% pipes/bibtex/pipes.bib for fluid dynamics articles
% $Author: predrag $ $Date: 2012-10-18 08:30:25 -0400 (Thu, 18 Oct 2012) $
% alphabetic by the main author
% started from halcrow/bibtex/fluid.bib Apr 28 2011
@incollection{DeNa12,
author = {Deguchi, K. and Nagata, M.},
title = {Exact coherent state with travelling hairpin vortex in plane {Couette} flow},
booktitle = {Progress in Turbulence and Wind Energy IV},
editor = {Oberlack, M. and Peinke, J. and Talamelli, A. and Castillo, L. and H\"olling, M.},
publisher = "Springer",
address = "New York",
pages = {129-134},
url = {http://dx.doi.org/10.1007/978-3-642-28968-2_27},
year = {2012}
}
@ARTICLE{HaMe98,
author = {G. Haller and I. Mezi\'c},
title = {Reduction of three-dimensional, volume-preserving flows with symmetry},
journal = {Nonlinearity},
year = {1998},
volume = {11},
pages = {319--339}
}
@ARTICLE{Xia92,
author = {Xia, Z.},
title = {Existence of invariant tori in volume-preserving diffeomorphisms},
journal = {Ergod. Theor. Dynam. Syst.},
year = {1992},
volume = {12},
pages = {621--31}
}
@ARTICLE{CheSu90,
author = {Cheng, C. Q. and Sun, Y. S.},
title = {Existence of invariant tori in three-dimensional measure-preserving
mappings},
journal = {Celestial Mech. Dynam. Astronom.},
year = {1990},
volume = {47},
pages = {275--92}
}
@ARTICLE{MeWi94,
author = {I. Mezi\'c and S. Wiggins},
title = {On the integrability and perturbation of three-dimensional fluid
flows with symmetry},
journal = {J. Nonlin. Sci.},
year = {1994},
volume = {4},
pages = {157--194},
abstract = {The purpose of this paper is to develop analytical methods for studyingparticle
paths in a class of three-dimensional incompressible fluid flows.
In this paper we study three-dimensionalvolume preserving vector
fields that are invariant under the action of a one-parameter symmetry
group whose infinitesimal generator is autonomous and volume-preserving.
We show that there exists a coordinate system in which the vector
field assumes a simple form. In particular, the evolution of two
of the coordinates is governed by a time-dependent, one-degree-of-freedom
Hamiltonian system with the evolution of the remaining coordinate
being governed by a first-order differential equation that depends
only on the other two coordinates and time. The new coordinates depend
only on the symmetry group of the vector field. Therefore they arefield-independent.
The coordinate transformation is constructive. If the vector field
is time-independent, then it possesses an integral of motion. Moreover,
we show that the system can be further reduced toaction-angle-angle
coordinates. These are analogous to the familiar action-angle variables
from Hamiltonian mechanics and are quite useful for perturbative
studies of the class of systems we consider. In fact, we show how
our coordinate transformation puts us in a position to apply recent
extensions of the {Kolmogorov-Arnold-Moser} {(KAM)} theorem for three-dimensional,
volume-preserving maps as well as three-dimensional versions of Melnikov's
method. We discuss the integrability of the class of flows considered,
and draw an analogy with Clebsch variables in fluid mechanics.}
}
@ARTICLE{EaPoSa10,
author = {J. Early and J. Pohjanpelto and R. Samelson},
title = {Group foliation of equations in geophysical fluid dynamics},
journal = {Discrete Contin. Dynam. Systems},
year = {2010},
volume = {27},
pages = {1571--1586}
}
@ARTICLE{Low58,
author = {F. E. Low},
title = {A {{Lagrangian}} formulation of the {{Boltzmann-Vlasov}} equation
for plasmas},
journal = {Proc. R. Soc. London A},
year = {1958},
volume = {248},
pages = {282--287},
abstract = {A variational principle is found for the {Boltzmann-Vlasov} equation
for an ionized gas in an electromagnetic field. The principle leads
to a new formulation of the problem of small oscillations about equilibrium.}
}
@ARTICLE{Morr98,
author = {Morrison, P. J.},
title = {Hamiltonian description of the ideal fluid},
journal = {Rev. Mod. Phys.},
year = {1998},
volume = {70},
pages = {467--521}
}
@ARTICLE{Morr80,
author = {Morrison, P. J.},
title = {The {Maxwell-Vlasov} equations as a continuous {Hamiltonian} system},
journal = {Phys. Lett. A},
year = {1980},
volume = {80},
pages = {383--386}
}
@ARTICLE{MorrGree80,
author = {Morrison, P. J. and J. M. Greene},
title = {Noncanonical {Hamiltonian} Density Formulation of Hydrodynamics and
Ideal Magnetohydrodynamics},
journal = {Phys. Rev. Lett.},
year = {1980},
volume = {45},
pages = {790--794},
note = {see also Phys. Rev. Lett. 48, 569 (1982)}
}
@ARTICLE{FowlerCLE82,
author = {Fowler , A. C. and {Gibbon}, J. D. and {McGuinness}, M. J.},
title = {The complex {Lorenz} equations},
journal = {Physica D},
year = {1982},
volume = {4},
pages = {139-163}
}
@ARTICLE{GibMcCLE82,
author = {Gibbon, J. D. and McGuinness, M. J.},
title = {The real and complex {Lorenz} equations in rotating fluids and lasers},
journal = {Physica D},
year = {1982},
volume = {5},
pages = {108--122}
}
@ARTICLE{GibMcCLE83,
author = {Gibbon, J. D. and McGuinness, M. J.},
title = {The real and complex {Lorenz} equations},
journal = {Proc. R. Irish. Acad., Ser. A},
year = {1983},
volume = {83},
pages = {17--22}
}
@ARTICLE{FowlerCLE83,
author = {Fowler , A. C. and Gibbon, J. D. and McGuinness, M. J.},
title = {The real and complex {Lorenz} equations and their relevance to physical
systems},
journal = {Physica D},
year = {1983},
volume = {7},
pages = {139--163}
}
@ARTICLE{ToDe98,
author = {Toronov, V. Y. and Derbov, V. L.},
title = {Topological properties of laser phase},
journal = {J. Optical Soc. of America B},
year = {1998},
volume = {15},
pages = {1282--1290}
}
@ARTICLE{VlToDe98a,
author = {Vladimirov, A. G. and Toronov, V. Y. and Derbov, V. L.},
title = {Properties of the phase space and bifurcations in the complex {Lorenz}
model},
journal = {Technical Physics},
year = {1998},
volume = {43},
pages = {877--884}
}
@ARTICLE{RaHaAb96,
author = {Rauh, A. and Hannibal, L. and Abraham, N. B.},
title = {Global stability properties of the complex {Lorenz} model},
journal = {Physica D},
year = {1996},
volume = {99},
pages = {45--58}
}
@article{FaHa12,
author = {M. Farazmand and G. Haller},
title = {Computing {Lagrangian} coherent structures from their variational theory},
year = {2012},
journal = {Chaos},
volume = {22},
numpages = {12},
pages = {013128},
keywords = {differential equations; nonlinear dynamical systems; tensors},
doi = {10.1063/1.3690153}
}
@INPROCEEDINGS{BayOrt04,
author = {E. Bayro-Corrochano and J. Ortegon-Aguilar},
title = {Lie algebra template tracking},
booktitle = {International Conference on Pattern Recognition},
year = {2004},
volume = {2},
pages = {56--59},
doi = {10.1109/ICPR.2004.1334036}
}
@ARTICLE{DruCip00,
author = {T. Drummond and R. Cipolla},
title = {Application of {Lie} algebras to visual servoing},
journal = {Intern. J. Computer Vision},
year = {2000},
volume = {37},
pages = {21--41}
}
@ARTICLE{CHHM98,
author = {Cendra, H. and Holm, D. D. and Hoyle, M. J. W. and Marsden, J. E.},
title = {The {Maxwell-Vlasov} equations in {Euler-Poincar\'e} form},
journal = {J. of Math. Phys.},
year = {1998},
volume = {39},
pages = {3138--3157}
}
@Article{ROC07,
author = "F. Rincon and G. I. Ogilvie and C. Cossu",
title = "On self-sustaining processes in Rayleigh-stable rotating
plane Couette flows and subcritical transition to turbulence
in accretion disks",
journal = "Astron. Astrophys.",
volume = "463",
pages = "817--832",
year = "2007",
}
@Article{HRCLOL11,
author = "J. Herault and F. Rincon and C. Cossu and G. I. Ogilvie
and P.-Y. Longaretti",
title = "Periodic magnetorotational dynamo action as a prototype
of nonlinear magnetic-field generation in shear flows",
journal = "Phys. Rev. E",
volume = "84",
pages = "036321",
year = "2011",
}
@Article{WHC10,
author = "A. P. Willis and Y. Hwang and C. Cossu",
title = "Optimally amplified large-scale streaks and drag
reduction in turbulent pipe flow",
journal = "Phys. Rev. E",
volume = "82",
pages = "036321",
year = "2010",
}
%%%% please prepend new references HERE: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% (so we can eventually transfer them back to the master
% file halcrow/bibtex/fluid.bib)
@book{Harter93,
author="Harter, W. G.",
title="Principles of Symmetry, Dynamics, and Spectroscopy",
address = "New York",
publisher="Wiley",
year="1993",
}
@Book{DS,
author = "{J.E. Dennis, Jr.,} and R. B. Schnabel",
title = "Numerical Methods for Unconstrained Optimization
and Nonlinear Equations",
publisher = "SIAM",
year = "1996",
address = "Philadelphia"
}
@Article{KawKida05,
author = "G. Kawahara and S. Kida",
title = "Elementary coherent structures in turbulent flows",
journal = "Parity",
volume = "20",
pages = "12",
year = "2005",
}
@Article{KawKiVeen06,
author = "G. Kawahara and S. Kida and L. van Veen",
title = "Unstable periodic motion in turbulent flows",
journal = "Nonlinear Process in Geophysics",
volume = "13",
year = "2006",
pages = "499--507",
}
@Article{KaUhlVeen12,
author = "G. Kawahara and M. Uhlmann and L. van Veen",
title = "The significance of simple invariant solutions
in turbulent flows",
journal = "Annu. Rev. Fluid Mech",
volume = "44",
year = "2012",
pages = "203--225",
}
@TechReport{ SpiekerBlog,
author = "Spieker, D. W.",
title = "{Gone fishin' - a research blog on fluid flows}",
institution = "School of Physics, Georgia Inst. of Technology",
year = "2009",
note = "{\tt Channelflow.org/dokuwiki/doku.php/gtspring2009:spieker\_blog}",
}
@article{duguet08,
author = {Duguet, Y. and Pringle, C. C. T. and Kerswell, R. R.},
title = {Relative periodic orbits in transitional pipe flow},
year = {2008},
journal = {Phys.\ Fluids},
volume = {20},
pages = {114102},
note = {{\tt arXiv:0807.2580}}
}
@Article{Palacios07,
author = {A. Palacios},
title = {Heteroclinic cycles},
journal = {Scholarpedia},
year = {2007},
volume = {2},
number = {1},
pages = {2352},
}
@Book{Pan96,
author = "R. L. Panton",
title = "Incompressible flow",
publisher = "Wiley-Interscience",
address = "New York",
year = "1996"
}
@Book{Pan97,
editor = "R. L. Panton",
title = "Self-sustaining mechanisms of wall turbulence",
publisher = "Computational Mechanics Publications",
address = "Southhampton",
year = "1997"
}
@Article{Pasq87,
author = "F. Pasquarelli and A. Quarteroni and G. Sacchi-Landriani",
title = "Spectral approximations of the {S}tokes problem by divergence-free functions",
journal = "J.\ Comp.\ Sci.",
volume = "2",
number = "3",
pages = "195--226",
year = "1987"
}
@Article{PexMull06,
author = {J. Peixinho and T. Mullin},
title = {Decay of turbulence in pipe flow},
journal = {Phys.\ Rev.\ Lett.},
year = {2006},
volume = {96},
pages = {094501},
}
@Article{PexMull07,
author = {J. Peixinho and T. Mullin},
title = {Finite-amplitude thresholds for transition in pipe flow},
journal = {J.\ Fluid Mech.},
year = {2007},
volume = {582},
pages = {169-178},
}
@Article{PenDai07,
author = "J. Peng and J. O. Dabiri",
title = "A potential-flow, deformable-body model for fluid-structure
interactions with compact vorticity:
application to animal swimming measurements",
journal = "Experiments in Fluids, to appear",
year = "2007",
URL = "\weblink{dabiri.caltech.edu/people/dabiri.html}"
}
@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.}
}
@Article{Petr40,
author = "Petrov, G.I.",
title = "Application of {Galerkin}'s method to the problem of the
stability of the flow of a viscous fluid",
journal = "Prikl.\ Mat.\ Mekh.",
volume = "4",
pages = "3--12",
year = "1940"
}
@Book{Peyret02,
author = "R. Peyret",
title = "Spectral Methods for Incompressible Flows",
year = "2002",
publisher = "Springer",
addr = "New York"
}
@Book{poinper,
author = "H. Poincar{\'e}",
title = "New Methods in Celestial Mechanics",
series = "History of Modern Physics and Astronomy",
volume = "13",
publisher = "Springer",
address = "New York",
year = "1992"
}
@Book{poincare,
author = "H. Poincar{\'e}",
title = "Les m{\'e}thodes nouvelles de la m{\'e}chanique c{\'e}leste",
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 \refref{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."
