
author = {Y. Lan and N. Garnier and P. Cvitanovi\'{c}},
title = {Stationary modulated-amplitude waves in the 1D complex
{G}inzburg-{L}andau equation},
journal = {Physica D},
volume = {188},
pages = {193-212},
year = {2004}
}
@BOOK{LM94,
AUTHOR = {A. Lasota and M. MacKey},
TITLE = {Chaos, Fractals, and Noise; Stochastic Aspects of Dynamics},
YEAR = 1994,
PUBLISHER = {Springer-Verlag},
ADDRESS = {Berlin}
}
@INCOLLECTION{lauw86,
AUTHOR = {H.A. Lauwerier},
YEAR = 1986,
PAGES = {39},
EDITOR = {V. Holden},
PUBLISHER = {Princeton University Press},
BOOKTITLE = {Chaos}
}
@ARTICLE{lorenz1,
author = {E. N. {L}orenz},
title = {Deterministic Nonperiodic Flow},
journal = {J. Atmos. Sci.},
volume = {20},
pages = {130},
year = {1963}
}
************** MMMMMMMMMMM ***************************
@ARTICLE{NLSmal91,
author = {B. A. Malomed and L. Stenflo},
title = {Modulational instabilities and soliton solutions of a generalized
nonlinear Schr\"{o}dinger equation},
journal = {J. Phys. A},
volume = {24},
pages = {L1149-L1153},
year = {1991},
abtract = {A Hamiltonian equation is proposed for the generalized nonlinear
Schr\"{o}dinger equation. Soliton solutions are found and the stability of the
plane waves is investigated.}
}
@ARTICLE{msr,
AUTHOR = {P. C. Martin and E. D. Siggia and H. A. Rose},
TITLE = {Statistical Mechanics of Classical Systems},
JOURNAL = {Physical Review A},
YEAR = 1973,
VOLUME = 8,
NUMBER = 1,
PAGES = {423-437}
}
@BOOK{mosersr,
author = {J. Moser},
title = {Stable and Random Motions in Dynamical Systems},
series = {Ann. Math. Stud.},
number = {77},
publisher = {Princeton University Press},
address = {Princeton},
year = {1973}
}
@ARTICLE{moser96,
AUTHOR = {J. Moser},
TITLE = {A rapidly converging iteration method and nonlinear partial
differential equations},
JOURNAL = {Ann. Scuola Norm. Super. Pisa},
VOLUME = {20},
PAGES = {265-315},
YEAR = 1966
}
@ARTICLE{moser96a,
AUTHOR = {J. Moser},
JOURNAL = {Ann. Scuola Norm. Super. Pisa},
VOLUME = {20},
PAGES = {499-535},
YEAR = 1966
}
************** NNNNNNNNNNN ***************************
@ARTICLE{temam85ks,
author = {B. Nicolaenko and B. Scheurer and R. Temam},
title = {Some global dynamical properties of the {K}uramoto-{S}ivashinsky
equations:nonlinear stability and attractors},
journal = {Physica D},
volume = {16},
pages = {155-183},
year = {1985}
}
@ARTICLE{NSTks85,
author = {B. Nicolaenko and B. Scheurer and R. Temam},
title = {Some global dynamical properties of the {K}uramoto-{S}ivashinsky equations:
nonlinear stability and attractors},
journal = {Physica D},
volume = {16},
pages = {155-183},
year = {1985},
abstract = {The KSe shows coherent spatial patterns but temporal chaos. The
anti-diffusion term destablizes the solutin while stability is achived by the collective
action of all modes through the nonlinear convection term. An estimate of the L^2 norm of
the solution in the antisymmetric periodic function space is given. The number of
determining Fourier modes is calculated together with fractal dimension of the strange
attractor.}
}
************** OOOOOOOOOOO ***************************
@ARTICLE{GiacoOtto05,
author = {Lorenzo Giacomelli and Felix Otto},
title = {New bounds for the Kuramoto-Sivashinsky equation},
journal = {Comm. Pure Appl. Math.},
volume = {58},
pages = {297-318},
year = {2005},
}
% http://www-mathphys.iam.uni-bonn.de/~otto/web/main-en.htm
************** PPPPPPPPPPP ***************************
@ARTICLE{NLSper77,
author = {N. R. Pereira and L. Stenflo},
title = {Nonlinear Schr\"{o}dinger equation including growth and damping},
journal = {Phys. Fluid},
volume = {20},
number = {10},
pages = {1733-1734},
year = {1977},
abstract = {A nonlinear Schr\"{o}dinger equation is found with growth and damping
terms in the weakly nonlinear regime. An exact solitary solution is found for the
equation.}
}
@BOOK{poinper,
author = {H. Poincar\'{e}},
title ={New Methods in Celestial Mechanics},
series= {History of Modern Physics and Astronomy},
volume = {13},
publisher = {Springer-Verlag},
address = {New York},
year = {1992}
}
@BOOK{poincare,
AUTHOR = {H. Poincar\'e},
TITLE = {Les m\'ethodes nouvelles de la
m\'echanique c\'eleste},
YEAR = 1899,
PUBLISHER = {Guthier-Villars},
ADDRESS = {Paris},
NOTE = {For a very readable exposition of Poincar\'e's work and
the development of the dynamical systems theory up to 1920's
see \rf{JBG97}.}
}
@BOOK{JBG97,
AUTHOR = {J. Barrow-Green},
TITLE = {Poincar\'e and the Tree Body Problem},
YEAR = 1997,
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence R.I.}
}
@Book{Press:96,
author = { W. H. Press and B. P. Flannery and S. A. Teukolsky and W. T. Vetterling},
title = {Numerical Recipes in Fortran},
publisher = {Cambridge University Press},
year = {1996},
address = {Cambridge},
}
@ARTICLE{pugh67cl,
author = {C. Pugh},
title = {An improved closing lemma and a general density theorem},
journal = {Amer. J. Math.},
volume = {89},
year = {1967}
}
************** QQQQQQQQQQQ ***************************
************** RRRRRRRRRRR ***************************
@ARTICLE{reimann1,
AUTHOR = {P. Reimann},
TITLE = {Noisy one-dimensional maps near a crisis 1: Weak
Gaussian white and colored noise},
JOURNAL = {Journal of Statistical Physics},
YEAR = 1996,
VOLUME = 82,
PAGES = {1467-1501}
}
@ARTICLE{reimann2,
AUTHOR = {P. Reimann},
TITLE = {Noisy one-dimensional maps near a crisis 2: General
uncorrelated weak noise},
JOURNAL = {Journal of Statistical Physics},
YEAR = 1996,
VOLUME = 85,
PAGES = {403-425}
}
@BOOK{Risken96,
author = {H. Risken and H. Haken},
title ={The Fokker-Planck Equation},
publisher = {Springer-Verlag},
year = {1996}
}
@ARTICLE{Robinson91,
author = {{Robinson}, S.~K.},
title = "{Coherent motions in the turbulent boundary layer}",
journal = {Annual Review of Fluid Mechanics},
year = 1991,
volume = 23,
pages = {601-639},
}
@ARTICLE{ross,
author = {O. E. R\"{o}ssler},
title = {An equation for continuous chaos},
journal = {Phys. Lett. A},
volume = {57},
pages = {397},
year = {1976}
}
@BOOK{ruelle,
author = {D. Ruelle},
title = {Statistical Mechanics, Thermodynamic Formalism},
publisher = {Addison-Wesley},
address = {Reading, MA},
year = {1978}
}
@ARTICLE{ruell71,
author = {D. Ruelle and F. Takens},
title = {On the nature of turbulence},
journal = {Commun. Math. Phys.},
volume = {20},
pages = {167},
year = {1971}
}
@ARTICLE{ruelext,
author = {D. Ruelle},
title = {Large volume limit of the distribution
of characteristic exponents in turbulence},
journal = {Commun. Math. Phys.},
volume = {87},
pages = {287-302},
year = {1982}
}
@ARTICLE{rue04ne,
author = {D. Ruelle},
title = {Conversations on nonequilibrium physics with
an extraterrestrial},
journal = {Physics Today},
volume = {57},
number = {5},
pages = {48-53},
year = {2004},
abstract = {New ideas of nonequilibrium physics are introduced. Emphasize
the SRB measure approach and energy fluctuation theorem.}
}
@ARTICLE{hhrugh92,
author = {H.H. Rugh},
title = {The correlation spectrum for hyperbolic analytic maps},
journal = {Nonlinearity},
volume = {5},
pages = {1237},
year = {1992}
}
************** SSSSSSSSSSS ***************************
@BOOK{lyschm,
author = {N. Sidorov and B. Loginov and A. Sinitsyn and M. Falaleev},
title = {{L}yapunov-{S}chmidt Methods in Nonlinear Analysis and Applications},
series = {mathematics and its applications},
volume = {550},
publisher = {Kluwer Academic Publishers},
address = {Dordrecht},
year = {2002}
}
@unpublished{SCD06,
author = {E. Siminos and P. Cvitanovi\'{c} and R. L. Davidchack},
title = {Relative periodic orbits in {K}uramoto-{S}ivashinsky equation},
year = {2006},
NOTE ={In preparation}
}
@ARTICLE{sinai,
AUTHOR = {Ya. G. Sinai},
TITLE = {Gibbs measures in ergodic theory},
JOURNAL = {Russian Mathematical Surveys},
YEAR = 1972,
VOLUME = 166,
PAGES = 21
}
@ARTICLE{siegel42,
AUTHOR = {C. L. Siegel},
TITLE = {Iteration of analytic functions},
JOURNAL = {Ann. Math.},
VOLUME = {43},
YEAR = 1942,
PAGES = {607-612}
}
@ARTICLE{sirovcgl90,
author = {J. D. Rodriguez and L. Sirovich},
title = {Low-dimensional dynamics for the complex {G}inzburg-{L}andau equation},
journal = {Physica D},
volume = {43},
pages = {77-86},
year = {1990}
}
@ARTICLE{siv,
author = {G. I. Sivashinsky},
title = {Nonlinear analysis of hydrodynamical instability in laminar
flames - {I}. {D}erivation of basic equations},
journal = {Acta Astr.},
volume = {4},
pages = {1177},
year = {1977},
abstract = {A integro-differential equation (KSe) is derived directly from the flame
propagation equations near the bifurcation point.}
}
@ARTICLE{michsiv77,
author = {D. M. Michelson and G. I. Sivashinsky},
title = {Nonlinear analysis of hydrodynamic instability in laminar flames - II.