}
@Article{Poinc1896,
author = "H. Poincar\'e",
title = "Sur les solutions p\'eriodiques et
le principe de moindre action",
journal = "C. R. Acad. Sci. Paris",
volume = "123",
pages = "915--918",
year = "1896",
}
@Article{PoBaGr88,
author = "A. Politi and R. Badii and P. Grassberger",
title = "On the geometric structure of non-hyperbolic attractors",
journal = "J. Phys. A",
volume = "15",
pages = "L763",
year = "1988"
}
@article{GrBaPo88,
author = {P. Grassberger and R. Badii and A. Politi},
title = {Scaling laws for
invariant measures on hyperbolic and nonhyperbolic attractors},
journal = {J. Stat. Phys.},
volume = {51},
year = {1988},
pages = {135},
}
@MISC{Postl08,
author = {C. M. Postlethwaite},
title = {Stabilisation of long-period periodic orbits using
time-delayed feedback control},
note = "{\tt arXiv:0804.2903}",
year = {2008}
}
@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 Univ.\ Press",
year = "1996",
address = "Cambridge"
}
@Article{Pringle07,
author = {Pringle, C. C. T. and Kerswell, R. R.},
title = {Asymmetric, helical, and mirror-symmetric traveling waves
in pipe flow},
journal = "Phys.\ Rev.\ Lett.",
year = {2007},
volume = {99},
pages = {074502}
}
@Article{Pringle09,
author = {Pringle, C. C. T. and Duguet, Y. and Kerswell, R. R.},
title = {Highly symmetric travelling waves in pipe flow},
journal = {Phil. Trans. Royal Soc. A},
volume = {367},
pages = {457--472},
year = {2009},
note = {{\tt arXiv:0804.4854}}
}
% NOTES for transfer back:
% replace Kerswell07 by KeTu06 everywhere
% replace DuguetARX07 by duguet07 everywhere
@article{AHMNPTB03,
author = {M. E. Brachet and M. Abid and C. Huepe and S. Metens and C.
Nore and C. T. Pham and L. S. Tuckerman},
title = {{Gross–Pitaevskii} dynamics of {Bose–Einstein} condensates and
superfluid turbulence},
journal = {Fluid Dyn. Research},
volume = {33},
year = {2003},
pages = {509--544},
doi = {10.1016/j.fluiddyn.2003.09.001},
}
@article{Absh05,
title = {Mode competition of rotating waves
in reflection-symmetric {Taylor-Couette} flow},
volume = {540},
journal = {J.\ Fluid Mech.},
author = {J. Abshagen and J. M. Lopez and F. Marques and G. Pfister},
year = {2005},
pages = {269--300}
}
@article{Absh05a,
title = {Symmetry breaking via global bifurcations
of modulated rotating waves in hydrodynamics},
volume = {94},
journal = {Phys.\ Rev.\ Lett.},
author = {J. Abshagen and J. M. Lopez and F. Marques and G. Pfister},
year = {2005},
pages = {074501}
}
@Article{AS87,
author = "M. S. Acarlar and C. R. Smith",
title = "A study of hairpin vortices in a laminar boundary layer",
journal = "J.\ Fluid Mech.",
year = "1987",
volume = "175",
pages = "1--41, 45--83"
}
@Article{AllgGeorg88,
author = "Eugene L. Allgower and Kurt Georg",
title = "Numerically stable homotopy methods without an extra dimension",
journal = "Lect.\ Appl.\ Math.",
volume = "26",
pages = "1--13",
year = "1990",
URL = "www.math.colostate.edu/emeriti/georg/homExtra.pdf"
}
@Book{AnAr88,
author = "D. V. Anosov and V. I. Arnol'd",
title = "Dynamical systems {I}: {Ordinary} Differential Equations
and Smooth Dynamical Systems",
publisher = "Springer",
address = "New York",
year = "1988",
}
@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{AGHks89,
author = "D. Armbruster and J. Guckenheimer and P. Holmes",
title = "{Kuramoto-Sivashinsky} Dynamics on the Center-Unstable manifold",
journal = "SIAM J.\ Appl.\ Math.",
volume = "49",
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{Ascher95,
author = "U. Ascher and S. Ruuth and B. Wetton",
title = "Implicit-explicit methods for time-dependent partial differential equations",
journal = "SIAM J.\ Numer.\ Anal.",
year = "1995",
volume = "32",
pages = "797--823"
}
@Article{ARS,
author = "U. M. Ascher and S. J. Ruuth and R. J. Spiteri",
title = "Implicit-explicit {R}unge-{K}utta methods for
time-dependent partial differential equations",
journal = "Applied Numerical Mathematics",
year = "1997",
volume = "25",
pages = "151--167"
}
@Article{Aubry88,
author = "N. Aubry and P. Holmes and J. L. Lumley and E. Stone",
title = "The dynamics of coherent structures in the wall region
of turbulent boundary layer",
journal = "J.\ Fluid Mech.",
year = "1988",
volume = "192",
pages = "115--173"
}
% main AHLS paper on ODE BL models
@article{AMdABH11,
author = {Avila, K. and Moxey, D. and de Lozar, A. and Avila, M.
and Barkley, D. and Hof, B.},
title = {The onset of turbulence in pipe flow},
journal = {Science},
volume = {333},
pages = {192-196},
year = {2011},
doi = {10.1126/science.1203223},
abstract ={Shear flows undergo a sudden transition from laminar to
turbulent motion as the velocity increases, and the onset of turbulence
radically changes transport efficiency and mixing properties. Even for
the well-studied case of pipe flow, it has not been possible to determine
at what Reynolds number the motion will be either persistently turbulent
or ultimately laminar. We show that in pipes, turbulence that is
transient at low Reynolds numbers becomes sustained at a distinct
critical point. Through extensive experiments and computer simulations,
we were able to identify and characterize the processes ultimately
responsible for sustaining turbulence. In contrast to the classical
Landau-Ruelle-Takens view that turbulence arises from an increase in the
temporal complexity of fluid motion, here, spatial proliferation of
chaotic domains is the decisive process and intrinsic to the nature of
fluid turbulence.},
}
@article{avila2010,
title={On the transient nature of localized pipe flow turbulence},
author={Avila, M. and Willis, A.P. and Hof, B.},
journal={J. Fluid Mech.},
volume={646},
pages={127--136},
year={2010}
}
@article{hof2008,
title={Repeller or attractor? {Selecting} the dynamical model for the
onset of turbulence in pipe flow},
author={Hof, B. and De Lozar, A. and Kuik, D.J. and Westerweel, J.},
journal={Phys. Rev. Lett.},
volume={101},
pages={214501},
year={2008},
}
@Unpublished{deLMeAvHo12,
author = {A. de Lozar and F. Mellibovsky and M. Avila and B. Hof},
title = {Experimental observation of the edge state in pipe flow},
note = {In preparation},
year = "2012",
}
@Unpublished{ ACHKW11,
author = "Willis, A. P. and Cvitanovi{\'c}, P. and Avila, M.",
title = "Revealing the state space of turbulent pipe flow by symmetry reduction",
year = "2012",
note = "{\tt arXiv:1203.3701}, submitted to J. Fluid Mech.",
}
@Unpublished{ WiAvCv12,
author = "Willis, A. P. and Avila, M. and Cvitanovi{\'c}, P.",
title = "Symmetries of turbulent flows: How to slice them",
year = "2012",
note = "In preparation"
}
@inproceedings{ CvWiAv12,
author = "Cvitanovi{\'c}, P. and Willis, A. P. and Avila, M.",
title = "Revealing the state space of turbulent pipe flow by symmetry reduction",
editor = "Jianxiang Wang",
booktitle = "Proceed. ICTAM 2012 Intern. Congr. Theor. and Appl. Mech.",
year = "2012",
}
@Book{SchGer00,
author = "Schlichting, H. and Gersten, K.",
title = "Boundary-Layer Theory",
year = "2000",
keywords = "Boundary-Layer Theory",
publisher = "Springer",
address = "New York"
}
% isbn = "3-540-66270-7",
@article{BCFLR08,
author="R. Bachelard and C. Chandre and D. Fanelli
and X. Leoncini and S. Ruffo",
title="Abundance of regular orbits and out-of-equilibrium phase
transitions in the thermodynamic limit for long-range systems",
journal="Phys. Rev. Lett.",
volume="101",
pages="260603",
year="2008"
}
@ARTICLE{BGMWW92,
author = {A. Back and J. Guckenheimer and M. R. Myers and F. J. Wicklin and P. A. Worfolk},
title = {DsTool: {Computer} assisted exploration of dynamical systems},
journal = {Notices of the AMS},
year = {1992},
volume = {39},
pages = {303--309}
}
@Article{BarChor98,
author = "G. I. Barenblatt and A. J. Chorin",
title = "New Perspectives in Turbulence: Scaling Laws, Asymptotics, and Intermittency",
journal = "SIAM Review",
volume = "40",
number = "2",
pages = "265--291",
year = "1998",
}
@ARTICLE{ Barkley99,
author = {Barkley, D. and {Tuckerman}, L.~S.},
title = "Stability analysis of perturbed plane {Couette} flow",
journal = {Phys.\ Fluids},
note = "{\tt arXiv:physics/0312048}",
year = 1999,
volume = 11,
pages = {1187-1195},
}
@article{TuckBar03,
author = {L. S. Tuckerman and D. Barkley},
title = {Symmetry breaking and turbulence in perturbed plane {Couette} flow},
abstract = {Perturbed plane Couette flow containing a thin
spanwise-oriented ribbon undergoes a subcritical bifurcation at
Re = 230 to a steady 3D state containing streamwise vortices.
This bifurcation is followed by several others giving rise to a
fascinating series of stable and unstable steady states of
different symmetries and wavelengths. First, the
backwards-bifurcating branch reverses direction and becomes
stable near Re = 200. Then, the spanwise reflection symmetry is
broken, leading to two asymmetric branches which are themselves
destabilized at Re = 420. Above this Reynolds number, time
evolution leads first to a metastable state whose spanwise
wavelength is halved and then to complicated time-dependent
behavior. These features are in agreement with experiments.},
year = {2002},
journal = {Theor. Comp. Fluid Dyn.},
volume = 16,
pages = {91-97},
note = "{\tt arXiv:physics/0312051}",
}
@ARTICLE{ Barkley05,
author = {{Barkley}, D. and {Tuckerman}, L.~S.},
title = "Computational study of turbulent laminar patterns in {Couette} flow",
journal = {Phys.\ Rev.\ Lett.},
note = "{\tt arXiv:physics/0403142}",
year = 2005,
volume = 94,
pages = {014502-+},
}
@article{TuckBar07,
title = {mean flow and modeling of turbulent-laminar patterns in
plane {Couette} flow},
volume = {117},
journal = {Springer Proc. Physics},
author = {L. S. Tuckerman and D. Barkley},
year = {2007},
pages = {224--226}
}
@article{BarTuck07,
title = {Mean flow of turbulent-laminar patterns in
plane {Couette} flow},
volume = {576},
journal = {J.\ Fluid Mech.},
author = {D. Barkley and L. S. Tuckerman},
year = {2007},
pages = {109--138}
}
@article{BLZ08,
author = "P. Bates, K. Lu, and C. Zeng",
title = "Approximately invariant manifolds and global dynamics of spike states",
journal = "Inventiones mathematicae",
volume = "55",
pages = "355--433",
year = "2008"
}
@Article{benks66,
author = "D. J. Benney",
title = "Long waves in liquid films",
journal = "J.\ Math.\ Phys",
volume = "45",
pages = "150--155",
year = "1966"
}
@Article{BG81,
author = "D. J. Benney and L. H. Gustavsson",
title = "A new mechanism for linear and nonlinear hydrodynamic instability",
journal = "Stud. Appl. Math.",
year = "1981",
volume = "64",
pages = "185--209"
}
@Article{B84,
author = "D. J. Benney",
title = "The evolution of disturbances in shear flows at high {R}eynolds numbers",
journal = "Stud. Appl. Math.",
year = "1984",
volume = "70",
pages = "1--19"
}
@Article{BC89,
author = "D. J. Benney and K. A. Chow",
title = "A mean flow first harmonic theory for hydrodynamic instabilities",
journal = "Stud. Appl. Math.",
year = "1989",
volume = "80",
pages = "37--73"
}
@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",
volume = "6",
number = "4",
pages = "1574--1578"
}
@Article{BerlinPF94,
author = "S. Berlin and A. Lundbladh and D. Henningson",
title = "Spatial simulations of oblique transition in a boundary layer",
journal = "Phys.\ Fluids",
year = "1994",
month = "June",
volume = "6",
number = "6",
pages = "1949--1951"
}
@Article{BK90,
author = "M. V. Berry and J. P. Keating",
title = "A rule for quantizing chaos",
journal = "J.\ Phys.\ A",
year = "1990",
volume = "23",
number = "21",
pages = "4839--4849"
}
@Article{BeTh04,
author = {W.-J. Beyn and V. Th\"ummler},
title = {Freezing solutions of equivariant evolution equations},
journal = {SIAM J. Appl. Dyn. Syst.},
year = {2004},
volume = {3},
pages = {85--116},
}
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author = "G. D. Birkhoff",
title = "Recent advances in dynamics",
journal = "Science",
year = "1920",
volume = "51",
pages = "51--55"
}
@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"
}
@article{blackwelder76,
AUTHOR="Blackwelder, R. F. and Kaplan, R.",
TITLE="On the wall structure of the turbulent boundary layer",
JOURNAL="J. Fluid Mech.",
YEAR="1976",
VOLUME="76",
PAGES="89"}
@article{blackwelder87,
AUTHOR="Swearingen, J. D. and Blackwelder, R. F.",
TITLE="The growth and breakdown of streamwise vortices in the presence
of a wall",
JOURNAL="J. Fluid Mech.",
YEAR="1987",
VOLUME="182",
PAGES="255"}
@Book{BH86,
author = "N. Bleistein and R. A. Handelsman",
title = "Asymptotic Expansions of Integrals",
year = "1986",
publisher = "Dover",
address = "New York"
}
@article{Bowen72,
author = "R. Bowen",
title = "Periodic orbits for hyperbolic flows",
journal = "Amer. J. Math.",
volume = "94",
pages = "1-30",
year = "1972"
}
@article{Bowen73,
author = "R. Bowen",
title = "Symbolic dynamics for hyperbolic flows",
journal = "Amer. J. Math.",
volume = "95",
pages = "429-460",
year = "1973"
}
%Flows as suspensions introduced here; correct Fredholm
%determinants, but restricted to stable manifold dimensions.
% http://math.ucsd.edu/~awong/handouts/boyce.html
@Book{BoydiP06,
author = "W. E. Boyce and R. C. DiPrima",
title = "Elementary Differential Equations and Boundary Value Problems",
publisher = "Wiley-Interscience",
address = "New York",
year = "2006"
}
@article{Briggs91,
author={Briggs, K.},
title={A precise calculation of the {Feigenbaum} constants},
volume={57},
journal={Mathematics of Computation}, year={1991},
pages={435--439}
}
@Article{BQT91,
author = "K. M. Briggs and G. R. W. Quispel and C. J. Thompson",
title = "Feigenvalues for {Mandelsets}",
journal = "J. Phys. A",
volume = "24",
pages = "3363--3368",
year = "1991",
keywords = "Feigenbaum constant, Mandelbrot set",
}
@article{BDS98,
author={Briggs, K. M. and Dixon, T. W. and Szekeres, G.},
title={Analytic solutions of the {Cvitanovic-Feigenbaum} and
{Feigenbaum-Kadanoff-Shenker} equations},
volume={8},
journal={Int. J. Bif. Chaos},
year={1998},
pages={347--357}
}
@Unpublished{Bouch03,
author={E. Bouchbinder},
title={The nonlinear {Schr\"odinger} equation},
note = {\HREF{http://ChaosBook.org/library/Bouch03.pdf}
{unpublished notes}},
year = "2003"
}
@article{BudMez12,
author = "M. Budi\v{s}i\'c and I. Mezi\'c",
title = "Geometry of the ergodic quotient reveals coherent structures in flows",
journal = "Physica D",
volume = "241",
pages = "1255--1269",
year = "2012",
doi = "10.1016/j.physd.2012.04.006",
note = {{\tt arXiv:1204.2050}},
abstract = "Dynamical systems that exhibit diverse behaviors can rarely
be completely understood using a single approach. However, by identifying
coherent structures in their state spaces, i.e., regions of uniform and
simpler behavior, we could hope to study each of the structures
separately and then form the understanding of the system as a whole. The
method we present in this paper uses trajectory averages of scalar
functions on the state space to: (a) identify invariant sets in the state
space, and (b) to form coherent structures by aggregating invariant sets
that are similar across multiple spatial scales. First, we construct the
ergodic quotient, the object obtained by mapping trajectories to the
space of the trajectory averages of a function basis on the state space.
Second, we endow the ergodic quotient with a metric structure that
successfully captures how similar the invariant sets are in the state
space. Finally, we parametrize the ergodic quotient using intrinsic
diffusion modes on it. By segmenting the ergodic quotient based on the
diffusion modes, we extract coherent features in the state space of the
dynamical system. The algorithm is validated by analyzing the
Arnold-Beltrami-Childress flow, which was the test-bed for
alternative approaches: the Ulam's approximation of the transfer
operator and the computation of Lagrangian Coherent Structures.
Furthermore, we explain how the method extends the Poincar\'e map analysis
for periodic flows. As a demonstration, we apply the method to a
periodically-driven three-dimensional Hill's vortex flow, discovering
unknown coherent structures in its state space. Finally, we discuss
differences between the ergodic quotient and alternatives, propose a
generalization to analysis of (quasi-)periodic structures, and lay out
future research directions."
}
@Article{Busse04,
author = "F. H. Busse",
title = "Visualizing the dynamics of the onset of turbulence",
journal = "Science",
volume = "305",
pages = "1574--1575",
year = "2004"
}
@Article{Clever74,
author = "R. Clever and F. Busse",
title = "Large wavelength convection rolls in low Prandt number
fluids",
journal = "J.\ Fluid Mech.",
volume = "65",
pages = "665",
year = "1974",
keywords = "fluid dynamics",
}
@Article{CB92,
author = "{Clever}, R. M. and {Busse}, F. H.",
title = "Three-dimensional convection in a horizontal layer subjected to constant shear",
journal = "J.\ Fluid Mech.",
volume = "234",
number = "",
pages = "511--527",
year = "1992",
}
@InBook{BC96,
author = {F.~H {Busse} and R.~M. {Clever}},
title = {Waves and Nonlinear Processes in Hydrodynamics},
publisher = {Springer},
year = {1996},
chapter = {Bifurcation sequences in problems of thermal convection and of
plane {Couette} flow},
pages = {209-226}
}
% editor= {J. Grue and B. Gjevik and J. E. Weber},
@InProceedings{BodEck11,
author = "E. Bodenschatz and M. Eckert",
title = "Prandtl and the {G\"ottingen School}",
booktitle = "Highlights in the History of Turbulence",
editor= {P. A. Davidson and Y. Kaneda and
H. K. Moffatt and K. R. Sreenivasan},
publisher = {Springer},
addr = "Berlin",
year = {2011},
}
@MISC{Bridges_priv,
author = {T. J. Bridges},
note = {private communication}
}
@ARTICLE{bronski2005,
author = {J. C. Bronski and T. N. Gambill},
title = {Uncertainty estimates and {$L_2$} bounds for the {Kuramoto-Sivashinsky}
equation},
journal = {Nonlinearity},
year = {2006},
volume = {19},
pages = {2023--2039},
note = {{\tt arXiv:math/0508481}}
}
@InProceedings{BC96a,
title = {Bifurcation sequences in fluid flows and coherent structures in
the turbulent state},
author = {Busse, F. H. and Clever, R. M.},
year = {1996},
booktitle = {Advances in Turbulence VI},
editor = {Gavrilakis, S. and Machiels, L. and Monkewitz, P. A.},
series = {Proceedings of the Sixth European Turbulence Conference},
pages = {309-312}
}
@ARTICLE{ CB97,
author = {Clever, R. M. and Busse, F. H.},
title = "{Tertiary and quaternary solutions for plane {Couette} flow}",
journal = {J.\ Fluid Mech.},
year = 1997,
volume = 344,
pages = {137-153},
abstract = {The manifold of Nagata steady solutions is
explored in the parameter space of the problem and their
instabilities are investigated. These instabilities usually
lead to time-periodic solutions whose properties do not
differ much from those of the steady solutions except that
the amplitude varies in time. In some cases travelling wave
solutions which are asymmetric with respect to the midplane
of the layer are found as quaternary states of flow.
Similarities with longitudinal vortices recently observed in
experiments are discussed.}
}
@book{grue_waves_1996,
author = {Grue, J. and Gjevik, B. and Weber, J. E.},
title = {Waves and Nonlinear Processes in Hydrodynamics},
publisher = {Springer},
addr = "Berlin",
year = {1996},
}
% pages = {424}
@Book{Canuto88,
author = "C. Canuto and M. Y. Hussaini and A. Quarteroni and T. A. Zang",
title = "Spectral Methods in Fluid Dynamics",
year = "1988",
publisher = "Springer",
addr = "Berlin",
}
@Book{Canuto2006,
author = "C. Canuto and M. Y. Hussaini and A. Quarteroni and T. A. Zang",
title = "Spectral Methods: Fundamentals in Single Domains",
year = "2006",
publisher = "Springer",
addr = "Berlin",
}
@Book{Canuto2007,
author = "C. Canuto and M. Y. Hussaini and A. Quarteroni and T. A. Zang",
title = "Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics",
year = "2007",
publisher = "Springer",
addr = "Berlin",
}
@Article{C02,
author = "S. J. Chapman",
title = "Subcritical transition in channel flows",
journal = "J.\ Fluid Mech.",
year = "2002",
volume = "451",
pages = "35--97"
}
@article{cherh96,
title = {Existence and stability of finite-amplitude states
in plane {Couette} flow},
volume = {36},
issn = {0926-5112},
journal = {Fluid Mechanics and its Applications},
author = {A. Cherhabili},
year = {1996},
pages = {317--320}
}
@article{cherh00,
title = {Nonlinear equilibrium states in plane {Couette} flow:
Continuation from {P}oiseuille flow to {Couette} flow},
volume = {74},
issn = {0179-9614},
journal = {Notes in Numerical Fluid Mechanics},
author = {A. Cherhabili},
year = {2000},
pages = {69--80}
}
@article{cherh95,
title = {Spatially localized two-dimensional finite-amplitude states
in plane {Couette} flow},
volume = {14},
issn = {0997-7546},
number = {6},
journal = {Eur. J. Mech. B/Fluids},
author = {A. Cherhabili},
year = {1995},
pages = {677--696}
}
@ARTICLE{ Cherh97,
author = {{Cherhabili}, A. and {Ehrenstein}, U.},
title = "{Finite-amplitude equilibrium states in plane {Couette} flow}",
journal = {J.\ Fluid Mech.},
year = 1997,
volume = 342,
pages = {159-177},
adsurl = {http://adsabs.harvard.edu/abs/1997JFM...342..159C},
}
@article{CouTre78,
author = {P. Coullet and C. Tresser},
title = {It\'erations d'endomorphismes et groupe de renormalisation},
journal = {J. de Physique Colloque C},
volume = {39},
pages = {5-25},
year = {1978},
}
@article{CouTre78a,
author = {P. Coullet and C. Tresser},
title = {It\'erations d'endomorphismes et groupe de renormalisation},
journal = {C. R. Acad. Sci. Paris A},
volume = {287},
pages = {577-580},
year = {1978},
}
@ARTICLE{ Mehta05,
author = {{Mehta}, P.~G. and {Healey}, T.~J.},
title = "{On steady solutions of symmetry-preserving perturbations
of the two-dimensional {Couette} flow problem}",
journal = {Phys.\ Fluids},
year = 2005,
volume = 17,
pages = {4108-+},
abstract = {We obtain numerical solutions for extremely
small symmetry-preserving perturbations of the nonlinear
equations governing 2D {Couette} flow.},
}
@ARTICLE{ Rincon07,
author = {{Rincon}, F.},
title = "{On the existence of two-dimensional nonlinear steady states
in plane {Couette} flow}",
journal = {Phys.\ Fluids},
note = "{\tt arXiv:0706.1165}",
year = 2007,
volume = 19,
pages = {4105-+},
abstract ={It is not possible to obtain the nonlinear plane
Couette flow solutions reported by Cherhabili and
Ehrenstein [Eur. J.\ Mech.\ B/Fluids 14, 667 (1995)] using
their Poiseuille-Couette homotopy. We also demonstrate that
the steady solutions obtained by Mehta and Healey [Phys.
Fluids 17, 094108 (2005)] for small symmetry-preserving
perturbations are influenced by an artefact of the modified
system of equations used in their paper.},
}
@ARTICLE{choi94,
author = {H. Choi and P. Moin and J. Kim},
title = {Active turbulence control for drag reduction in wall-bounded flows},
journal = {J. Fluid Mech.},
volume = {262},
pages = {75},
year = {1994},
}
@Book{Chossat94,
author = "P. Chossat and G. Iooss",
title = "The {T}aylor-{Couette} Problem",
publisher = "Springer",
year = "1994"
}
@Article{Christiansen97,
author = "F. Christiansen and P. Cvitanovi\'{c} and V.
Putkaradze",
title = "Spatio-temporal chaos in terms of unstable recurrent
patterns",
journal = "Nonlinearity",
year = "1997",
volume = "10",
pages = "55--70",
note = "{\tt arXiv:chao-dyn/9606016}",
}
@article{ChrPol95,
author = {F. Christiansen and A. Politi},
title = {A generating partition for the standard map},
volume = {51},
journal = {Phys. Rev. E},
year = {1995},
pages = {3811},
note = "{\tt arXiv:chao-dyn/9411005}",
}
@Article{ChriPol96,
author = "F. Christiansen and A. Politi",
title = "Symbolic encoding in symplectic maps",
journal = "Nonlinearity",
year = "1996",
volume = "9",
pages = "1623--1640",
}
@Article{ChriPol97,
author = "F. Christiansen and A. Politi",
title = "Guidelines for the construction of a generating
partition in the standard map",
journal = "Physica D",
volume = "109",
pages = "32--41",
year = "1997"
}
@article{CCKFNRS12,
title = {Neural population dynamics during reaching},
volume = {487},
doi = {10.1038/nature11129},
number = {7405},
journal = {Nature},
author = {Churchland, M. M. and Cunningham, J. P. and Kaufman,
M. T. and Foster, J. D. and Nuyujukian, P. and Ryu, S. I.
and Shenoy, K. V.},
year = {2012},
pages = {51--56},
}
@PHDTHESIS{ Coff76,
author = "Coffee, T. P.",
title = "{The stability of plane {Couette} flow}",
school = "City Univ.~of New York, NY.",
year = 1976,
}
@ARTICLE{collis04,
author = {S. S. Collis and R. D. Joslin and A. Seifert and V. Theofilis},
title = {Issues in active flow control:
{Theory}, control, simulation, and experiment},
journal = {Prog. Aerosp. Sci.},
year = {2004},
volume = {40},
pages = {237},
}
@article{CroDav06,
author = {J. J. Crofts and R. L. Davidchack},
title = {Efficient detection of periodic orbits in chaotic systems by
stabilizing transformations},
year = {2006},
journal = {SIAM J. Sci. Comp.},
volume = {28},
pages = {1275-1288},
}
@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",
}
@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 Inst.",
year = "2012",
address = "Copenhagen",
note = "{\tt ChaosBook.org}"
}
@MISC{DasBuchMirror,
author = {P. Cvitanovi{\'c}},
year = "2008",
note = {chapter {\em ``World in a mirror''}, in Ref.~\cite{DasBuch}}
}
@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."