Numerical experiments},
journal = {Acta Astr.},
volume = {4},
pages = {1207-1221},
year = {1977},
abstract = {Consider the effect of hydrodynamics instability and diffusion thermal
instability through the numerical calculation of the KSe.}
}
@ARTICLE{sivmich80,
author = {G. I. Sivashinsky and D. M. Michelson},
title = {On irregular wavy flow of a liquid down a vertical plane},
journal = {Prog. Theor. Phys. Suppl.},
volume = {66},
pages = {2112},
year = {1980}
}
@ARTICLE{smale,
author = {S. Smale},
title = {Differentiable dynamical systems},
journal = {Bull. Amer. Math. Soc.},
volume = {73},
pages = {747-817},
year = {1967}
}
publisher = {AMS},
address = {Providence, RI},
@ARTICLE{EAS87,
author = {E. A. Spiegel},
title = {Chaos: a mixed metaphor for turbulence},
journal = {Proc. Roy. Soc.},
volume = {A413},
pages = {87},
year = {1987}
}
% this paper is one of the primordial ``chaology'' papers and it (or another
% Moore and Spiegel paper) might or
% might not contain method for finding periodic orbits not invented by
% Vackhtang
@ARTICLE{MS66,
author = {D. W. Moore and E. A. Spiegel},
title = {A thermally excited nonlinear oscillator},
journal = {Astrophys. J.},
volume = {143},
pages = {871},
year = {1966}
}
% the method for calculating periodic orbits is laid out
@ARTICLE{BMS71,
author = {N. H. Baker and D. W. Moore and E. A. Spiegel},
title = {Aperiodic behavior of a non-linear oscillator},
journal = {Quatr. J. Mech. and Appl. Math.},
volume = {24},
pages = {391},
year = {1971}
}
@ARTICLE{NLSstn03,
author = {L. Stenflo and M. Y. Yu},
title = {Oscillons at a plasma surface},
journal = {Phys. Plasmas},
volume = {10},
number = {3},
pages = {912-913},
year = {2003},
abstract = {Under the external driving force, an oscillon solutions is found.}
}
@ARTICLE{sten98osc,
author = {L. Stenflo and M. Y. Yu},
title = {Plasma oscillons in spherically bounded magnetized plasmas},
journal = {Phys. Plasmas},
volume = {5},
pages = {3122},
year = {1998}
}
@ARTICLE{sternb57,
AUTHOR = {S. Sternberg},
JOURNAL = {Amer. J. Math.},
VOLUME = {79},
PAGES = {809},
YEAR = 1957
}
@ARTICLE{sternb58,
AUTHOR = {S. Sternberg},
JOURNAL = {Amer. J. Math.},
VOLUME = {80},
PAGES = {623},
YEAR = 1958
}
@ARTICLE{sternb59,
AUTHOR = {S. Sternberg},
JOURNAL = {Amer. J. Math.},
VOLUME = {81},
PAGES = {578},
YEAR = 1959
}
@BOOK{strogb,
author = {S. H. Strogatz},
title = {Nonlinear Dynamics and Chaos},
series = {Studies in Nonlinearity},
publisher = {Perseus Publishing},
address = {Cambridge Massachusetts},
year = {2000}
}
************** TTTTTTTTTTT ***************************
@BOOK{infdymnon,
author = {R. Temam},
title = {Infinite-Dimensional Dynamical Systems in Mechanics and Physics},
Series = {Appl. Math. Sci.},
number = {68},
publisher = {Springer-Verlag},
address = {New York},
year = {1988}
}
************** UUUUUUUUUUU ***************************
@unpublished{VKK05,
author = {L. van Veen and S. Kida and G. Kawahara},
title = {Periodic motion representing isotropic turbulence},
note = {preprint, 2005}
}
************** VVVVVVVVVVV ***************************
@ARTICLE{decar3,
author = {D. Viswanath},
title = {The {L}indstedt-{P}oincar\'e technique as an algorithm for
finding periodic orbits},
journal = {SIAM Review},
volume = {43},
pages = {478-496},
year = {2002}
}
@ARTICLE{decar,
author = {D. Viswanath},
title = {Symbolic Dynamics and Periodic Orbits of the {L}orenz Attractor},
journal = {Nonlinearity},
volume = {16},
pages = {1035-1056},
year = {2003}
}
@unpublished{decar2,
author = {Divakar Viswanath},
title = {An Extension of the {L}indstedt-{P}oincar\'{e} Algorithm for
Computing Periodic Orbits},
note = {unpublished}
}
************** WWWWWWWWWWW ***************************
************** XXXXXXXXXXX ***************************
************** YYYYYYYYYYY ***************************
************** ZZZZZZZZZZZ ***************************
@ARTICLE{ZG96,
author = {S. M. Zoldi and H. S. Greenside},
title = {Spatially localized unstable periodic orbits
of a high-dimensional chaotic system},
journal = {Phys. Rev. E},
volume = {57},
pages = {R2511},
year = {1998}
}
************** TEXTBOOKS ***************************
@BOOK{rtb,
author = {V. Szebehely},
title = {Theory of Orbits},
publisher = {Academic Press},
address = {New York},
year = {1967}
}
@BOOK{henonrtb2,
author = {M. H\'{e}non},
title = {Generating Families in the Restricted Three-Body
Problem II. Quantitative Study of Bifurcations},
publisher = {Springer},
address = {New York},
year = {2001}
}
volume = {m65},
% series = {Lecture Notes in Physics},
@BOOK{mandel,
author = {B. B. Mandelbrot},
title = {The Fractal Geometry of Nature},
publisher = {WH Freeman Co.},
address = {New York},
year = {1982}
}
@Book{perg,
author = {P. Walters},
title = {An introduction to ergodic theory},
publisher = {Speringer-Verlag},
address = {New York},
year = {1981}
}
@BOOK{sinai76,
author = {Y. G. Sinai},
title = {Introduction to Ergodic Theory},
publisher = {Princeton University Press},
address = {Princeton},
year = {1976}
}
@BOOK{frisch,
author = {U. Frisch},
title = {Turbulence},
publisher = {Cambridge University Press},
address = {Cambridge, England},
year = {1996}
}
@BOOK{liede,
author = {P. J. Olver},
title = {Applications of Lie Groups to Differential Equations},
series = {GTM},
volume = {107},
publisher = {Spring-Verlag},
address = {New York},
year = {1998}
}
@BOOK{spfunc1,
author = {N. M. Temme},
title = {Special functions: an introduction to the classical functions
of mathematical physics},
publisher = {Wiley-Interscience},
address = {New York},
year = {1996}
}
@BOOK{jeffb,
author = {H. Jeffreys and B. Jeffreys},
title = {Methods of Mathematical Physics},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {1999}
}
@BOOK{haleob,
author = {J. Hale},
title ={Ordinary Differential Equations},
publisher = {Wiley-Interscience},
address = {New York},
year = {1969}
}
series = {Pure and Appl. Math.},
volume = {21},
@BOOK{haledb,
author = {J. Hale and H. Ko\c{c}ak},
title ={Dynamics and Bifurcations},
publisher = {Springer-Verlag},
address = {New York},
year = {1991}
}
%PC removed this:
series = {Texts in Appl. Math.},
volume = {3},
@BOOK{guckb,
author = {J. Guckenheimer and P. Holmes},
title = {Nonlinear Oscillations, Dynamical Systems, and Bifurcations
of Vector Fields},
publisher = {Springer-Verlag},
address = {New York},
year = {1983}
}
series = {Appl. Math. Sci.},
volume = {42},
@BOOK{wiggb,
author = {S. Wiggins},
title = {Global Bifurcations and Chaos},
publisher = {Springer-Verlag},
address = {New York},
year = {1988}
}
series = {Appl. Math. Sci.},
volume = {73},
@BOOK{perkob,
author = {L. Perko},
title = {Differential Equations and Dynamical Systems},
publisher = {Springer-Verlag},
address = {New York},
year = {1991}
}
%PC removed this
series = {Texts in Appl. Math.},
volume = {7},
@BOOK{marsdenbb,
author = {J. E. Marsden and M. McCracken},
title = {The {H}opf Bifurcation and its Applications},
publisher = {Springer-Verlag},
year = {1976}
}
series = {Appl. Math. Sci.},
volume = {19},
@BOOK{hartmanb,
author = {P. Hartman},
title ={Ordinary Differential Equations},
publisher = {Wiley},
address = {New York},
year = {1964}
}
@BOOK{ablowbook,
author = {M. J. Ablowitz and P. A. Clarkson},
title ={Solitions, Nonlinear Evolution Equations and Inverse Scattering},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {1992}
}
series = {London Math. Soc. Lect. Notes Ser.},
volume = {149},
@BOOK{contebook,
author = {R. Conte},
title = {The {P}ainlev\'{e} property: one century later},
publisher = {Springer},
address = {New York},
year = {1999}
}
series = {CRM Series in Mathematical Physics},
@BOOK{hille,
author = {E. Hille},
title = {Ordinary Differential Equations in the Complex Domain},
publisher = {Dover},
address = {New York},
year = {1997}
}
@BOOK{ince,
author = {E. L. Ince},
title = {Ordinary Differential Equations},
publisher = {Dover},
address = {New York},
year = {1956}
}
@BOOK{devnmap,
author = {R. L. Devaney},
title = {An Introduction to Chaotic Dynamical systems},
publisher = {Addison-Wesley},
address = {Red-wood City},
year = {1989}
}
@BOOK{jhos,
author = {J. K. Hale},
title = {Oscillations in Nonlinear Systems},
publisher = {McGraw-Hill},
address = {New York},
year = {1963}
}
series = {McGraw-Hill Series in Advanced Mathematics
with Applications},
@BOOK{haleinf84,
author = {J. K. Hale and L. T. Magalhaes},
title = {An Introduction to Infinite Dimensional Systems-Geometric Theory},
publisher = {Springer-Verlag},
address = {New York},
year = {1984}
}
@BOOK{ellip,
author = {D. F. Lawden},
title = {Elliptic Functions and Applications},
publisher = {Springer-Verlag},
address = {New York},
year = {1989}
}
series = {Appl. Math. Sci.},
volume ={80},
#########################
@ARTICLE{lamb98,
author = {J. S. W. Lamb and J. A. G. Roberts},
title = {Time reversal symmetry in dynamical systems: A survey},
journal = {Physica D},
volume = {112},
pages = {1},
year = {1998}
}
@ARTICLE{kruskal65,
author = {N. J. Zabusky and M. D. Kruskal},
title = {Interaction of ``solitons" in a collisionless plasma and
the recurrence of initial states},
journal = {Phys. Rev. Lett.},
volume = {15},
pages = {240},
year = {1965}
}
@ARTICLE{weiss82,
author = {J. Weiss and M. Tabor and G. Carnevale},
title ={The {P}ainlev\'{e} property for partial differential equations},
journal = {J. Math. Phys.},
volume = {24},
pages = {522},
year = {1983}
}
@ARTICLE{weiss83,
author = {J. Weiss},
title ={The {P}ainlev\'{e} for partial differential equations.II:
B\"{a}cklund transformation, Lax pairs,
and the Schwarzian derivative},
journal = {J. Math. Phys.},
volume = {24},
pages = {1405},
year = {1983}
}
@ARTICLE{weiss86,
author = {J. Weiss},
title ={{B}\"{a}cklund transformation and the {P}ainlev\'{e} property},
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author = {M. C. Cross and P. C. Hohenberg},
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volume = {404},
pages = {733},
year = {2000},
abstract = {Numerical study on Boussinesq equation(pseudospectral method
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is breaking or connecting rolls mediated by defects at the domain boundary
and exsisting for only a short time.}
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volume = {369},
pages = {129},
year = {1994},
abstract = {Numerical study on CGLe. Show that even though the fractal
dimension increases with system size the simple scaling relation is not
correct, i.e. the correlation length can not be used to directly calculate
the fractal dimension.}
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author = {D. Rothstein and E. Henry and J. P. Gollub},
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volume = {401},
pages = {770-772},
year = {1999},
abstract = {Experimentally generate a 2-d periodic(or weakly turbulent)
velocity field using glycerol-water on magnets driven by electric current.
Persistent spatial patterns are observed, which decays exponentially.}
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pages = {254501},
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poincar\'{e} section are identified as well as their stable and unstable manifolds.
These information can be related to the gradient of the stretching field and the
distribution of passive scalar quantities. It seems that on these manifolds the
stretching field gains its maximum.}
}
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author = {J. P. Gollub},
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year = {1995},
abstract = {A brief review of experiments on film flows, surface waves, and thermal convection.
In 1-d, the transition from cellular states to spatiotemporal chaos is described. In 2-d,
periodic and quasiperiodic patterns including defect-mediating ones are discussed.
Statistical emphasis on the transport and mixing phenomena in fluids is put and some open
problems are stated. }
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author = {B. P. Luce},
title = {Homoclinic explosions in the complex {G}inzburg-{L}andau equation},
journal = {Physica D},
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pages = {553-581},
year = {1995}
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author = {A. Doelman},
title = {Finite-dimensional models of the {G}inzburg-{L}andau equation},
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pages = {231-250},
year = {1991}
}
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title = {Scale and space localization in the {K}uramoto-{S}ivashinsky
equation},
journal = {Chaos},
volume = {9},
number = {2},
pages = {452},
year = {1999},
abstract = {Using a wavelet basis, the spatiotemporally chaotic regime of
the KSe is explored where a good seperation of scales is observed. In large
scales, the dynamics is Gaussian. In the intermediate scales, the dynamics is
reminiscent of travelling waves and heteroclinic cyces which is the typical
behavior for small system size. In the small scales, the dynamics is
intermittent. Through investigation of the interaction between different
scales, we see the intermediate structures give the defining shape of the
cell and the large scales trigger the spatiotemporal chaos. The small scales
dissipate energy and modify the background in a average sense.}
}
@ARTICLE{marcio03,
author = {M. Gameiro and W. Kalies and K. Mischaikow},
title = {Topological characterization of spatial-temporal chaos},
journal = {submitted},
volume = {1},
number = {1},
pages = {1},
year = {2003},
abstract = {Use the Betti number to characterize the spatiotemporal
dynamics of a system. A lyapunov exponent is suggested to measure the
space as well as temporal complexity of the evolution.}
}
@ARTICLE{ksgreene88,
author = {J. M. Greene and J. -S. Kim},
title = {The steady states of the {K}uramoto-{S}ivashinsky equation},
journal = {Physica D},
volume = {33},
pages = {99-120},
year = {1988},
abtract = {Use Fourier modes to discuss the energy transport from
long wavelength modes to short ones. The generation of steady states with
periodic boundary condition is studied systematically with a bifurcation
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large system size is presented.}
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number = {3},
pages = {1242},
year = {1995},
abstract = {The stationary KSe can be transformed into a time-depent 1-d
Hamiltonian system. There exist a large class of initial conditions under which
the system will explode in a finite time like 120/(x-x_0)^3.}
}
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author = {A. P. Hooper and R. Grimshaw},
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journal = {Wave Motion},
volume = {10},
pages = {405-420},
year = {1988},
abstract = {Numerically calculated connections of the KSe}
}
@ARTICLE{ksyang97,
author = {T. -S. Yang},
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equation},
journal = {Physica D},
volume = {110},
pages = {25-42},
year = {1997},
abstract = {Using asymptotic analysis (beyond all orders)in the weak
shock limit, the oscillatory
shocks and the solitary saves are constructed. It shows that the oscillatory
shocks can be only be anti-symmetric and numurical results support the
anlytical calculation.}
}
@ARTICLE{ksgrim91,
author = {R. Grimshaw and A. P. Hooper},
title = {The non-existence of a certain class of travelling wave
solutions of the {K}uramoto-{S}ivashinsky equation},
journal = {Physica D},
volume = {50},
pages = {231-238},
year = {1991},
abstract = {If the effect of short-wave stability is small, there are no
regular shocks.}
}
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author = {P. Kent and J. Elgin},
title = {Travelling-waves of the {K}uramoto-{S}ivashinsky equation:
period-multiplying bifurcations},
journal = {Nonlinearity},
volume = {5},
pages = {899-919},
year = {1992},
abstract = {Some properties of the KSe are discussed. A unified argument
about the connections and periodic orbits is given. In particular, the
K-bifurcation of the system is conjectured to arises in the 1:n resonances of
a fixed point.}
}
@ARTICLE{kschang86,
author = {H.-C. Chang},
title = {Travelling waves on fluid interfaces: {N}ormal form analysis of
the {K}uramoto-{S}ivashisky equation},
journal = {Phys. Fluids},
volume = {29},
number = {10},
pages = {3142},
year = {1986},
abstract = {The lowest order normal form analysis is give to the KSe and
the approximate analytical wavelength-amplitude and wavespeed-amplitude
relation are given. They agree quite well with the numerical calculations.}
}
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author = {W. C. Troy},
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equation},
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volume = {82},
pages = {269-313},
year = {1989},
abstract = {For c=1, the existence of two periodic orbits and two
heteroclinic orbits are proved and suggestions for proving the existence of
more complicated orbits are given.}
}
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author = {J. Jones and W. C. Troy and A. D. MacGillivary},