}
@ARTICLE{pre88top,
author = {P. Cvitanovi\'{c} and G. H. Gunaratne and I. Procaccia},
title = {Topological and metric properties of {H\'enon}-type strange
attractors},
journal = {Phys. Rev. A},
volume = {38},
pages = {1503},
year = {1988}
}
@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{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{chfield,
author = "P. Cvitanovi{\'c}",
title = "Chaotic field theory: {A} sketch",
journal = "Physica A",
volume = "288",
pages = "61",
year = "2000",
note = "{\tt arXiv:nlin.CD/0001034}"
}
@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"
}
% sometimes also called {pexp, % PAGES = ``L237--L241'',
@Article{CE91,
author = "P. Cvitanovi{\'c} and B. Eckhardt",
title = "Periodic orbit expansions for classical smooth flows",
journal = "J. Phys. A",
volume = "24",
pages = "L237",
year = "1991"
}
@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 arXiv:chao-dyn/9303016}"
}
@ARTICLE{CE89,
AUTHOR = "Cvitanovi\'c, P. and Eckhardt, B.",
TITLE = "Periodic orbit quantization of chaotic systems",
JOURNAL = "Phys. Rev. Lett.",
volume = "63",
PAGES = "823--826",
year = "1989",
}
@Article{PC88,
author = "P. Cvitanovi{\'c}",
title = "Invariant measurement of strange sets in terms of cycles",
journal = "Phys. Rev. Lett.",
volume = "61",
pages = "2729",
year = "1988"
}
% sometimes called ARTICLE{inv,
@INCOLLECTION{CERRS,
author = {B. Eckhardt and G. Russberg and P. Cvitanovi\'c and P. E.
Rosenqvist and P. Scherer},
title = {Pinball scattering},
pages = {483},
booktitle = {Quantum Chaos: Between Order and Disorder},
year = {1995},
editor = {G. Casati and B. Chirikov},
publisher = {Cambridge Univ. Press},
address = {Cambridge},
}
@TechReport{ Halc04,
author = "Halcrow, J.",
title = "{Implementation of a {Pressure Poisson Equation}
method for plane {Couette} flow}",
institution = "School of Physics, Georgia Inst. of Technology",
year = "2004",
note = "{\tt ChaosBook.org/projects}",
}
@Article{GHCW07,
author = "J. F. Gibson and J. Halcrow and P. Cvitanovi{\'c}",
title = "Visualizing the geometry of
state space in plane {Couette} flow",
journal ="J.\ Fluid Mech.",
year = "2008",
volume = "611",
pages = "107--130",
note = "{\tt arXiv:0705.3957}"
}
@Article{HGC08,
author = "J. F. Gibson and J. Halcrow and P. Cvitanovi{\'c}",
title = "Equilibrium and traveling-wave solutions of plane {Couette} flow",
journal ="J.\ Fluid Mech.",
year = "2009",
volume = "638",
pages = "1--24",
note = "{\tt arXiv:0808.3375}"
}
@Article{GHCV08,
author = "J. Halcrow and J. F. Gibson
and P. Cvitanovi{\'c} and D. Viswanath",
title = "Heteroclinic connections in plane {Couette} flow",
journal = "J.\ Fluid Mech.",
year = "2009",
volume = "621",
pages = "365--376",
note = "{\tt arXiv:0808.1865}"
}
@Article{CviGib10,
author = "Cvitanovi{\'c}, P. and Gibson, J. F.",
title = "Geometry of turbulence in wall-bounded shear flows:
{Periodic} orbits",
journal = "Phys. Scr. T",
volume = "142",
pages = "014007",
year = "2010",
}
% note = "in
% S. I. Abarzhi and K. R. Sreenivasan, eds.,
% {\em Turbulent Mixing and Beyond, Proc. 2nd Int. Conf.
% ICTP, Trieste}",
@Unpublished{ GHC08a,
author = "Gibson, J. F. and Cvitanovi{\'c}, P.",
title = "Periodic orbits of plane {Couette} flow",
year = "2010",
note = "In preparation"
}
@Article{SGB10,
author = "T. M. Schneider and J. F. Gibson and J. Burke",
title = "Snakes and ladders:
{L}ocalized solutions of plane {Couette} flow",
journal = "Phys. Rev. Lett.",
volume = "104",
pages = "104501",
year = "2010",
}
@inproceedings{etc12,
author = {Cvitanovi{\'c}, P. and Gibson, J. F.},
title = {Geometry of state space in plane {Couette} flow},
booktitle = {Advances in Turbulence XII},
pages = {41-43},
year = 2009,
editor = {A. V. Johansson and B. Eckhardt},
series = {Proc.\ 12th EUROMECH Eur.\ Turb.\ Conf.},
publisher = "Springer",
address = "Berlin",
}
@PhdThesis{ HalcrowThesis,
author = "Halcrow, J.",
title = "Geometry of turbulence: An exploration of the
state-space of plane {Couette} flow",
school = "School of Physics, Georgia Inst.\ of Technology",
address = "Atlanta",
year = "2008",
note = "{\tt ChaosBook.org/projects/theses.html}"
}
@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 arXiv:nlin.CD/0308006}"
}
@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 arXiv: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 arXiv:nlin.CD/0508026}"
}
% formerly lanCvit06
@Article{lanCvit07,
author = "Y. Lan and P. Cvitanovi{\'c}",
title = "Unstable recurrent patterns in {Kuramoto-Sivashinsky} dynamics",
journal = "Phys. Rev. E",
volume = "78",
pages = "026208",
year = "2008",
note = "{\tt arXiv.org:0804.2474}",
}
@unpublished{Cvi07,
author = {P. Cvitanovi\'{c}},
title = {Continuous symmetry reduced trace formulas},
year = {2007},
note = "\\ \wwwcb{/$\sim$predrag/papers/trace.pdf}"
}
@ARTICLE{LipCvi08,
author = {D. Lippolis and P. Cvitanovi\'c},
title = {How well can one resolve
the state space of a chaotic map?},
journal = {Phys. Rev. Lett.},
volume = {104},
pages = {014101},
year = {2010},
note = "{\tt arXiv:0902.4269}",
}
@unpublished{LipCvi07,
author = {D. Lippolis and P. Cvitanovi\'c},
title = {Optimal resolution of the state space of a chaotic flow
in presence of noise},
note = {In preparation, 2011}
}
@inproceedings{CviLip12,
author = {P. Cvitanovi\'c and D. Lippolis},
title = {Knowing when to stop: {How} noise frees us from determinism},
booktitle="Let's Face Chaos through Nonlinear Dynamics",
editor="M. Robnik and V. G. Romanovski",
year="2012",
publisher = {Am. Inst. of Phys.},
address = {Melville, New York},
pages="1-31",
}
@inproceedings{CviLip12,
author = {P. Cvitanovi\'c and D. Lippolis},
title = {Knowing when to stop: {How} noise frees us from determinism},
booktitle="Let's Face Chaos through Nonlinear Dynamics",
editor="M. Robnik and V. G. Romanovski",
pages = {82--126},
year="2012",
publisher = {Am. Inst. of Phys.},
address = {Melville, New York},
doi = {10.1063/1.4745574},
note = {{\tt arXiv:1206.5506}},
}
@Article{SCD07,
author = "Cvitanovi{\'c}, P. and Davidchack, R. L. and Siminos, E.",
title = "On the state space geometry of the {Kuramoto-Sivashinsky} flow
in a periodic domain",
journal = "SIAM J. Appl. Dyn. Syst.",
volume = "9",
pages = "1--33",
year = "2009",
note = "{\tt arXiv:0709.2944}"
}
@Article{SiCvi10,
author = "Siminos, E. and Cvitanovi{\'c}, P.",
title = "Continuous symmetry reduction
and return maps for high-dimensional flows",
journal = "Physica D",
year = "2011",
volume = "240",
pages = "187--198",
}
@Article{FrCv11,
author = "Froehlich, S. and Cvitanovi{\'c}, P.",
title = "Reduction of continuous symmetries of chaotic flows
by the method of slices",
journal = "Comm. Nonlinear Sci. and Numerical Simulation",
volume = "17",
pages = "2074-2084",
year = "2011",
note = "{\tt arXiv:1101.3037}",
}
@Unpublished{ SCD09b,
author = "Siminos, E. and Cvitanovi{\'c}, P. and Davidchack, R. L.",
title = "Recurrent spatio-temporal structures of translationally
invariant {Kuramoto-Sivashinsky} flow",
year = "2011",
note = "In preparation",
}
@PhdThesis{ SiminosThesis,
author = "Siminos, E.",
title = "Recurrent Spatio-temporal Structures in Presence
of Continuous Symmetries",
school = "School of Physics, Georgia Inst. of Technology",
year = "2009",
address = "Atlanta",
note = "\\ \wwwcb{/projects/theses.html}"
}
@Unpublished{atlas12,
author = "Cvitanovi\'c, P. and Borrero-Echeverry, D. and Carroll, K. and
Robbins, B. and Siminos, E.",
title = {Cartography of high-dimensional flows: {A} visual guide to sections and slices},
year = {2012},
note = "{\tt arXiv:1209.4915}; Chaos J., to appear."
}
@Book{cvt89b,
author = "P. Cvitanovi{\'c}",
title = "Universality in Chaos",
publisher = "Adam Hilger",
address = "Bristol",
year = "1989"
}
@Article{lanmaw03,
author = "Y. Lan and N. Garnier and P. Cvitanovi{\'c}",
title = "Stationary modulated-amplitude waves in the 1{D} complex {Ginzburg-Landau} equation",
journal = "Physica D",
volume = "188",
pages = "193--212",
year = "2004"
}
@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{DauBer12,
title = {Subcritical transition to turbulence:
{What} we can learn from the physics of glasses},
author = {Dauchot, O. and Bertin, E.},
journal = {Phys. Rev. E},
volume = {86},
pages = {036312},
numpages = {10},
year = {2012},
doi = {10.1103/PhysRevE.86.036312},
}
@Article{DV2000,
author = "O. Dauchot and N. Vioujard",
title = "Phase space analysis of a dynamical model
for the subcritical transition to turbulence in plane {Couette} flow",
journal = "European Physical J.\ B",
year = "2000",
volume = "14",
pages = "377--381"
}
@Article{DHB92,
author = "F. Daviaud and J. Hegseth and P. Berge",
title = "Subcritical transition to turbulence in plane {Couette} flow",
journal = "Phys.\ Rev.\ Lett.",
volume = "69",
year = "1992",
pages = "2511--2514"
}
@Article{DLBconv92,
author = "F. Daviaud and J. Lega and P. Berg{\'e} and P. Coullet and M. Dubois",
title = "Spatio-temporal intermittency in a 1{D} convective pattern:
{T}heoretical 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"
}
@InProceedings{ HDB92,
author = "J. Hegseth and F. Daviaud and P. Berge",
title = "Intermittent turbulence in plane and circular {Couette} flow",
booktitle = "Ordered and Turbulent Patterns in Taylor-{Couette} Flow",
editor = "C. D. Andereck and F. Hayot",
publisher = "Plenum Press",
address = "New York",
year = "1992",
pages = "159--166"
}
@Article{BDMD98,
author = "S. Bottin and F. Daviaud and P. Manneville and O. Dauchot",
title = "Discontinuous transition to spatiotemporal intermittency in
plane {Couette} flow",
journal = "Europhys.\ Lett.",
volume = "43",
year = "1998",
pages = "171--176"
}
@ARTICLE{ Bottin98,
author = {{Bottin}, S. and {Dauchot}, O. and {Daviaud}, F. and {Manneville}, P.
},
title = "{Experimental evidence of streamwise vortices
as finite amplitude solutions in transitional
plane {Couette} flow}",
journal = {Phys.\ Fluids},
year = 1998,
volume = 10,
pages = {2597-2607},
doi = {10.1063/1.869773},
adsurl = {http://adsabs.harvard.edu/abs/1998PhFl...10.2597B},
}
@Article{DD95,
author = "O. Dauchot and F. Daviaud",
title = "Finite-amplitude perturbation and spots growth mechanism in plane {Couette} flow",
journal = "Phys.\ Fluids",
volume = "7",
year = "1995",
pages = "335--343"
}
@Article{DDM98,
author = "O. Dauchot and F. Daviaud and P. Manneville",
title = "Experimental evidence of streamwise vortices as finite amplitude solutions in transitional plane {Couette} flow",
journal = "Phys.\ Fluids",
volume = "10",
year = "1998",
pages = "2597--2607"
}
@Book{DS83,
author = "{J.E. Dennis, Jr.,} and R. B. Schnabel",
title = "Numerical Methods for Unconstrained Optimization and Nonlinear Equations",
publisher = "Prentice-Hall",
year = "1983",
address = "Englewood Cliffs, New Jersey"
}
@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 {Ginzburg-Landau} 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 Ginzburg-Landau} equation",
journal = "Phys.\ Rev.\ Lett.",
volume = "59",
number = "26",
pages = "2911--2914",
year = "1987"
}
@UNPUBLISHED{Dragt11,
author = {A. J. Dragt},
title = {Lie Methods for Nonlinear Dynamics with Applications to Accelerator
Physics},
note = {www.physics.umd.edu/dsat/dsatliemethods.html},
year = {2011},
institution = {Physics Department, U. of Maryland}
}
@TECHREPORT{Dragt11bad,
author = {Dragt, A. J.},
title = {Lie methods for nonlinear dynamics with applications to accelerator
physics},
institution = {{Physics Department, U. of Maryland}},
year = {2011},
note = {{www.physics.umd.edu/dsat/dsatliemethods.html}}
}
@ARTICLE{Dragt05,
author = {Dragt, A. J.},
title = {The symplectic group and classical mechanics},
journal = {Ann. New York Acad. Sci.},
year = {2005},
volume = {1045},
pages = {291--307}
}
@UNPUBLISHED{DraHab08,
author = {Dragt, A. J. and Habib, S.},
title = {How {Wigner} functions transform under symplectic maps},
note = {{\tt arXiv:quant-ph/9806056}},
year = {1998}
}
%%%%%%%%%%%%%% plane Couette flow stability:
% sometimes called book{Reid81,
@Book{DR81,
author = "P. Drazin and W. H. Reid",
title = "Hydrodynamic Stability",
publisher = "Cambridge Univ.\ Press",
address = "Cambridge",
year = "1981"
}
@Book{DR04,
author = "P. Drazin and W. H. Reid",
title = "Hydrodynamic Stability",
publisher = "Cambridge Univ.\ Press",
address = "Cambridge",
edition = "Second",
year = "2004"
}
@Book{Drazin02,
author= {P.G. Drazin},
title = {Introduction to Hydrodynamic Stability},
publisher = {Cambridge Univ.\ Press},
year = {2002}
}
@Article{Romanov73,
author = "V. A. Romanov",
title = "Stability of plane-parallel {Couette} flow",
journal = "Functional Anal.\ Applics.",
year = "1973",
volume = "7",
pages = "137--146"
}
@Article{EGP70,
author = "T. Ellingsen and B. Gjevik and E. Palm",
title = "On the non-linear stability of plane {Couette} flow",
journal = "J.\ Fluid Mech.",
year = "1970",
volume = "40",
pages = "97--112"
}
@Article{MS51,
author = "D. Meksyn and J. T. Stuart",
title = "Stability of viscous motion between parallel planes for finite disturbances",
journal = "Proc.\ Roy.\ Soc. A",
pages = "517",
year = "1951",
volume = "208"
}
@Article{Cra72,
author = "A. D. D. Craik",
title = "Eigenvalue bounds in linear inviscid stability theory",
journal = "J.\ Fluid Mech.",
pages = "657--670",
year = "1972",
volume = "53",
abstract = "New eigenvalue bounds are derived for the linear
stability of inviscid parallel flows, both for homogeneous and for
stratified fluids. The usefulness of these bounds, as compared with
that of previous results, is assessed for several examples. For
homogeneous fluids the new upper bounds for the imaginary part ci of
the complex phase velocity are sometimes better than previous
criteria. For both homogeneous and stratified flows, the new upper
bounds for the wave-number ? of neutrally stable disturbances
improve on previous results, giving values within 10% of the known
exact solution in several cases."
}
@Article{Cra89,
author = "A. D. D. Craik",
title = "The stability of unbounded two- and three-dimensional flows subject to body forces: some exact solutions",
journal = "J.\ Fluid Mech.",
pages = "275--292",
year = "1989",
volume = "198",
abstract = "A formulation, previously employed to find exact Navier-Stokes solutions for planar disturbances in two- and three-dimensional flows with spatially uniform rates of strain, is here adapted to incorporate the contribution of various types of body force. In the absence of body forces, it is known that unbounded flows with constant vorticity and elliptical streamlines are unstable to certain planar disturbances, which are amplified by a Floquet mechanism. The influence of a Coriolis force upon this instability mechanism is here described in detail, as an illustration of the general formulation. The results are likely to be of geophysical interest and may also have relevance to the breakdown of closed-eddy structures in turbulence. The final section of the paper reviews other systems for which analogous exact solutions may be obtained."
}
@Unpublished{ DKY05,
author = "Hua-Shu Dou and Boo Cheong Kho and Khoon Seng Yeo",
title = "Flow transition in plane {Couette} flow",
year = "2005",
note = "{\tt arXiv:nlin.CD/0501048}",
abstract = "The energy gradient theory has been proposed
with the aim of better understanding the mechanism of
flow transition from laminar flow to turbulent flow. In
this theory, it is suggested that the transition to
turbulence depends on the relative magnitudes of the
energy gradient amplifying the disturbance and the
viscous friction damping that disturbance. For a given
flow geometry and fluid properties, when the maximum of K
(the ratio of the energy gradient in the transverse
direction to that in the streamwise direction) in the
flow field is larger than a certain critical value, it is
expected that instability would occur for some initial
disturbances. In this paper, using the energy analysis,
the equation for calculating K for plane Couette flow is
derived. It is demonstrated that the critical value of K
at subcritical transition is about 370 for plane Couette
flow. This value is about the same as for plane
Poiseuille flow and pipe Poiseuille flow (385-389).
Therefore, it is concluded that the critical value of K
at subcritical transition is about 370-389 for wall
bounded parallel shear flows which include both pressure
and shear driven flows."
}
@Article{Da73,
author = "A. Davey",
title = "On the stability of plane {Couette} flow to infinitesimal disturbances",
journal = "J.\ Fluid Mech.",
pages = "369--380",
year = "1973",
volume = "57",
abstract = "conjectured for many years that plane Couette
flow is stable to infinitesimal disturbances although
this has never been proved. In this paper we use a,
combination of asymptotic analysis and numerical
computation to examine the associated Orr-Sommerfeld
differential problem in a systematic manner. We obtain
new evidence that the conjecture is, in all probability,
correct."
}
@MISC{cresson-2006,
author = {J. Cresson and C. Guillet},
title = {Hyperboliity versus partial-hyperbolicity
and the transversality-torsion phenomenon},
note = "{\tt arXiv:math/0601701}",
year = {2006}
}
@Article{ET02,
author = "M. Embree and L. Trefethen",
title = "Generalizing eigenvalue theorems to pseudospectra theorems",
journal = {SIAM J.\ Sci.\ Comp.},
pages = "83--590",
year = "2002",
volume = "23"
}
@Article{SB95,
author = "R. Sureshkumar and A. Beris",
title = "Linear stability analysis of viscoelastic {P}oiseuille flow using an {A}rnoldi-based orthogonalization algorithm",
journal = "J.\ Non-Newton.\ Fluid Mech.",
pages = "151--182",
year = "1995",
volume = "56"
}
@Article{Tre97,
author = "L. Trefethen",
title = "Pseudospectra of linear operators",
journal = "SIAM Rev.",
pages = "383--406",
year = "1997",
volume = "39"
}
@TechReport{ Wri06,
author = "T. Write",
title = "Eigtool, a graphical tool for nonsymmetric eigenproblems",
institution = "Oxford Univ.",
year = "2006",
note = "{\tt web.comlab.ox.ac.uk/projects/pseudospectra/eigtool}"
}
@Article{RP67,
author = "W. C. Reynolds and M. C. Potter",
title = "Finite-amplitude instability of parallel shear flows",
journal = "J.\ Fluid Mech.",
pages = "465",
year = "1967",
volume = "27"
}
@Article{SW60,
author = "J. T. Stuart and J. Watson",
title = "On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows",
journal = "J.\ Fluid Mech.",
pages = "353--370, 371--389",
year = "1960",
volume = "9"
}
@Article{Sq33,
author = "H. B. Squire",
title = "On the stability of three-dimensional distribution of viscous fluid between parallel walls",
journal = "Proc.\ Roy.\ Soc.\ A",
pages = "621",
year = "1933",
volume = "142"
}
@Article{Dea63,
author = "J. W. Deardorff",
title = "On the stability of viscous plane {Couette} flow",
journal = "J.\ Fluid Mech.",
pages = "623--630",
year = "1963",
volume = "15",
abstract = "The problem of the stability of plane Couette
flow to infinitesimal disturbances is carried numerically
to larger Reynolds numbers than heretofore. The flow is
definitely stable up to R = 1430"
}
@Article{GuHu98,
author = "L. H. Gustavsson and L. S. Hultgren",
title = "A resonance mechanism in plane {Couette} flow",
journal = "J.\ Fluid Mech.",
pages = "149--159",
year = "1980",
volume = "98",
annote = "resonant response leading to algebraic growth for small times.
It occurs when the eigenvalues of the Orr-Sommerfeld equation coincide
with the eigenvalues of the homogeneous operator for the streamwise
velocity component. The resonance has been investigated numerically for
plane Couette flow. The aspect ratio of the most rapidly growing wave increases
with the Reynolds number, with its spanwise wavelength approaching
a constant value of about 3 channel heights."
}
@Article{Koch89,
author = "W. Koch",
title = "Direct resonances in {O}rr-{S}ommerfeld problems",
journal = "Acta Mechanica",
pages = "11",
year = "1986",
volume = "59"
}
@Article{Rem03,
author = "D. Rempfer",
title = "Low-dimensional modeling and numerical simulation of transition in simple shear flows",
journal = "Review of Fluid Mechanics",
pages = "229",
year = "2003",
volume = "35"
}
%%%%%%%%%%%%%% plane Couette flow stability finished %%%%%%%
@Article{EM99,
author = "B. Eckhardt and A. Mersmann",
title = "Transition to turbulence in a shear flow",
journal = "Phys.\ Rev.\ E",
year = "1999",
volume = "60",
pages = "509--517"
}
@Article{ESHW07,
author = "B. Eckhardt and T. M. Schneider and B. Hof and J. Westerweel",
title = "Turbulence transition in pipe flow",
journal = "Ann.\ Rev.\ Fluid Mech.",
year = "2007",
volume = "29",
pages = "447--468"
}
@Article{SE00,
author = "A. Schmiegel and B. Eckhardt",
title = "Persistent turbulence in annealed plane {Couette} flow",
journal = "Europhys.\ Lett.",
year = "2000",
volume = "51",
pages = "395--400"
}
@Article{SE97,
author = "A. Schmiegel and B. Eckhardt",
title = "Fractal stability border in plane {Couette} flow",
journal = "Phys.\ Rev.\ Lett.",
year = "1997",
volume = "79",
pages = "5250",
abstract = "plane {Couette} flow: For Re > 200,
chaotic transients appear with lifetimes longer than the viscous one.
Depending on the type of the perturbation, isolated initial conditions
with infinite lifetime appear for Reynolds numbers
larger than about 270--305. In this third regime, the lifetime
as a function of Reynolds number and amplitude is fractal.
These results suggest that in the transition region
the turbulent dynamics is characterized by a chaotic repeller
rather than an attractor.",
note = "{\tt arXiv:chao-dyn/9704018}"
}
@phdthesis{ Schmi99,
author = "A. Schmiegel",
title = "Transition to turbulence in linearly stable shear flows",
year = "1999",
school = {Philipps-Universit{\"a}t Marburg},
note = "Available on {\tt archiv.ub.uni-marburg.de/diss/z2000/0062}"
}
@Article{SGLDE08,
author = "T. M. Schneider and J. F. Gibson and M. Lagha
and F. De Lillo and B. Eckhardt",
title = "Laminar-turbulent boundary in plane {Couette} flow",
journal = "Phys.\ Rev.\ E.",
year = "2008",
volume = "78",
pages = "037301",
note = "{\tt arXiv:0805.1015}"
}
@inproceedings{KraOsi00,
author = {B. Krauskopf and H. M. Osinga},
title = {Investigating torus bifurcations in
the forced {Van der Pol} oscillator},
booktitle = {Numerical Methods for
Bifurcation Problems and Large-Scale Dynamical Systems},
pages = {199--208},
year = 2000,
editor = {E. Doedel and L. Tuckerman},
series = {The IMA Volumes in Mathematics and its Applications, vol. 119},
publisher = "Springer",
address = "Berlin",
}
@Article{EKO04,
author = {J. P. England and B. Krauskopf and H. M. Osinga},
title = {Computing one-dimensional stable and
unstable sets of planar maps without the inverse},
journal = {SIAM J. Appl. Dyn. Syst.},
year = {2004},
volume = {3},
pages = {161--190},
}
@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 = "P. Collet and J.-P. Eckmann",
title = "Iterated Maps on the Interval as Dynamical Systems",
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"
}
@Article{ETFS,
author = "W. S. Edwards and L. S. Tuckerman and R. A. Friesner and D. C. Sorensen",
title = "Krylov methods for the incompressible {Navier-Stokes} equations",
journal = "J.\ Comput.\ Physics",
year = "1994",
volume = "110",
pages = "82--102"
}
@ARTICLE{feigen78,
author = {M. J. Feigenbaum},
title = {Quantitative universality for a class of nonlinear
transformations},
journal = {J. Stat. Phys.},
volume = {19},
pages = {25--52},
year = {1978}
}
@ARTICLE{feigen79,
author = {M. J. Feigenbaum},
title = {The universal metric properties of nonlinear transformations},
journal = {J. Stat. Phys.},
volume = {21},
pages = {669--706},
year = {1979},
note = {Reprinted in \refref{cvt89b}.}
}
@article{feigen79scal,
author = {M. J. Feigenbaum},
title = {The onset spectrum of turbulence},
journal = {Phys. Lett. A},
year = {1979},
volume = {74},
pages = {375--378}
}
@ARTICLE{FeHa82,
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and H. Wedin and R. R. Kerswell and F. Waleffe",
title = "Experimental observation of nonlinear traveling waves
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journal = "Science",
volume = "305",
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.",
}
%% note = "{\tt www.sciencemag.org/cgi/reprint/305/5690/1594.pdf}"
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pages = "015105"
}
% pages = "015105" This cannot be right, yet it matches what I find on
% the web. jfg 2007-05-02
% sometimes called article{JKSN05,
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YEAR="2007",
VOLUME="367",
PAGES="715"}
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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."