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pages = {28-55},
year = {1992},
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any c>0, there existes no monotone bounded travelling wave solution. The
normal form analysis may not be able to give correct results to all orders.}
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year = {1994}
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pages = {543-558},
year = {1992}
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year = {1998}
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H. Richter and L. Kramer},
title = {The {E}ckhaus instability for traveling waves},
journal = {Physica D},
volume = {55},
pages = {269},
year = {1992}
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number = {8},
pages = {2804},
year = {1994},
abstract = {A nonlinear equation of the plasma surface is derived for the
small deformation. Possible solitary solutions are discussed.}
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year = {1994}
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journal = {Phys. Lett. A},
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pages = {459},
year = {1994}
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journal = {Phys. Fluids B},
volume = {5},
pages = {2888},
year = {1993}
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volume = {1},
pages = {96},
year = {1994}
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@ARTICLE{chate94,
author = {H. Chat\'{e}},
title = {Spatiotemporal intermittency regimes of the one-dimensional complex
{G}inzburg-{L}andau equation},
journal = {Nonlinearity},
volume = {7},
pages = {185-204},
year = {1994},
abstract = {In the Benjamin-Feir stable region, spatiotemporal intermittency
regimes are identified which consists of patches of linearly stale stable plane
waves separated by localized objects with a well defined dynamics. In the transition
reginon, asymptotic states with an irregular, frozen spatial structure are shown to
occur. The bistable region beyond the Benjamin-Feir stable region is of the same
spatiotemporal intermittency type with phase turbulence as the ``laminar" state.}
}
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volume = {80},
pages = {1896},
year = {1998}
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volume = {85},
pages = {86},
year = {2000}
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author = {L. Brusch and A. Torcini and M. van Hecke and
M. G. Zimmermann and M. B\"{a}r},
title = {Modulated amplitude waves and defect formation in the
one-dimensional complex {{G}inzburg-{L}andau} equation},
journal = {hysica D},
volume = {160},
pages = {127-148},
year = {2001}
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author = {P. K. {N}ewton and L. Sirovich},
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journal = {Quart. Appl. Math.},
volume = {XLIV},
number = {1},
pages = {49-58},
year = {1986}
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author = {P. K. {N}ewton and L. Sirovich},
title = {Instabilities of {Ginzburg}-{{L}andau} equation: secondary
bifurcation},
journal = {Quart. Appl. Math.},
volume = {XLIV},
number = {2},
pages = {367-374},
year = {1986}
}
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author = {W. van Saarloos and P. C. Hohenberg},
title = {Fronts, pulses, sources and sinks in generalized complex
{G}inzburg-{L}andau equations},
journal = {Physica D},
volume = {56},
pages = {303-367},
year = {1992}
}
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volume = {74},
pages = {99-143},
year = {2002}
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title = {Long waves in liquid films},
journal = {J. Math. Phys},
volume = {45},
pages = {150-155},
year = {1966}
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title = {Nonlinear instability at the interface between two viscous fluids},
journal = {Phys. Fluids},
volume = {28},
pages = {37-45},
year = {1985}
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author = {B. I. Cohen and J. A. Krommes and W. M. Tang and M. N. Rosenbluth},
title = {Non-linear saturation of the dissipative trapped-ion mode by mode coupling},
journal = {Nucl. Fusion},
volume = {16},
pages = {971-991},
year = {1972}
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pages = {172-175},
year = {1994}
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@ARTICLE{eckcgl,
author = {B. Janiaud and A. Pumir and D. Bensimon and V. Croquette
and H. Richter and L. Kramer},
title = {The {E}ckhaus instability for travelling waves},
journal = {Physica D},
volume = {55},
pages = {269-286},
year = {1992}
}
@ARTICLE{spercgl,
author = {W. Sch\"{o}pf and L. Kramer},
title = {Small-Amplitude Periodic and Chaotic Solutions of the Complex
{G}inzburg-{L}andau Equation for a Subcritical Bifurcation},
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volume = {66},
number = {18},
pages = {2316},
year = {1991}
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author = {M. C. Cross and P. C. Hohenberg},
title = {Pattern formation outside of equilibrium},
journal = {Rev. Mod. Phys. },
volume = {65},
number = {3},
year = {1993}
}
@ARTICLE{cross94sp,
author = {M. C. Cross and P. C. Hohenberg},
title = {Spatiotemporal Chaos},
journal = {Science},
volume = {263},
number = {5153},
pages = {1569-1570},
year = {1994},
abstract = {The phenomena of spatiotermporal chaos are introduced based on both
rotating and regular Rayleigh-Bernard convection. The enormous number of available
degrees of freedom is a vivid feature. A set of interesting questions have
been raised, including how to characterize a spatiotemporal chaotic state. The
importance of the study of small subsystems is emphasized.}
}
@ARTICLE{bknk,
author = {N. Bekki and K. Nozaki},
title = {Formations of Spatial Patterns And Holes In The Generalized
{G}inzburg-{L}andau Equation},
journal = {Phys. Lett.},
volume = {110A},
pages = {133-135},
year = {1985}
}
@ARTICLE{bknk84,
author = {K. Nozaki and N. Bekki},
title = {Exact Solutions of the Generalized {G}inzburg-{L}andau Equation},
journal = {J. Phys. Soc. Japan},
volume = {53},
number = {5},
pages = {1581-1582},
year = {1984}
}
@ARTICLE{bknk83,
author = {K. Nozaki and N. Bekki},
title = {Pattern Selection and Spatiotemporal Transition to Chaos in the
{G}inzburg-{L}andau Equation},
journal = {Phys. Rev. Lett.},
volume = {51},
number = {24},
pages = {2171},
year = {1983}
}
@ARTICLE{BTBmaw03,
author = {L. Brusch and A. Torcini and M. B\"{a}r},
title = {Nonlinear analysis of the {E}ckhaus instability: modulated amplitude waves
and phase chaos with nonzero average phase gradient},
journal = {Physica D},
volume = {174},
number = {1-4},
pages = {152-167},
year = {2003}
}
@ARTICLE{BTHmaw01,
author = {L. Brusch and A. Torcini and M. van Hecke and M. G. Zimmermann
and M. B\"{a}r},
title = {Modulated amplitude waves and defect formation in the one-dimensional
complex {G}inzburg-{L}andau equation},
journal = {Physica D},
volume = {160},
pages = {127-148},
year = {2001}
}
@ARTICLE{BZMmaw00,
author = {L. Brusch and M. G. Zimmermann and M. van Hecke and M. B\"{a}r and
A. Torcini},
title = {Modulated Amplitude Waves and the Transition from Phase to Defect Chaos},
journal = {Phys. Rev. Lett.},
volume = {85},
number = {1},
pages = {86},
year = {2000},
abstract = {MAWs of various periods P occur in phase chaotic states. The pair of
MAWs cease to exist via a saddle-node bifurcation. For periods beyond this bifurcation,
incoherent near-MAW evolve toward defects.}
}
@ARTICLE{SHcgl90,
author = {W. van Saarloos and P. C. Hohenberg},
title = {Pulses and Fronts in the Complex {G}inzburg-{L}andau Equation near a
Subcritical Bifurcation},
journal = {Phys. Rev. Lett.},
volume = {64},
number = {7},
pages = {749},
year = {1990}
}
@ARTICLE{SPScgl92,
author = {B. I. Shraiman and A. Pumir and W. van Saarloos and P. C. Hohenberg
and H. Chat\'{e} and M. Holen},
title = {Spatiotemporal chaos in the one-dimensional complex {G}inzburg-{L}andau
equation},
journal = {Physica D},
volume = {57},
pages = {241-248},
year = {1992}
}
@ARTICLE{MvH98,
author = {M. van Hecke},
title = {Building Blocks of Spatiotemporal Intermittency},
journal = {Phys. Rev. Lett.},
volume = {80},
number = {9},
pages = {1896},
year = {1998}
}
@ARTICLE{torc96,
author = {A. Torcini},
title = {Order Parameter for the Transition from Phase to Amplitude Turbulence},
journal = {Phys. Rev. Lett.},
volume = {77},
number = {6},
pages = {1047},
year = {1996},