}
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author = "Kerswell, R. R. and Tutty, O.R.",
title = "Recurrence of travelling waves in transitional pipe flow",
journal = "J.\ Fluid Mech.",
volume = "584",
pages = "69--102",
year = "2007",
abstract = " We find that travelling waves with low wall shear
stresses (lower branch solutions) are on a surface which separates initial
conditions which uneventfully relaminarise and those which lead to a
turbulent evolution. Evidence for recurrent travelling wave visits is
found in both 5D and 10D long periodic pipes but only for those travelling
waves with low-to-intermediate wall shear stress and for less than about
10\% of the time in turbulent flow. Dynamical structures such as periodic
orbits need to be isolated and included in any such expansion.",
note = "{\tt arXiv:physics/0611009}",
}
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author = {A.P. Willis and J. Peixinho and R.R. Kerswell and T. Mullin},
title = {Experimental and theoretical progress in pipe flow
transition},
journal = {Phil. Trans. R. Soc. A},
year = {2008},
volume = {366},
pages = {2671-2684},
}
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author = {Duguet, Y. and Willis, A. P. and Kerswell, R. R.},
title = {Transition in pipe flow:
the saddle structure on the boundary of turbulence},
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year = {2008},
volume = {613},
pages = {255-274},
note = "{\tt arXiv:0711.2175}",
}
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author = "Chandler, G. J. and Kerswell, R. R.",
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year = 2012,
note = "{\tt arXiv:1207.4682}",
}
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author = "Kline, S. J. and {R}eynolds, W. C. and Schraub, F. A. and Runstadler, P. W.",
title = "The structure of turbulent boundary layers",
journal = "J. Fluid Mech.",
year = "1967",
volume = "30",
pages = "741--773"
}
% also cited as Article{Kim87,
@Article{KMM87,
author = "J. Kim and P. Moin and R. Moser",
title = "Turbulence statistics in fully developed channel flow at low {Reynolds} number",
journal = "J. Fluid Mech.",
year = "1987",
volume = "177",
pages = "133--166"
}
@ARTICLE{kim99,
author = "K. C. Kim and R. J. Adrian",
year = "1999",
title = "Very large-scale motion in the outer layer",
journal = "Phys.\ Fluids",
volume = "11",
pages = "417-422"
}
% tau correction paper
@InProceedings{ Kleiser80,
author = "L. Kleiser and U. Schumann",
title = "Treatment of incompressibility and boundary conditions
in 3-{D} numerical spectral simulations of plane channel flows",
booktitle = "Proc. 3rd GAMM Conf. Numerical Methods in Fluid Mechanics",
organization = "GAMM",
year = "1980",
editor = "E. Hirschel",
address = "Viewweg, Braunschweig",
pages = "165--173"
}
@Article{KZ,
author = "L. Kleiser and T. A. Zang",
title = "Numerical simulation of transition in wall-bounded shear flows",
journal = "Annu.\ Rev.\ Fluid Mech.",
year = "1991",
volume = "23",
pages = "495--537"
}
@ARTICLE{klewicki95,
author = {Klewicki, J. C. and Metzger, M. M. and Kelner, E. and E. M. Thurlow},
title = "{Viscous sublayer flow visualizations at $R_{\theta} \sim 1500000$}",
journal = {Phys. Fluids},
year = 1995,
volume = 7,
pages = {857},
}
@Article{KLH94,
author = "G. Kreiss and A. Lundbladh and D. S. Henningson",
title = "{Bounds for threshold amplitudes in subcritical shear flows}",
journal = "J.\ Fluid Mech.",
year = "1994",
volume = "270",
pages = "175--198"
}
@Unpublished{KreEck12,
author = "T. Kreilos and B. Eckhardt",
title = {Periodic orbits in plane {Couette} flow},
year = {2012},
note = "{\tt arXiv:1205.0347}; Chaos J., to appear."
}
@Unpublished{KrVeSchEck12,
author = {{Kreilos}, T. and {Veble}, G. and {Schneider}, T.~M. and
{Eckhardt}, B.},
title = "Edge states for the turbulence transition in the asymptotic
suction boundary layer",
note = "{\tt arXiv:1209.0593}",
year = 2012,
}
@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."
}
@article{KuKuKnMo04,
author = {A. Yu. Kuznetsova and A. P. Kuznetsov and C. Knudsen
and E. Mosekilde},
title = {Catastrophe theoretic classification of nonlinear oscillators},
journal = {Int. J. Bif. Chaos},
volume = {14},
year = {2004},
pages = {1241--1266},
doi = {10.1142/S0218127404009995},
}
@BOOK{lapack,
AUTHOR = {Anderson, E. and Bai, Z. and Bischof, C. and
Blackford, S. and Demmel, J. and Dongarra, J. and
Du Croz, J. and Greenbaum, A. and Hammarling, S. and
McKenney, A. and Sorensen, D.},
TITLE = {{LAPACK} Users' Guide},
EDITION = {Third},
PUBLISHER = {Society for Industrial and Applied Mathematics},
YEAR = {1999},
ADDRESS = {Philadelphia},
ISBN = {0-89871-447-8 (paperback)}
}
@PhdThesis{ LanThesis,
author = "Y. Lan",
title = "Dynamical systems approach to 1-$d$ spatiotemporal chaos
-- {A} cyclist's view",
school = "School of Physics, Georgia Inst.\ of Technology",
address = "Atlanta",
year = "2004",
note = "{\tt ChaosBook.org/projects/theses.html}"
}
@Book{LM94,
author = "A. Lasota and M. MacKey",
title = "Chaos, Fractals, and Noise; Stochastic Aspects of Dynamics",
year = "1994",
publisher = "Springer",
address = "Berlin",
}
@book{Lau05,
author = "Lautrup, B.",
title = "Physics of Continuous Matter:
Exotic and Everyday Phenomena in the Macroscopic World",
publisher = "{CRC Press}",
address = "{Boca Raton, FL}",
year = "2011"
}
@InCollection{ lauw86,
author = "H. A. Lauwerier",
year = "1986",
pages = "39",
editor = "V. Holden",
publisher = "Princeton University Press",
booktitle = "Chaos"
}
@ARTICLE{lee01,
author = {K. Lee and L. Cortelezzi and J. Kim and J. L. Speyer},
title = {Application of reduced order controller to turbulent flows for drag
reduction},
journal = {Phys. Fluids},
volume = {13},
pages = {1321},
year = {2001},
}
@Article{Li08,
author = "Li, Y. C.",
title = "Persistence of steady 3D {Euler} solutions
for 3D {Navier-Stokes} equations",
note = "{\tt arXiv:0811.0383}"
}
@Article{LY75,
author = "T. Li and J. A. Yorke",
title = "Period 3 implies chaos",
journal = "Amer.\ Math.\ Monthly",
volume = "82",
pages = "985",
year = "1975"
}
@Book{Lin55,
author = "Lin, C. C.",
title = "The Theory of Hydrodynamic Stability",
publisher = "Cambridge Univ.\ Press",
address = "Cambridge",
year = "1955"
}
@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 = ""
}
@Article{lixigraham06,
author = "W. Li and L. Xi and M.D. Graham",
title = "Nonlinear travelling waves as a framework for understanding turbulent drag reduction",
journal = "J.\ Fluid Mech",
volume = "565",
number = "",
pages = "353-362",
year = "2006",
abstract = ""
}
@Article{lixigraham07,
author = "W. Li and M.D. Graham",
title = "Polymer induced drag reduction in exact coherent structures of plane Poiseuille flow",
journal = "Phys.\ Fluids",
volume = "19",
number = "083101",
pages = "1-15",
year = "2007",
abstract = ""
}
@article{LoCaCoPeGo11,
author = {Lombardi, M. and Caulfield, C.P. and Cossu, C.
and Pesci, A.I. and Goldstein, R.E.},
title = {Growth and instability of a laminar plume in a strongly
stratified environment},
journal = "J.\ Fluid Mech.",
volume = {671},
pages = {184-206},
year = {2011},
doi = {10.1017/S0022112010005574},
}
@Article{LopezPhysD05,
author = "J. M. Lopez and F. Marques",
title = "Finite aspect ratio {Taylor-Couette} flow:
{Shil'nikov} dynamics of 2-tori",
journal = "Physica D",
volume = "211",
pages = "168--191",
year = "2005"
}
@article{LopMarq05,
title = "Finite aspect ratio {Taylor-Couette} flow:
{Shil'nikov} dynamics of 2-tori",
volume = {211},
journal = {Physica D},
author = {J. M. Lopez and F. Marques},
year = {2005},
pages = {168--191}
}
@ARTICLE{lumley67,
author = {{Bakewell}, P. and {Lumley}, J. L.},
title = "Viscous sublayer and adjacent wall region in turbulent pipe flow",
journal = {Phys. Fluids},
year = 1967,
volume = 10,
pages = {1880},
}
@ARTICLE{lumley98,
author = {Lumley, J. L. and Blossey, P.},
title = "{Control of turbulence}",
journal = {Ann. Rev. Fluid Mech.},
year = 1998,
volume = 30,
pages = {311},
}
@Article{Ma02,
author = {X. Ma and G.E. Karniadakis},
title = {A low-dimensional model for simulating
3-dimensional cylinder flow},
journal = {J. Fluid Mech.},
year = {2002},
volume = {458},
pages = {181--190}
}
@ARTICLE{May76,
author = {R. May},
title = {Simple mathematical models with very complicated dynamics},
journal = {Nature},
volume = {261},
pages = {459-467},
year = {1976}
}
@Article{MeseguerANM07,
author = "A. Meseguer and F. Mellibovsky",
title = "On a solenoidal {Fourier-Chebyshev} spectral method
for stability analysis of the {Hagen--Poiseuille} flow",
journal = "Appl.\ Num.\ Math.",
volume = "57",
pages = "920--938",
year = "2007"
}
@article{mellibovsky11,
title={{Takens--Bogdanov} bifurcation of travelling-wave solutions in
pipe flow},
author={Mellibovsky, F. and Eckhardt, B.},
journal={J. Fluid Mech.},
volume={670},
pages={96--129},
year={2011},
}
@article{mellibovsky12,
title={From travelling waves to mild chaos: {A} supercritical
bifurcation cascade in pipe flow},
author={Mellibovsky, F. and Eckhardt, B.},
journal={Arxiv preprint arXiv:1110.1222},
year={2011}
}
@article{Lumley_1989,
journal={Whither turbulence Turbulence at the crossroads}, publisher={Springer-Verlag},
author={}, editor={Lumley, J. L.}, year={}}
@incollection{Moffatt90,
author = {H. K. Moffatt},
title={Fixed points of turbulent dynamical systems and suppression of
nonlinearity, {Comment} 1},
booktitle = {Whither Turbulence! Turbulence at the Crossroads},
editor = {Lumley, J. L.},
publisher = "Springer",
address = "New York",
pages = {250--257},
year = {1990}
}
@book{Moon87,
author = {F. C. Moon},
title = {Chaotic Vibrations: An Introduction for Applied Scientists and
Engineers},
publisher = "Wiley",
address = "New York",
year = 1987 }
@book{Mees81,
author = "Mees, A. I.",
title = "Dynamics of Feedback Systems",
publisher = "Wiley",
address = "New York",
year = "1981",
}
@article{MFKM10,
author = {Morita, Y. and Fujiwara, N.
and Kobayashi, M. U. and Mizuguchi, T.},
title = {Scytale decodes chaos: {A} method for estimating
unstable symmetric solutions},
journal = {Chaos},
volume = {20},
pages = {013126},
year = {2010},
abstract = "A method for estimating a period of unstable periodic solutions
is suggested in continuous dissipative chaotic dynamical
systems. The measurement of a minimum distance between a
reference state and an image of transformation of it exhibits a
characteristic structure of the system, and the local minima of
the structure give candidates of period and state of
corresponding symmetric solutions. Appropriate periods and
initial states for the Newton method are chosen efficiently by
setting a threshold to the range of the minimum distance and
the period.",
}
% sometimes {lorenz1,
@Article{lorenz63,
author = "E. N. Lorenz",
title = "Deterministic nonperiodic flow",
journal = "J.\ Atmos.\ Sci.",
volume = "20",
pages = "130--141",
year = "1963"
}
@Book{sparrow,
author = {C. Sparrow},
title = {The {L}orenz Equations: Bifurcations, Chaos and Strange
Attractors},
publisher = {Springer},
address = {New York},
year = {1982}
}
@Book{froyland,
author = {Fr\o yland, J.},
title = {Chaos and coherence},
publisher = "Inst.\ of Phys.\ Publ.",
address = {Bristol},
year = {1992}
}
@Article{froyland_alfsen,
author = {Fr\o yland, J. and Alfsen, K. H.},
journal = {Phys.\ Rev.\ A},
volume = {29},
pages = 2928,
year = {1984}
}
@book{seydel,
author = {R\"{u}diger Seydel}, year = 1988,
title = {From equilibrium to chaos: Practical bifurcation and stability
analysis}, publisher = {Elsevier}, address = "New York"}
@book{abraham:shaw,
author = {R. H. Abraham and C. D. Shaw},
year = 1992,
title = {Dynamics - The geometry of behavior},
publisher = "Addison-Wesley", address = "Reading, MA"
}
@book{AbSh92,
author = {R. H. Abraham and C. D. Shaw},
year = 1992,
title = {Dynamics - The Geometry of Behavior},
publisher = "Addison-Wesley", address = "Reading, MA"
}
@BOOK{AbrMars78,
title = {Foundations of Mechanics},
publisher = {Benjamin-Cummings},
year = {1978},
author = {R. Abraham and J. E. Marsden},
address = {Reading, Mass.}
}
@book{jackson89,
author = {Jackson, E. Atlee}, year = 1989,
publisher = "Cambridge Univ.\ Press",
address = {Cambridge},
title = {Perspectives of nonlinear dynamics: Vol.\ 1 and 2}
}
@Article{NP,
author = "K. Lust and D. Roose",
title = "An adaptive {Newton-Picard} algorithm with subspace
iteration for computing periodic solutions",
journal = "SIAM J.\ Sci.\ Comput.",
volume = "19",
year = "1998",
ISSN = "1064-8275",
pages = "1188--1209",
doi = "http://dx.doi.org/10.1137/S1064827594277673",
}
@article{noack03,
author={B. R. Noack and K. Afanasiev and M. Morzynski and F. Thiele},
year={2003},
title={A hierarchy of low-dimensional models for
the transient and post-transient cylinder wake},
journal={J. Fluid Mech.},
volume={497},
pages={335-363},
}
@Misc{ RMChaosHist,
author = "R. Mainieri",
note = "Appendix {\em ``A brief history of chaos''}, in \refref{DasBuch}"
}
@Book{man90b,
author = "P. Manneville",
title = "Dissipative structures and weak turbulence",
publisher = "Academic Press",
address = "Boston",
year = "1990"
}
@Article{Mann04,
author = "P. Manneville",
title = "{Spots and turbulent domains in a model of
transitional plane {Couette} flow}",
journal = "Theor. Comp. Fluid Dyn.",
year = "2004",
volume = "18",
pages = "169--181",
doi = "10.1007/s00162-004-0142-4",
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"
}
@Article{Mann08,
author = "P. Manneville",
title = "{Understanding the sub-critical transition to turbulence in wall flows}",
journal = "Pramana",
year = "2008",
volume = "70",
pages = "1009--1021",
}
@ARTICLE{ LagMann07,
author = {{Lagha}, M. and {Manneville}, P.},
title = "{Modeling of plane {Couette} flow.\ I. Large scale flow around turbulent spots}",
journal = {Phys.\ Fluids},
year = 2007,
volume = 19,
pages = {4105},
doi = {10.1063/1.2768946},
adsurl = {http://adsabs.harvard.edu/abs/2007PhFl...19i4105L},
}
@ARTICLE{ Lagha07,
author = {{Lagha}, M.},
title = "{Modeling of plane {Couette} flow. II. On the
spreading of a turbulent spot}",
journal = {Phys.\ Fluids},
year = 2007,
volume = 19,
pages = {4108},
doi = {10.1063/1.2793143},
adsurl = {http://adsabs.harvard.edu/abs/2007PhFl...19j4108L},
}
@ARTICLE{ Lagha07b,
author = "M. Lagha and P. Manneville",
title = "Modeling transitional plane {Couette} flow",
journal = "European Physical Journal B",
year = "2007",
volume = "58",
pages = "433--447",
doi = "10.1140/epjb/e2007-00243-y",
}
@Article{lop05rel,
author = "V. L{\'o}pez and P. Boyland and M. T. Heath and R. D.
Moser",
title = "Relative periodic solutions of the {Complex
Ginzburg-Landau} equation",
journal = "SIAM J. Appl. Dyn. Syst.",
volume = "4",
pages = "1042--1075",
year = "2006",
abstract = "Define relative periodic orbits and use Fourier modes
to find them.",
note = "{\tt arXiv:nlin/0408018}",
doi = "10.1137/040618977"
}
@Article{LBHM05,
title = "Relative periodic solutions of the {Complex
Ginzburg-Landau} equation",
author = "V. L{\'o}pez and P. Boyland and M. T. Heath and R. D. Moser",
journal = "SIAM J.\ Appl.\ Dynam.\ Systems",
pages = "1042--1075",
volume = "4",
year = "2006",
abstract = "Define relative periodic orbits and use Fourier modes
to find them.",
note = "{\tt arXiv:nlin/0408018}",
doi = "10.1137/040618977"
}
@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 = "Phys.\ Rev.\ A",
year = "1973",
volume = "8",
number = "1",
pages = "423--437"
}
@Article{M86,
author = "S. A. Maslowe",
title = "Critical layers in shear flows",
journal = "Ann.\ Rev.\ Fluid Mech.",
year = "1986",
volume = "18",
pages = "405--432"
}
@Book{MH,
author = "K. R. Meyer and G. R. Hall",
title = "Introduction to Hamiltonian Dynamical Systems and the {N}-body Problem",
publisher = "Springer",
address = "New York",
year = "1992"
}
@Article{MFE04,
author = "J. Moehlis and H. Faisst and B. Eckhardt",
title = "A low-dimensional model for turbulent shear flows",
journal = "New Journal of Physics",
year = "2004",
volume = "6",
pages = "56"
}
@Article{MFE04b,
author = "J. Moehlis and H. Faisst and B. Eckhardt",
title = "Periodic orbits and chaotic sets in a low-dimensional model for shear flows",
journal = "SIAM J.\ Appl.\ Dyn.\ Sys.",
year = "2005",
volume = "4",
pages = "352--376"
}
@Article{MEF04,
author = "J. Moehlis and B. Eckhardt and H. Faisst",
title = "Fractal lifetimes in the transition to turbulence",
journal = "CHAOS",
year = "2004",
volume = "14",
pages = "S11"
}
% criticisnm of AHLS models for lack of streamwise variation
@inproceedings{Moffat90,
author = {H. K. Moffatt},
title = {Fixed points of turbulent dynamical systems
and suppression of nonlinearity},
booktitle = {Whither Turbulence? Turbulence at the Crossroads},
organization = {Springer Lect. Notes in Physics},
year = {1990},
editor = {J. L. Lumley},
publisher = "Springer",
address = "Berlin",
pages = {250--257}
}
@inproceedings{FeigKen83,
author = {M. J. Feigenbaum and R. D. Kenway},
title = {The onset of chaos},
booktitle = {Statistical and Particle Physics},
organization = {Proc. 26th Scottish Universities Summer School in Physics},
year = {1983},
editor = {K. C. Bowler and A. J. McKane},
publisher = "SUSSP Publ.",
address = "Edinburgh",
pages = {1--100}
}
@Misc{petsc-web-page,
Author = "S. Balay and K. Buschelman and W. D. Gropp
and D. Kaushik and M. G. Knepley
and L. C. McInnes and B. F. Smith and H. Zhang",
Title = "{PETSc} web page",
Note = "www.mcs.anl.gov/petsc",
Year = "2001"}
@TechReport{petsc-user-ref,
Author = "S. Balay and K. Buschelman and V. Eijkhout
and W. D. Gropp and D. Kaushik and M. G. Knepley
and L. C. McInnes and B. F. Smith and H. Zhang",
Title = "{PETS}c Users Manual",
Number = "ANL-95/11 - Revision 2.1.5",
Institution = "Argonne National Laboratory",
Year = "2004"}
@InProceedings{petsc-efficient,
Author = "S. Balay and W. D. Gropp
and L. C. McInnes and B. F. Smith",
Title = "Efficient management of parallelism in
object oriented numerical software libraries",
Booktitle = "Modern Software Tools in Scientific Computing",
Editor = "E. Arge and A. M. Bruaset and H. P. Langtangen",
Pages = "163--202",
Publisher = "Birkh{\"{a}}user Press",
Year = "1997"}
%also mcke
@article{McKean74,
author={McKean, H. P.},
title={Selberg's trace formula as applied to a compact {Riemann} surface},
volume={25}, journal={Comm. Pure Appl. Math.},
year={1972}, pages={225--246}}
@Article{MM,
author = "P. Moin and K. Mahesh",
title = "Direct numerical simulation: a tool in turbulence research",
journal = "Annu. Rev.\ Fluid Mech.",
year = "1998",
volume = "30",
pages = "539--78"
}
@Book{MonYag71,
author = "Monin, A.S. and Yaglom, A.M.",
title = "Statistical Fluid Mechanics",
publisher = "MIT Press",
address = "Cambridge",
year = "1971"
}
@article{MoMeCa84,
author = {P. J. Morrison and J. D. Meiss and J. R. Cary},
title = {Scattering of regularized-long-wave solitary waves},
volume = {11},
abstract = {The Lagrangian density for the regularized-long-wave equation
(also known as the BBM equation) is presented. Using the trial
function technique, ordinary differential equations that describe
the time dependence of the position of the peaks, amplitudes,
and widths for the collision of two solitary waves are obtained.
These equations are analyzed in the Born and ``equal-width''
approximations and compared with numerical results obtained
by direct integration utilizing the split-step fast Fourier-transform
method. The computations show that collisions are inelastic and
that production of solitary waves may occur.},
journal = {Physica D},
year = {1984},
pages = {324--336}
}
@Article{Morrison02,
author = "J. F. Morrison and W. Jiang and B. J. McKeon and A. J. Smits",
title = "Reynolds Number Dependence of Streamwise Velocity Spectra in Turbulent Pipe Flow",
journal = "Phys.\ Rev.\ Lett.",
volume = "88",
number = "",
pages = "214501",
year = "2002",
abstract = ""
}
@Article{Morrison04,
author = "J. F. Morrison and B. J. McKeon and W. Jiang and A. J. Smits",
title = "Scaling of the streamwise velocity component in turbulent pipe flow",
journal = "J.\ Fluid Mech.",
volume = "508",
number = "",
pages = "99--131",
year = "2004",
abstract = ""
}
@InProceedings{ Morkovin69,
author = "M. V Morkovin",
title = "On the many faces of transition",
booktitle = "Viscous Drag Reduction",
editor = "C. S. Wells",
year = "1969",
publisher = "Plenum, New York",
pages = "1--31"
}
@Book{mosersr,
author = "J. Moser",
title = "Stable and Random Motions in Dynamical Systems",
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"
}
@Article{Moser83,
author = "R. Moser and P. Moin and A. Leonard",
title = "A spectral method for the {N}avier-{S}tokes equations
with applications to {Taylor-Couette} flow",
journal = "J.\ Comp.\ Phys.",
volume = "52",
pages = "524--544",
year = "1983"
}
@Article{MoserPF99,
author = "R. Moser and J. Kim and N.N. Mansour",
title = "Direct numerical simulation of turbulent channel flow up to $Re_{\tau} = 590$",
journal = "Phys.\ Fluids",
volume = "11",
number = "4",
pages = "943--946",
year = "1999"
}
%%%%%%%%%%%%%% plane Couette flow bifurcation:
@Article{NaBu83,
author = "Nagata, M. and Busse, F. H.",
title = "Three-dimensional tertiary motions in a plane shear layers",
journal = "J.\ Fluid Mech.",
volume = "135",
pages = "1--26",
year = "1983"
}
@Article{N90,
author = "M. Nagata",
title = "Three-dimensional finite-amplitude solutions in
plane {Couette} flow: {Bifurcation} from infinity",
journal = "J.\ Fluid Mech.",
volume = "217",
pages = "519--527",
year = "1990"
}
@Article{N96,
author = "M. Nagata",
title = "Nonlinear solutions of modified plane {Couette} flow
in the presence of a transverse magnetic field",
journal = "J.\ Fluid Mech.",
volume = "307",
year = "1996",
pages = "231-243"
}
@Article{N97,
title={{Three-dimensional traveling-wave solutions in plane
{Couette} flow}},
author={Nagata, M.},
journal={Phys.\ Rev.\ E},
volume={55},
pages={2023--2025},
year={1997},
}
@Article{Ehren08,
author = "Ehrenstein, U. and Nagata, M. and Rincon, F.",
year = "2008",
title = "Two-dimensional nonlinear plane {P}oiseuille-{Couette}
flow homotopy revisited",
journal = "Phys.\ Fluids",
volume = "20",
pages = "064103-1--4"
}
% have this in ChaosBook.org/library
@InCollection{Narasimha89,
author = {R. Narasimha},
title = {The utility and drawbacks of traditional approaches},
booktitle = {Whither Turbulence? Turbulence at the Cross-Road},
pages = {13-48},
publisher = {Springer},
address = {Berlin},
year = {1989},
editor = {J. Lumley},
}
% sometimes called {temam85ks,
@Article{NSTks85,
author = "B. Nicolaenko and B. Scheurer and R. Temam",
title = "Some global dynamical properties of the {Kuramoto-Sivashinsky} 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."
}
@TechReport{ octave,
author = "J. Eaton",
title = "{GNU Octave}: a high-level language for numerical computation",
publisher = "J. Eaton",
institution = "University of Wisconsin",
year = "2007",
URL = "\weblink{www.octave.org}"
}
@article{Orr1907,
author={Orr, W. M. F.},
title={The stability or instability of the steady motions of a liquid.
{Part II: A} viscous liquid},
volume={27}, journal={Proc. Roy. Irish Acad. A}, year={1907},
pages={ 69--138}}
@Article{OK96,
author = "S. A. Orszag and L. C. Kells",
title = "Transition to turbulence in plane {Poiseuille} and plane {Couette} flow",
journal = "J.\ Fluid Mech.",
year = "1980",
volume = "96",
pages = "159--205"
}
@Article{Orszag71,
author = "S. A. Orszag",
title = "Accurate solution of the {O}rr-{S}ommerfeld stability equation",
journal = "J.\ Fluid Mech.",
volume = "50",
pages = "689--703",
year = "1971",
abstract = "The Orr-Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials and the QR matrix eigenvalue algorithm. It is shown that results of great accuracy are obtained very economically. The method is applied to the stability of plane Poiseuille flow. Expansions in Chebyshev polynomials are better suited to the solution of hydrodynamic stability problems than expansions in other, seemingly more relevant, sets of orthogonal functions."