abstract = {The maximal consersed phase gradient is introduced as an order parameter
to characterize the transition from phase to defect turbulence in the CGLe. It has a
finite value in the PT regime and decreases to zero when the transition to defect
turbulence is approached. A modified KSe is able to reproduce the main feature of the
stable waves and to explain their origin.}
}
@ARTICLE{MvHhole01,
author = {M. van Hecke and M. Howard},
title = {Ordered and Self-Disordered Dynamics of Holes and Defects in the
One-Dimensional Complex {G}inzburg-{L}andau Equation},
journal = {Phys. Rev. Lett.},
volume = {86},
number = {10},
pages = {2018},
year = {2001},
abtract = {Ordered hole-defect dynamics happens when an unstable hole invade a plane
wave state and periodically nucleates defects from which new holes are born. The
interaction between the holes and a self-disordered background are essential for the
occurrence of spatiotemporal chaos in hole-defect states.}
}
@ARTICLE{CM87tran,
author = {H. Chat\'{e} and P. Manneville},
title = {Transition to Turbulence via spatiotemporal Intermittency},
journal = {Phys. Rev. Lett.},
volume = {58},
number = {2},
pages = {112},
year = {1987}
}
@ARTICLE{MC96pha,
author = {P. Manneville and H. Chat\'{e}},
title = {Phase turbulence in the two-dimensional complex {G}inzburg-{L}andau equation},
journal = {Physica D},
volume = {96},
pages = {30-46},
year = {1996},
abtract = {instability of plane waves}
}
@ARTICLE{EGtran95,
author = {D. A. Egolf and H. S. Greenside},
title = {Characterization of the Transition from Defect to Phase Turbulence},
journal = {Phys. Rev. Lett.},
volume = {74},
number = {10},
pages = {1751},
year = {1995},
abtract = {For the CGLe on a large periodic interval, the transition from defect to
phase turbulence is continuous. The conclusion is based on the calculation of the defect
density, Lyapunov dimension density and correlation lengths.}
}
@ARTICLE{MHMwind96,
author = {R. Montagne and E. Hern\'{a}ndez-Garc\'{i}a and M. S. Miguel},
title = {Winding Number Instability in the Phase-Turbulence Regime of the Complex
{G}inzburg-{L}andau Equation},
journal = {Phys. Rev. Lett.},
volume = {77},
number = {2},
pages = {267},
year = {1996},
abstract = {There exist a band of winding numbers associated with stable states in
the phase turbulent regime. The states with large winding numbers decay to the ones
with smaller winding numbers. Defect chaos happens when the range of stable winding
numbers vanishes.}
}
@ARTICLE{Mcoh03,
author = {M. van Hecke},
title = {Coherent and Incoherent structures in systems described by the 1D
{CGLE}: Experiments and Identification},
journal = {Physica D},
volume = {174},
pages = {134},
year = {2003},
abstract = {Give an overview of the most important properties of the coherent
structures and address the question how such structures may be identified in the
experimental space-time data set.}
}
@ARTICLE{SHcoh92,
author = {W. van Saarloos and P. C. Hohenberg},
title = {Fronts, pulses, sources and sinks in generalized complex {G}inzburg-{L}andau
equations},
journal = {Physica D},
volume = {56},
pages = {303-367},
year = {1992},
abstract = {Propose the concept of coherent structures and study the competition
between them in chaotic dynamics. Specifically, the invasion of front to another state
is discussed.}
}
@ARTICLE{TFGphase97,
author = { A. Torcini und H. Frauenkron and P. Grassberger},
title = {Studies of Phase Turbulence in the One Dimensional Complex {G}inzburg-{L}andau
Equation},
journal = {Phys. Rev. E},
volume = {55},
pages = {5073},
year = {1997},
abstract = {The phase gradient is used as an order parameter. Different states are
identified. In the PT region, a modified KSe rules the phase dynamics of the CGLe.}
}
@ARTICLE{AKcgl02,
author = {I. S. Aranson and L. Kramer},
title = {The World of the Complex {G}inzburg-{L}andau Equation},
journal = {Rev. Mod. Phys.},
volume = {74},
pages = {99-143},
year = {2002}
}
@ARTICLE{SMcgl02,
author = {J. Scheuer and B. A. Malomed},
title = {Stable and chaotic solutions of the complex {G}inzburg-{L}andau equation with
periodic boundary conditions},
journal = {Physica D},
volume = {161},
pages = {102-115},
year = {2002},
abstrac ={The behavior of the solutions of the CGLe without diffusion is studied
with a periodic boundary condition.}
}
@ARTICLE{DLBconv92,
author = {F. Daviaud and J. Lega and P. Berg\'{e} and P. Coullet and M. Dubois},
title = {Spatio-temporal intermittency in a 1D convective pattern: theoretical
model and experiments},
journal = {Physica D},
volume = {55},
pages = {287-308},
year = {1992},
abstract = {Describe the occurrence of the spatio-temporal intermittency in a 1-d
convective system that shows time-independent patterns based on the amplitude equation
approach}
}
@ARTICLE{CEEks93,
author = {P. Collet and J. -P. Eckmann and H. Epstein and J. Stubbe},
title = {Analyticity for the {K}uramoto-{S}ivashinsky equation},
journal = {Physica D},
volume = {67},
pages = {321-326},
year = {1993},
abstract = {Proved the analyticity of the solution of the KSe on a periodic interval
near the real axis and the width of the region is estimated. Numerical calculations
show that this width should not depend on the size of the interval.}
}
@ARTICLE{CEEksgl93,
author = {P. Collet and J. -P. Eckmann and H. Epstein and J. Stubbe},
title = {A global attracting set for the {K}uramoto-{S}ivashinsky equation},
journal = {Commun. Math. Phys.},
volume = {152},
pages = {203-214},
year = {1993},
abstract = {New bounds are given for the L^2 norm of solutions of the KSe (not
necessarily antisymmetric).}
}
@ARTICLE{Lega:holes,
author = {J. Lega},
title = {Traveling hole solutions of the {C}omplex {G}inzburg {L}andau
equation: a review},
journal = {Physica D},
volume = {269},
pages = {152-153},
year = {2001}
}
@ARTICLE{temam91edd,
author = {R. Temam},
title = {Approximation of attractors, large eddy simulations and multiscale
methods},
journal = {Proc. R. Soc. Lond. A},
volume = {434},
number = {1890},
pages = {23-39},
year = {1991},
abstract = {Estimates on the dimension of the attractors and the decay rates of
Fourier mode of NSe are given. By approximation of the inertial manifold, the
interaction between the large and small eddies are derived. Such laws open the way to
numerically efficient algorithms via multiscale mthods.}
}
@ARTICLE{spald91,
author = {D. B. Spalding},
title = {Kolmogorov's two-equation model of turbulence},
journal = {Proc. R. Soc. Lond. A},
volume = {434},
number = {1890},
pages = {211-216},
year = {1991},
abstract = {A brief survey of modelling equations of fully developed homogeneous
turbulence. Kolmogorov's original paper is translated.}
}
@ARTICLE{kolm91,
author = {A. N. Kolmogorov},
title = {The local structure of turbulence in incompressible viscous fluid
for very large Reynolds numbers},
journal = {Proc. R. Soc. Lond. A},
volume = {434},
number = {1890},
pages = {9-13},
year = {1991},
abstract = {Study the distribution of velocity difference of neighboring points
based on the locally homogeneous and locally isotropic hypothesized poperties of
turbulence motion. Two more similarity hypothesis suggested the existence of a
universal distribution for different viscosity and energy dissipation rate. The
large distance asymptotic behavior of the distribution is also deduced.}
}
@ARTICLE{FJolly95,
author = {C. Foias and M. S. Jolly},
title = {On the numerical algebraic approximation of global attractors},
journal = {Nonlinearity},
volume = {8},
pages = {295-319},
year = {1995},
abstract = {An algebraic approximation of the global attractors is developed for
the {L}orenz and KS system.}
}
@ARTICLE{HNks86,
author = {J. M. Hyman and B. Nicolaenko},
title = {The {K}uramoto-{S}ivashinsky equation: a bridge between PDE's and dynamical
systems},
journal = {Physica D},
volume = {18},
pages = {113-126},
year = {1986},
abstract = {Numerical work on the transition to chaos of the KSe. The attracting
solution manifold undergoes a complex bifurcation sequence including fixed points, limit
cycles, travelling waves, homoclinic orbits, invariant tori.}
}
@ARTICLE{HNZks86,
author = {J. M. Hyman and B. Nicolaenko and S. Zaleski},
title = {Order and complexity in the {K}uramoto-{S}ivashinsky model of weakly turbulent
interfaces},
journal = {Physica D},
volume = {23},
pages = {265-292},
year = {1986},
abstract = {The properties of the KSe is discussed in even larger domains. Still
the dynamical behavior is dominated by the skeleton of low dimensional vector fields.
More complex dynamics happens in large domain where intermittency is accurately
identified only by high-precision numerical method. The presence of strange fixed points
overrun the cellular fixed points in large scales which generates complex spatial
structure to the solutions. The Large scale statistical behavior
of the system is computed.}
}
@ARTICLE{KNSks90,
author = {I. G. Kevrekidis and B. Nicolaenko and J. C. Scovel},
title = {Back in the saddle again: a computer assisted study of the
{K}uramoto-{S}ivashinsky equation},
journal = {SIAM J. Appl. Math.},
volume = {50},
number = {3},
pages = {760-790},
year = {1990},
abstract = {The initial bifurcations of the KSe are examined
analytically and numerically. The heteroclinic connections
between symmetric solutions proved to play
an important role in the dynamics.}
}
% also known as ARTICLE{saddks,
% author = {I. G. Kevrekidis and B. Nicolaenko and J. C. Scovel},
@ARTICLE{FSTks86,
author = {U. Frisch and Z. S. She and O. Thual},
title = {Viscoelastic behavior of cellular solutions to the {K}uramoto-{S}ivashinsky
model},
journal = {J. Fluid Mech.},
volume = {168},
pages = {221-240},
year = {1986},
abstract = {The KSe on the periodic interval has stable steady states of period
2*pi/N. These ``cellular" states exhibit elastic behavior under perturbations of
wavelength much longer than that of itself.}
}
@ARTICLE{AGHks88,
author = {D. Armbruster and J. Guckenheimer and P. Holmes},
title = {Heteroclinic cycles and modulated travelling waves in systems with
O(2) symmetry},
journal = {Physica D},
volume = {29},
pages = {257-282},
year = {1988},
abstract = {Consider the normal form on the center manifold for the KSe and
discuss bifurcations with intrinsic O(2) symmetry.}
}
@ARTICLE{Mks86,
author = {D. Michelson},
title = {Steady solutions of the {K}uramoto-{S}ivashinsky equation},
journal = {Physica D},
volume = {19},
pages = {89-111},
year = {1986},
abtract = {The variety of steady solution of the KSe is discussed. For large c,
there is only one odd front-like bounded solution. In decreasing c, odd solutions with
more zeros are born until finally a periodic solution is born. Associated with the
periodic solution, infinite many tori will appear in the elliptic case and Cantor-type
set of chaotic solutions, with infinite many homoclinic odd solutions.}
}
@ARTICLE{FNSTks85,
author = {C. Foias and B. Nicolaenko and G.R. Sell and R. Temam},
title = {Inertial manifold for the {K}uramoto-{S}ivashinsky equation},
journal = {C. R. Acad. Sci. I-Math},
volume = {301},
number = {6},
pages = {285-288},
year = {1985}
}
@ARTICLE{FNSTks88,
author = {C. Foias and B. Nicolaenko and G.R. Sell and R. Temam},
title = {Inertial manifold for the {K}uramoto-{S}ivashinsky equation and an estimate of
their lowest dimension},
journal = {J. Math. Pure Appl.},
volume = {67},
number = {3},
pages = {197-226},
year = {1988}
}
@ARTICLE{DHBEks96,
author = {H. Dankowicz and P. Holmes and G. Berkooz and J. Elezgaray},
title = {Local models of spatio-temporally complex fields},
journal = {Physica D},
volume = {90},
pages = {387-407},
year = {1996}
}
@ARTICLE{newenv74,
author = {A. C. Newell},
title = {Envelope equations},
journal = {Lect. Appl. Math.},
volume = {15},
pages = {157-163},
year = {1974}
}
@ARTICLE{ave99,
author = {V. M. Egu\'{i}luz and P. Alstr{\o}m
and E. Hern\'{a}ndez-Garc\'{\i}a and O. Piro},
title = {Average patterns of spatiotemporal chaos: A boundary effect},
journal = {Phys. Rev. E},
volume = {59},
number = {3},
pages = {2822},
year = {1999}
}
77777 *****
77777 Computational techniques
77777 *****
@BOOK{kellbv,
author = {H. B. Keller},
title = {Numerical Methods for Two-Point Boundary-Value Problems},
publisher = {Dover},
address = {New York},
year = {1992}
}
@BOOK{ns,
author = {J. W. Thomas},
title = {Numerical Partial Differential Equations},
publisher = {Spring-Verlag},
address = {New York},
year = {1995}
}
series = {Texts in Applied Mathematics},
volume = {22},
@BOOK{bl,
author = {J. Stoer and R. Bulirsch},
title = {Introduction to Numerical Analysis},
publisher = {Springer-Verlag},
address = {New York},
year = {1983}
}
@BOOK{nr,
author = {W. H. Press and S. A. Teukolsky and W. T. Vetterling and
B. P. Flannery},
title = {Numerical Recipes in {C}},
publisher = {Cambridge University Press},
address = {Cambridge, England},
year = {1992}
}
@BOOK{lls,
author = {B.L. Ro\v{z}destvenski\v{i} and N.N. Janenko},
title = {Systems of Quasilinear Equations and Their Applications to
Gas Dynamics},
publisher = {AMS},
address = {Providence, RI},
year = {1983}
}
% series = {Translations of Mathematical Monographs},
@INCOLLECTION{brand03,
author = {A. Brandenburg},
title = {Computational aspects of astrophysical {MHD} and turbulence},
booktitle = {Advances in Nonlinear Dynamos},
editor = {A. Ferriz-Mas and M. N{\'{u}\~{n}ez}},
note = {{\tt astro-ph/0109497}},
publisher = {Taylor \& Francis},
address = {London},
pages ={269-344},
year = {2003}
}
series = {The Fluid Mechanics of Astrophysics and Geophysics},
volume = {9},
address = {London and New York},
@ARTICLE{kooknewt,
author = {H-T Kook and J. D. Meiss},
title = {Application of {N}ewton's method to {L}agrangian mappings},
journal = {Physica D},
volume = {36},
pages = {317-326},
year = {1989},
abstract = {An algorithm of {N}ewton's method is presented to find periodic
orbits for {L}agrangian mappings. The method is based on block-diagonalization
of the Hessian Matrix of the action function.}
}
@ARTICLE{mis01ks,
author = {P. Zgliczynski and K. Mischaikow},
title = {Rigorous Numerics for Partial Differential Equations: the
Kuramoto-Sivashinsky equation},
journal = {Foundations of Comp. Math.},
volume = {1},
pages = {255-288},
year = {2001}
}
@ARTICLE{vp98,
author = {A. Varga and S. Pieters},
title = {Gradient-Based Approach to Solve Optimal Periodic Output
Feedback Control Problems},
journal = {Automatica},
volume = {34},
number = {4},
pages = {477-481},
year = {1998},
abstract = {A numerical method based on the gradient minimization and
Periodic Schur Decomposition (PSD) is devised to solve the linear-quadratic
(LQ) optimization problem the linear periodic discrete-time control systems.
It can be used as a discretized version of the similar problem in the
continuous regime.}
}
@INCOLLECTION{autod89,
author = {A. Griewank},
title = {On Automatic Differentiation},
booktitle = {Mathematical Programming: Recent Developments
and Applications},
editor = {M. Iri and K. Tanabe},
pages = {83-108},
publisher = {Kluwer Academic Publishers},
address = {Amsterdam},
year = {1989}
}
@INPROCEEDINGS{carsim,
author = {C. Simo},
title = {On the Analytical and Numerical Approximation of Invariant
Manifolds},
booktitle = {Les M\'{e}thodes Modernes de la M\'{e}canique C\'{e}leste
(Goutelas '89)},
editor = {D. Benest and C. Froeschl\'{e}},
pages = {285-329},
year = {1989}
}
@ARTICLE{GM00aut,
author = {J. Guckenheimer and B. Meloon},
title = {Computing Periodic Orbits and their Bifurcations with
Automatic Differentiation},
journal = {SIAM J. Sci. Comp.},
volume = {22},
number = {3},
pages = {951 - 985},
year = {2000}
}
@ARTICLE{pimyk,
author = {H. E. Nusse and J. A. York},
title = {A procedure for finding numerical trajectories on chaotic
saddles},
journal = {Physica},
volume = {36D},
pages = {137},
year = {1989},
abstract = {{PIM} (proper interior maximum) method is proposed to find chaotic
trajectories on a nonattracting set with one-d unstable manifold. By
partitioning a straight line straddling the stable manifolds, the point with
longer and longer escaping time is found in a sequel until a sufficiently
small interval is reached when the point is considered to be very close to the
stable manifolds. After that, evolving and restraining the line element
results in a NUMERICAL orbit is obtained.}
}
@ARTICLE{pimsimp,
author = {P. Moresco and S. P. Dawson},
title = {The {PIM}-simplex method: an extension of the {PIM}-triple method
to saddles with an arbitrary number of expanding directions},
journal = {Physica},
volume = {126D},
pages = {38},
year = {1999},
abstract = {They generilized the PIM method to treat the chaotic saddle with
more than one unstable direction by introducing a simplex to cross the
stable manifolds. Combining with one method for finding local extremum in
a simplex, the authors do essentially the same job as in the simple PIM
method.}
}
@ARTICLE{pimstag,
author = {D. Sweet and H. E. Nusse. and J. A. Yorke},
title = {Stagger-and-Step Method: Detecting and Computing Chaotic
Saddles in Higher Dimensions},
journal = {Phys. Rev. Lett.},
volume = {86},
number = {11},
pages = {2261},
year = {2001},
abstract = {This is a statistical extension of the PIM method. Here, the
representative points are selected according to the so-called Exponential
Stagger Distribution about the current point, instead of the points on the
line or vertices of a simplex. This is a brutal force method while the
authors claim that it is quite efficient.}
}
@ARTICLE{tompaid96,
author = {S. Tompaidis},
title = {Numerical Study of Invariant Sets of a Quasi-periodic
Perturbation of a Symplectic Map},
journal = {Experimental Mathematics},
volume = {5},
pages = {211-230},
year = {1996},
abstract = {{N}ewton method is used to construct periodic orbits of longer
and longer period to approach a invariant torus with specific rotation vector.
Behavior after the break of a torus is described.}
}
@ARTICLE{StMeiss98,
author = {D. Sterling and J. D. Meiss},
title = {Computing Periodic Orbits using the Anti-Integrable Limit},
journal = {Phys. Lett. A},
volume = {241},
pages = {46-52},
year = {1998},
abstract = {Singular perturbation method. Through scaling transformation,
a small parameter is introduced such that when it is equal to zero, the
map becomes simple and controlable though non-deterministic. The symbolic
dynamics is easy to build and the AI limit orbit can be extended in a unique
way to the small parameter case when the AI limit is nondegenerate. {H}\'{e}non map
is used to demonstrate the idea.}
}
@ARTICLE{aub95ant,
author = {S. Aubry},
title = {Anti-integrability in dynamical and variational problems},
journal = {Physica D},
volume = {86},
pages = {284-296},
year = {1995},
}
@ARTICLE{kev01ks,
author = {M. E. Johnson and M. S. Jolly and I. G. Kevrekidis},
title = {The Oseberg transition: visualization of global bifurcations
for the Kuramoto-Sivashinsky equation},
journal = {Int. J. Bifurc. Chaos},
volume = {11},
number = {1},
pages = {1-18},
year = {2001}
}
@ARTICLE{mart87lfa,
author = {C. C. Martens and M. J. Davis and G. S. Ezra},
title = {Local Frequency analysis of chaotic motion in multidimensional
systems: energy transport and bottlenecks in planar OCS},
journal = {Chem. Phys. Lett.},
volume = {142},
number = {6},
pages = {519},
year = {1987},
abstract = {local frequency analysis is used and action-angle coordinate
is parametrized by the frequency ratio. The transport zones and partial
barriers are identified according to the ``noble ratio" consideration.}
}
@ARTICLE{shch04fli,
author = {E. Shchekinova and C. Chandre and Y. Lan and T. Uzer},
title = {Analyzing intramolecular dynamics by fast Lyapunov indicators},
journal = {J. Chem. Phys.},
volume = {121},
pages = {1-7},
year = {2004},
abstract = {Fast Lyapunov indicator method is used to identify the
chaotic and regular regions in the phase space. Periodic orbits are determined
to characterized the resonance properties of these regions.}
}
@ARTICLE{chand03tf,
author = {C. Chandre and S. Wiggins and T. Uzer},
title = {Time-frequency analysis of chaotic systems},
journal = {Physica D},
volume = {181},
pages = {171},
year = {2003},
abstract = {Use wavelets to extract local frequencies and use the major
frequencies to identify the characteristics of phase space motion.}
}
ARTICLE{
author = {},
title = {},
journal = {},
volume = {},
pages = {},
year = {}
}
@ARTICLE{lfind,
author = {F. K. Diakonos and P. Schmelcher and O. Biham},
title = {Systematic Computation of the Least Unstable Periodic Orbits in
Chaotic Attractors},
journal = {Phys. Rev. Lett.},
volume = {81},
pages = {4349},
year = {1998},
note = {{\tt chao-dyn/9810022}}
}
@ARTICLE{PSDB00,
author = {Detlef Pingel and P. Schmelcher and F.K. Diakonos and O. Biham},
title = {Theory and Applications of the Systematic Detection of Unstable
Periodic Orbits in Dynamical Systems},
journal = {Phys. Rev. E},
volume = {62},
pages = {2119},
year = {2000},
note = {{\tt nlin.CD/0006011}}
}
@ARTICLE{afind,
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author = {},
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%% From ~gibson/turbulence.bib
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month = {April},
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%%%% From FW.bib
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@inproceedings{Morkovin69,
AUTHOR={Morkovin, M.V},
TITLE={On the many faces of transition},
BOOKTITLE="Viscous Drag Reduction",
EDITOR="Wells, C.S.",
YEAR="1969",
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author = {{Manneville}, P.},
title = "{Spots and turbulent domains in a model of transitional plane {C}ouette flow}",
journal = {Theoretical and Computational Fluid Dynamics},
year = 2004,
month = nov,
volume = 18,
pages = {169-181},
doi = {10.1007/s00162-004-0142-4},
adsurl = {http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2004ThCFD..18..169M&db_key=PHY},
adsnote = {Provided by the Smithsonian/NASA Astrophysics Data System}
}
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author={R.R. Kerswell},
title={Recent progress in understanding the transition to turbulence in a pipe},
journal={Nonlinearity},
volume={18},
number={6},
pages={R17-R44},
url={http://stacks.iop.org/0951-7715/18/R17},
year={2005},
abstract={The problem of understanding the nature of fluid flow through a circular straight pipe remains one of the oldest problems in fluid mechanics. So far no explanation has been substantiated to rationalize the transition process by which the steady unidirectional laminar flow state gives way to a temporally and spatially disordered three-dimensional (turbulent) solution as the flow rate increases. Recently, new travelling wave solutions have been discovered which are saddle points in phase space. These plausibly represent the lowest level in a hierarchy of spatio-temporal periodic flow solutions which may be used to construct a cycle expansion theory of turbulent pipe flows. We summarize this success against the backdrop of past work and discuss its implications for future research. }
}
%%%%%%%%%%%%%%%%%%%% end FW.bib
%%% From JFG.bib
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% tau correction paper
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% main AHLS paper on ODE BL models
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%%%%%%%%%%%%%%%%%% end JFG.bib
%%%%% From PC.bib
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%From HJ.bib
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of the pressure term},
journal = {J. Comput. Physics},
volume = {199},
number = {1},
pages = {221--259},
year = {2004},
abstract = {}
}
@ARTICLE{Orszag1,
author = {S.A. Orszag and M. Israeli and M. Deville},
title = {Boundary conditions for incompressible flows},
journal = {J. Sci. Comput.},
volume = {1},
number = {},
pages = {75--111},
year = {1986},
abstract = {}
}
@ARTICLE{liuliupego05,
author = {J.-G. Liu and J. Liu and R.L. Pego },
title = {Divorcing pressure from viscosity in incompressible {N}avier-{S}tokes dynamics},
journal = {ar{X}iv::math.{AP}/0502549},
volume = {},
number = {},
pages = {},
year = {2005},
abstract = {}
}
@BOOK{greshosani,
author = {P.M. Gresho and R.L. Sani},
title = {Incompressible Flow and the Finite Element Method},
publisher = {John Wiley and Sons},
address = { },
year = {2000}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%end HJ.bib
@article{RPM,
author = {G.M. Shroff and H.B. Keller},
title = {Stabilization of unstable procedures: the recursive projection method},
journal = {SIAM J. Numer. Anal.},
volume = {30},
number = {4},
year = {1993},
issn = {0036-1429},
pages = {1099--1120},
doi = {http://dx.doi.org/10.1137/0730057},
publisher = {Society for Industrial and Applied Mathematics},
address = {Philadelphia, PA, USA},
}
@ARTICLE{CCW05,
author = {G. Csordas and M. Charalambides and F. Waleffe},
title = {A new property of a class of {J}acobi polynomials},
journal = {Proc. Amer. Math. Soc.},
volume = {133 },
number = {},
pages = {3551-3560},
year = {2005},
abstract = {}
}
@article{NP,
author = {K. Lust and D. Roose},
title = {An Adaptive {N}ewton-{P}icard Algorithm with Subspace Iteration for Computing Periodic Solutions},
journal = {SIAM J. Sci. Comput.},
volume = {19},
number = {4},
year = {1998},
issn = {1064-8275},
pages = {1188--1209},
doi = {http://dx.doi.org/10.1137/S1064827594277673},
publisher = {Society for Industrial and Applied Mathematics},
address = {Philadelphia, PA, USA},
}
% Predrag jun 20 2006
@MISC{ChencinerLink,
author = {Chenciner, A.},
note = {www.imcce.fr/Equipes/ASD/person/chenciner/chenciner.html}
}
% reference to Poincare relative periodic orbits
@Article{Creagh91,
author = {Creagh, Stephen C. and Littlejohn, Robert G.},
title = {Semiclassical trace formulas
in the presence of continuous symmetries},
journal = {Phys. Rev. A},
volume = {44},
number = {2},
pages = {836--850},
numpages = {14},
year = {1991},
month = {Jul},
doi = {10.1103/PhysRevA.44.836},
abstract = {A semiclassical expression for the symmetry-projected Green?s function is obtained; it involves a sum over classical periodic orbits on a symmetry-reduced phase space, weighted by characters of the symmetry group. These periodic orbits correspond to trajectories on the full phase space which are not necessarily periodic, but whose end points are related by symmetry.
Examples: the stadium billiard, a particle in a periodic potential, the Sinai billiard, the quartic oscillator, and the rotational spectrum of SF6.},
publisher = {American Physical Society}
}
@Article{Creagh92,
author = {Creagh, Stephen C. and Littlejohn, Robert G.},
title = {Semiclassical trace formulas
for systems with non-Abelian symmetry},
journal = {J. Phys. A},
volume = {25},
pages = {1643--1669},
year = {1992},
abstract = {
generalizations of the trace formula valid in the presence of a non-Abelian continuous symmetry. The usual trace formula must be modified in such cases because periodic orbits occur in continuous families, whereas the usual trace formula requires that the periodic orbits be isolated at a given energy. These calculations extend the results of a previous paper, in which they considered Abelian continuous symmetries. The most important application of the results is to systems with full three-dimensional rotational symmetry, and they give this case special consideration.
}
}
@article{Creagh93,
author={S C Creagh},
title={Semiclassical mechanics of symmetry reduction},
journal={Journal of Physics A: Mathematical and General},
volume={26},
pages={95-118},
note={http://stacks.iop.org/0305-4470/26/95},
year={1993},
abstract={The author discusses semiclassical approximations that are adapted to given symmetry classes in quantum mechanics. Arbitrary abelian symmetries and also rotational symmetry are treated. Semiclassical approximations are derived for the projected propagator and energy dependent Green's function associated with a given irreducible representation of the symmetry group. From these they derive trace formulae, analogous to the usual trace formula, that determine the energy levels in a given symmetry class in terms of classical orbits. }
}
% url={http://stacks.iop.org/0305-4470/26/95},
% as in Creagh92, now with irre reps
@Article{BisPal98,
author = {Pal, Santanu and Biswas, Debabrata },
title = {Symmetry reduction and semiclassical analysis of axially symmetric systems},
journal = {Phys. Rev. E},
volume = {57},
number = {2},
pages = {1475--1484},
numpages = {9},
year = {1998},
month = {Feb},
doi = {10.1103/PhysRevE.57.1475},
abstract = {
semiclassical trace formula for a symmetry-reduced part of the spectrum in axially symmetric systems. The classical orbits that contribute are closed in (?,z,p?,pz) and have
$p_\phi=m\hbar$,
where m is the azimuthal quantum number.
For $m \neq 0$,
these orbits vary with energy and almost never lie on periodic trajectories in the full phase space in contrast to the case of discrete symmetries.
},
publisher = {American Physical Society}
}
@Article{robb,
author = {Robbins, Jonathan M.},
title = {Discrete symmetries in periodic-orbit theory},
journal = {Phys. Rev. A},
volume = {40},
number = {4},
pages = {2128--2136},
numpages = {8},
year = {1989},
month = {Aug},
doi = {10.1103/PhysRevA.40.2128},
publisher = {American Physical Society}
}
@article{SakWhe03,
author = {Jamal Sakhr and Niall D. Whelan},
title = {Semiclassical trace formulas
for noninteracting identical particles},
publisher = {APS},
year = {2003},
journal = {Physical Review E},
volume = {67},
eid = {066213},
numpages = {27},
pages = {066213},
keywords = {chaos; quantum theory},
url = {http://link.aps.org/abstract/PRE/v67/e066213}
}
@article{Cassanas05a,
author ={Roch Cassanas},
title = {Reduced Gutzwiller formula with symmetry: {C}ase of a finite group},
journal = {J. Math. Phys.},
volume = {47},
pages = {042102},
year = {2006},
url = {arXiv.org:math-ph/0506063},
abstract = {assuming that periodic orbits are nondegenerate in SigmaE/G,
we get a reduced Gutzwiller trace formula which makes
periodic orbits of the reduced space
}
}
@article{Cassanas05b,
author ={Roch Cassanas},
title = {Reduced Gutzwiller formula with symmetry: {C}ase of a Lie group},
url = {arXiv.org:math-ph/0509014},
abstract = {We consider a classical Hamiltonian $H$ on $\mathbb{R}^{2d}$,
invariant by a Lie group of symmetry $G$
}
}
@MISC{Laurent-Polz04,
author = {Laurent-Polz, Frederic},
title = {Relative periodic orbits in point vortex systems},
url = {arXiv.org:math/0401022},
year = {2004}
}
% published in Nonlinearity
@ARTICLE{lop05rel,
author = {V. L{\'o}pez and P. Boyland and M. T. Heath and R. D. Moser},
title = {Relative Periodic Solutions of the
{C}omplex {G}inzburg-{L}andau equation},
journal = {SIAM J. Appl. Dyn. Syst.},
volume = {4},
pages = {1042},
year = {2005},
abstract = {Define relative periodic orbits and use Fourier modes to find
them.}
}
@MISC{lopezLink,
author = {L{\'o}pez, Vanessa},
url = {www.cse.uiuc.edu/~vlopez}
}
% chris.mccord at uc.edu j.montaldi at umist.ac.uk
% m.roberts at surrey.ac.uk l.sbano at surrey.ac.uk
@INPROCEEDINGS{McCordMontaldi,
author = {C. McCord AND J. Montaldi AND M. Roberts AND L. Sbano},
title = {Relative periodic orbits of symmetric {L}agrangian systems},
booktitle = {Proceedings of ``{E}quadiff 2003" },
editor = {Freddy Dumortier and et.al.},
pages = {482-493},
year = {2004}
}
@article{shroff:1099,
author = {Gautam M. Shroff and Herbert B. Keller},
title = {Stabilization of Unstable Procedures: The Recursive Projection Method},
publisher = {SIAM},
year = {1993},
journal = {SIAM Journal on Numerical Analysis},
volume = {30},
number = {4},
pages = {1099-1120},
keywords = {stabilization procedures; projection methods; stabilized continuation},
}
url = {http://link.aip.org/link/?SNA/30/1099/1}
doi = {10.1137/0730057}
@inproceedings{Keller77,
author = {H. B. Keller},
title = {Numerical solution of bifurcation
and nonlinear eigenvalue problems},
booktitle = {Applications of Bifurcation Theory},
year = {1977},
editor = {P.H. Rabinowitz},
publisher = {Academic Press},
address = {New York},
pages = {359-384}
}
@inproceedings{Keller79,
author = {H. B. Keller},
title = {Global homotopies and Newton methods},
booktitle = {Recent advances in numerical analysis},
year = {1979},
editor = {C. de Boor and G. H. Golub},
publisher = {Academic Press},
address = {New York},
pages = {73--94}
}
MR 80f:65059
@article{AllgGeorg88,
author={Allgower, E. L. and Georg, K.},
title={Numerically stable homotopy methods without an extra dimension},
journal = {Lect. Appl. Math.},
volume = {26},
pages = {1-13},
year={1990},
note={http://www.math.colostate.edu/emeriti/georg/homExtra.pdf}
}
% url={http://www.math.colostate.edu/emeriti/georg/homExtra.pdf}
% ={Eugene L. Allgower and Kurt Georg},
@MISC{Vanderb,
author = {Andre Vanderbauwhede},
title = {On the continuation of relative periodic orbits in reversible systems},
url = {cage.ugent.be/~avdb/eng/frameseng.html}
}
% from Gent, Belgium
% Does not seem important (no published papers on this):
@MISC{ViswanathPC06,
author = {Viswanath, Divakar},
title = {Recurrent motions within plane Couette turbulence},
url = {arXiv.org:physics/0604062},
year = {2006}
}
@ARTICLE{Wulff00,
author = {C. Wulff},
title = {Transitions from relative equilibria to relative periodic orbits},
journal = {Documenta Mathematica},
volume = {},
number = {},
pages = {},
year = {2000},
abstract = {}
}
% emis.ams.org
% discusses spiral waves
%%%%%%%%%%%%%% GROUP THEORY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@BOOK{BlumanSDE89,
author = {G. W. Bluman and S. Kumei},
title = {Symmetries and {D}ifferential {E}quations},
abstract = {Group Invariance Theory},
year = 1989,
publisher = "Springer-Verlag",
address = "New York, NY"
}
% from http://mat520.unime.it/fazio/preprints/sc-gita.bib
@Book{PCgr,
author = {{Cvitanovi\'c}, Predrag},
title = {Group {T}heory},
publisher = {Princeton {U}niversity {P}ress},
year = 2006,
address = {Princeton, NJ},
note = {ChaosBook.org/GroupTheory}
}
URL = {ChaosBook.org/GroupTheory}
% pdehaye at math.stanford.edu
% to specialized to be useful
@MISC{Dehaye05,
author = {Dehaye, Paul-Olivier},
title = {Averages over classical compact {L}ie
groups and {W}eyl characters},
year = {2005}
}
@Book{gilmore74,
author = {Robert Gilmore},
title = {Lie Groups, {L}ie Algebras, and Some of Their
Applications},
publisher = {John Wiley \& Sons},
year = {1974},
address = {New York}
}
@Book{hamer,
author = {Morton Hamermesh},
title = {Group Theory and Its Application to Physical Problems},
publisher = {Dover},
year = 1962,
address = {New York},
}
@Book{mta,
author = "Abraham, R. and Marsden, J. E. and Ratiu, T.",
title = "Manifolds, Tensor Analysis, and Applications",
publisher = "Springer-Verlag",
year = "1988",
address = "New York, NY",
edition = "2nd"
}
% "Abraham, Ralph and Marsden, Jerrold E. and Ratiu, Tudor",
@MISC{bronski-2005,
author = {Bronski, Jared C. and Gambill, Thomas N.},
title = {Uncertainty estimates and
L\_2 bounds for the Kuramoto-Sivashinsky equation},
journal={Nonlinearity},
volume={19},
pages={2023-2039},
year={2006},
url = {arXiv.org:math/0508481},
year = {2005}
}
@BOOK{TelGr06,
author = {T\'el, T. and Gruiz, M.},
title = {Chaotic Dynamics:
An Introduction Based on Classical Mechanics},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {2006}
}