}
@Article{Orszag86,
author = "S. A. Orszag and M. Israeli and M. Deville",
title = "Boundary conditions for incompressible flows",
journal = "J.\ Sci.\ Comput.",
volume = "1",
number = "",
pages = "75--111",
year = "1986",
abstract = ""
}
%%%%%%%%%%%%%% plane Couette flow experiments:
% \rf{BDMD98,DD95,DDM98,ExpsPCtransition,%
%spotspreading,BTAA95,LundJohan91,HDB92,DHB92,MMG95}
@Article{ExpsPCtransition,
author = "N. Tillmark and P. H. Alfredsson",
title = "Experiments on transition in plane {Couette} flow",
journal = "J.\ Fluid Mech.",
volume = "235",
year = "1992",
pages = "89--102"
}
@Article{spotspreading,
author = "N. Tillmark",
title = "On the spreading mechanisms of a turbulent spot in plane {Couette} flow",
journal = "Europhys.\ Lett.",
volume = "32",
pages = "481--485",
year = "1995"
}
% also cited as Article{BTAA,
@Article{BTAA95,
author = "K. H. Bech and N. Tillmark and P. H. Alfredsson and H. I. Andersson",
title = "An investigation of turbulent plane {Couette} flow at low {R}eynolds numbers",
journal = "J.\ Fluid Mech.",
volume = "286",
year = "1995",
pages = "291--325"
}
@Article{LundJohan91,
author = "A. Lundbladh and A. Johansson",
title = "Direct simulation of turbulent spots in plane {Couette} flow",
journal = "J.\ Fluid Mech.",
volume = "229",
year = "1991",
pages = "499--516"
}
@InCollection{ LHR94,
author = "A. Lundbladh and and S. Henningson and S. C. Reddy",
title = "Threshold amplitudes for transition in channel flows",
booktitle = "Turbulence and Combustion",
editor = "Hussaini, M.Y. and Gatski, T.B. and Jackson, T.L.",
publisher = "Kluwer",
address = "Dordrecht",
year = "1994"
}
@Article{MMG95,
author = "S. Malerud and K. J. M\aa{}l\o{}y and W. I. Goldburg",
title = "Measurements of turbulent velocity fluctuations in a planar {Couette} cell",
journal = "Phys.\ Fluids",
volume = "7",
year = "1995",
pages = "1949--1955"
}
%%%%%%%%%%%%%% plane Couette flow experiments, END %%%%%%%%%%%%%%%%%%%
% for mechanical engineers and high Re flows
% replace everywhere \cite{nr} by {Press:96}
@MISC{lvov-2007,
author = {V. S. L'vov and I. Procaccia and O. Rudenko},
title = {Universal model of finite-{Reynolds} number turbulent flow
in channels and pipes},
note = "{\tt arXiv:0705.4592}",
year = {2007}
}
@Article{pugh67cl,
author = "C. Pugh",
title = "An improved closing lemma and a general density theorem",
journal = "Amer.\ J.\ Math.",
volume = "89",
year = "1967"
}
@Book{MarRat99,
author = {Marsden, Jerrold E. and Ratiu, Tudor S.},
title = {Introduction to Mechanics and Symmetry},
publisher = {Springer},
year = 1999,
address = {New York, NY},
}
@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"
}
@Article{RSBH98,
author = "S. C. Reddy and P. J. Schmid and J. S. Baggett and D. S. Henningson",
title = "On stability of streamwise streaks and transition thresholds in plane channel flows",
journal = "J.\ Fluid Mech.",
year = "1998",
volume = "365",
pages = "269--303"
}
@Article{R1883,
author = "O. Reynolds",
title = "An experimental investigation of the circumstances
which determine whether the motion of water shall
be direct or sinuous, and the law of resistance in parallel channels",
journal = "Proc.\ Roy.\ Soc.\ Lond.\ Ser A",
year = "1883",
volume = "174",
pages = "935--982"
}
@Article{R1895,
author = "O. Reynolds",
title = "On the dynamical theory of incompressible viscous flows
and the determination of the criterion",
journal = "Proc.\ Roy.\ Soc.\ Lond.\ Ser A",
year = "1895",
volume = "186",
pages = "123--164"
}
@Book{Risken96,
author = "H. Risken and H. Haken",
title = "The {F}okker-{P}lanck Equation",
publisher = "Springer",
year = "1996"
}
@Article{Robinson91,
author = "S. K. Robinson",
title = "{Coherent motions in the turbulent boundary layer}",
journal = "Ann.\ Rev.\ Fluid Mech.",
year = "1991",
volume = "23",
pages = "601--639"
}
@TechReport{ Robinson91b,
author = "S. K. Robinson",
title = "The Kinematics of turbulent boundary layer structure",
institution = "NASA",
year = "1991",
note = "103859"
}
@Article{ross,
author = {O. E. R{\"o}ssler},
title = "An equation for continuous chaos",
journal = "Phys.\ Lett.\ A",
volume = "57",
pages = "397",
year = "1976"
}
@ARTICLE{rowley_reduction_2003,
author = {Rowley, C. W. and Kevrekidis, I. G. and Marsden, J. E. and Lust,
K.},
title = {Reduction and reconstruction for self-similar dynamical systems},
journal = {Nonlinearity},
year = {2003},
volume = {16},
pages = {1257--1275},
abstract = {We present a general method for analysing and numerically solving
partial differential equations with self-similar solutions. The method
employs ideas from symmetry reduction in geometric mechanics, and
involves separating the dynamics on the shape space (which determines
the overall shape of the solution) from those on the group space
(which determines the size and scale of the solution). The method
is computationally tractable as well, allowing one to compute self-similar
solutions by evolving a dynamical system to a steady state, in a
scaled reference frame where the self-similarity has been factored
out. More generally, bifurcation techniques can be used to find self-similar
solutions, and determine their behaviour as parameters in the equations
are varied. The method is given for an arbitrary Lie group, providing
equations for the dynamics on the reduced space, for reconstructing
the full dynamics and for determining the resulting scaling laws
for self-similar solutions. We illustrate the technique with a numerical
example, computing self-similar solutions of the Burgers equation}
}
@ARTICLE{rowley_reconstruction_2000,
author = {Rowley, C. W. and Marsden, J. E.},
title = {Reconstruction equations and the {Karhunen-Lo\'eve} expansion for
systems with symmetry},
journal = {Physica D},
year = {2000},
volume = {142},
pages = {1--19},
abstract = {We present a method for applying the {Karhunen-Lo\'eve} decomposition
to systems with continuous symmetry. The techniques in this paper
contribute to the general procedure of removing variables associated
with the symmetry of a problem, and related ideas have been used
in previous works both to identify coherent structures in solutions
of {PDEs,} and to derive low-order models via Galerkin projection.
The main result of this paper is to derive a simple and easily implementable
set of reconstruction equations which close the system of {ODEs}
produced by Galerkin projection. The geometric interpretation of
the method closely parallels techniques used in geometric phases
and reconstruction techniques in geometric mechanics. We apply the
method to the {Kuramoto-Sivashinsky} equation and are able to derive
accurate models of considerably lower dimension than are possible
with the traditional {Karhunen-Lo\'eve} expansion.}
}
@ARTICLE{rowley04,
author = {C. W. Rowley},
title = {Model reduction for fluids, using balanced proper orthogonal decomposition},
journal = {Int. J. Bif. Chaos},
volume = {15},
pages = {997},
year = {2005}
}
@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{Ruelle76,
author = "D. Ruelle",
title = "Zeta-functions for expanding maps and {Anosov} flows",
journal = "Inv. Math.",
volume = "34",
pages = "231--242",
year = "1976",
}
@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{ruelle86,
author = {D. Ruelle},
title = {Locating resonances for {Axiom A} dynamical systems},
journal = {J. Stat. Phys.},
volume = {44},
year = {1986},
pages = {281--292},
abstract = {For a class of differentiable dynamical systems (called Axiom
A systems) it has been shown by Pollicott and the author that correlation
functions have Fourier transforms which are meromorphic in a strip. The
poles (or resonances) are, however, not easy to locate. This note reviews
the results which are known and discusses a simple model where the
position of resonances can be estimated. The effect of noise is also
discussed.}
}
@article{ruelle86a,
author = {D. Ruelle},
title = {Resonances of chaotic dynamical systems},
journal = {Phys. Rev. Lett.},
volume = {56},
year = {1986},
pages = {405--407},
}
@Article{hhrugh92,
author = "H. H. Rugh",
title = "The correlation spectrum for hyperbolic analytic maps",
journal = "Nonlinearity",
volume = "5",
pages = "1237",
year = "1992"
}
@Article{SNGS,
author = "J. Sanche\'{z} and M. Net and B. Gar{\'c}ia-Archilla and C. Sim{\'o}",
title = "{Newton-Krylov} continuation of periodic orbits for {Navier-Stokes} flows",
journal = "J.\ Comput.\ Physics",
year = "2004",
volume = "201",
pages = "13--33"
}
@Article{SchEck06,
author = "T. M. Schneider and B. Eckhardt",
title = "Edge of chaos in pipe flow",
journal = "CHAOS",
year = "2006",
volume = "16",
pages = "041103"
}
@Article{SchEckVoll07,
author = "Schneider, T. M. and Eckhardt, B. and Vollmer, J.",
title = "Statistical analysis of coherent structures
in transitional pipe flow",
journal = "Phys.\ Rev.\ E",
volume = "75",
pages = {066313},
year = "2007"
}
@Article{VSE98,
author = {J. Vollmer and T. M. Schneider and B. Eckhardt},
title = {Basin boundary, edge of chaos,
and edge state in a two-dimensional model},
year = {2007},
note = {{\tt arXiv:0808.2636}},
}
@article{Selberg56,
author = {Selberg, A.},
title = {Harmonic analysis and discontinuous groups in weakly
symmetric {Riemannian} spaces with applications to {Dirichlet}
series},
journal = {J. Indian Math. Soc. (N.S.)},
pages = {47--87},
volume = {20},
year = {1956}
}
@ARTICLE{ShWi06,
author = {Shapere, A. and Wilczek, F.},
title = {Geometry of self-propulsion at low {Reynolds} number},
journal = {J. Fluid Mech.},
year = {2006},
volume = {198},
pages = {557}
}
@ARTICLE{ShWi89,
author = {Shapere, A. and Wilczek, F.},
title = {Efficiencies of self-propulsion at low {Reynolds} number},
journal = {J. Fluid Mech.},
year = {1989},
volume = {198},
pages = {587--599}
}
@ARTICLE{ShWi89a,
author = {Shapere, A. and Wilczek, F.},
title = {Gauge kinematics of deformable bodies},
journal = {Amer. J. Physics},
year = {1989},
volume = {57},
pages = {514--518}
}
@ARTICLE{ShWi87,
author = {Shapere, A. and Wilczek, F.},
title = {Self-propulsion at low {Reynolds} number},
journal = {Phys. Rev. Lett.},
year = {1987},
volume = {58},
pages = {2051--2054}
}
@Article{Shte69,
author = "Shtern, V. N.",
title = "Stability of plane {Couette} flow ",
journal = "J.\ Appl.\ Mechanics and Tech.\ Physics",
volume = "10",
year = "1969",
pages = "793-796"
}
@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",
publisher = "Kluwer",
address = "Dordrecht",
year = "2002"
}
@BOOK{bowen,
AUTHOR = {R. Bowen},
TITLE = {Equilibrium States and the Ergodic Theory of
{Anosov} Diffeomorphisms},
YEAR = 1975,
PUBLISHER = {Springer},
ADDRESS = {Berlin}
}
@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 {Ginzburg-Landau} equation",
journal = "Physica D",
volume = "43",
pages = "77--86",
year = "1990"
}
@Article{Zhou92,
author = "X. Zhou and L. Sirovich",
title = "Coherence and Chaos in a model of turbulent boundary layer",
journal = "Phys.\ Fluids A",
year = "1992",
volume = "4",
pages = "2855--2874"
}
@Article{Sirovich94,
author = "L. Sirovich and X. Zhou",
title = "Reply to ``{O}bservations regarding '{C}oherence and chaos in a model of turbulent boundary layer' by {X}. {Z}hou and {L}. {S}irovich''",
journal = "Phys.\ Fluids",
year = "1994",
volume = "6",
number = "4",
pages = "1579--1582"
}
@Article{siv,
author = "G. I. Sivashinsky",
title = "Nonlinear analysis of hydrodynamical instability in laminar flames - {I}. {D}erivation of basic equations",
journal = "Acta Astronaut.",
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 Astronaut.",
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{Mks86,
author = "D. Michelson",
title = "Steady solutions of the {Kuramoto-Sivashinsky} equation",
journal = "Physica D",
volume = "19",
pages = "89--111",
year = "1986",
abstract = "The variety of steady solution of the KSe is discussed.
For large c, there is only one odd front-like bounded solution.
In decreasing c, odd solutions with more zeros are born until finally
a periodic solution is born. Associated with the periodic solution,
infinite many tori will appear in the elliptic case
and Cantor-type set of chaotic solutions,
with infinite many homoclinic odd solutions."
}
@Article{SYE05,
author = "J. D. Skufca and J. A. Yorke and B. Eckhardt",
title = "Edge of chaos in a parallel shear flow",
journal = "Phys.\ Rev.\ Lett.",
volume = "96",
pages = "174101",
year = "2006",
URL = "http://yorke.umd.edu/papers/Skufca-Eckhardt-Yorke\%20PRL\%20submission.pdf",
abstract = "identify and follow the edge of chaos and provide evidence that it is a surface. For low Reynolds numbers we find that the edge of chaos coincides with the stable manifold of a periodic orbit, whereas at higher Reynolds numbers it is the stable set of a higher-dimensional chaotic object.",
number = "17"
}
@phdthesis{ Skuf05,
author = "J. D. Skufca",
title = "Understanding the chaotic saddle with focus on a 9-variable model of planar {Covette} flow",
year = "2005",
school = "U. Maryland",
URL = "https://drum.umd.edu/dspace/handle/1903/2897"
}
@Article{SchEckYor07,
author = "T. M. Schneider and B. Eckhardt and J. Yorke",
title = "Turbulence, transition, and the edge of chaos in pipe flow",
journal = "Phys.\ Rev.\ Lett.",
volume = "99",
year = "2007",
pages = "034502"
}
@BOOK{Smale80,
title = {The Mathematics of Time},
publisher = {Springer},
year = {1980},
author = {S. Smale},
address = {New York},
annote = {Subtitle: Essays on Dynamical Systems, Economic Processes and related
topics}
}
@ARTICLE{Smale70I,
author = {S. Smale},
title = {Topology and mechanics, {I}.},
journal = {Invent. Math.},
year = {1970},
volume = {10},
pages = {305--331}
}
@ARTICLE{Smale70II,
author = {S. Smale},
title = {Topology and mechanics, {II}.},
journal = {Invent. Math.},
year = {1970},
volume = {11},
pages = {45--64}
}
@ARTICLE{smale,
author = {S. Smale},
title = {Differentiable dynamical systems},
journal = {Bull. Amer. Math. Soc.},
year = {1967},
volume = {73},
pages = {747--817}
}
@BOOK{Smale00,
title = {The collected papers of {Stephen Smale}},
publisher = {World Scientific},
year = {2000},
author = {S. Smale and F. Cucker and R. Wong},
address = {Singapore}
}
@Article{SM,
author = "C. R. Smith and S. P. Metzler",
title = "The study of hairpin vortices in a laminar boundary layer",
journal = "J.\ Fluid Mech.",
year = "1983",
volume = "129",
pages = "27"
}
@Article{SMH05,
author = "T. R. Smith and J. Moehlis and P. Holmes",
title = "Low-dimensional models for turbulent plane {Couette} flow in a minimal flow unit",
journal = "J.\ Fluid Mech.",
year = "2005",
volume = "538",
pages = "71--110"
}
@Article{Spalart91,
author = "P. R. Spalart and R. D. Moser and M. M. Rogers",
title = "Spectral Methods for the {N}avier-{S}tokes Equations with One Infinite and Two Periodic Directions",
journal = "J.\ Comp.\ Physics",
year = "1991",
volume = "96",
pages = "297--324"
}
@Article{EAS87,
author = "E. A. Spiegel",
title = "Chaos: a mixed metaphor for turbulence",
journal = "Proc.\ Roy.\ Soc. A",
volume = "413",
pages = "87",
year = "1987"
}
@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"
}
@Article{SWG02,
author = "P. A. Stone and F. Waleffe and M. D. Graham",
title = "Toward a structural understanding of turbulent drag reduction:
nonlinear coherent states in viscoelastic shear flows",
journal = "Phys.\ Rev.\ Lett.",
year = "2002",
volume = "89",
pages = "20831"
}
@Book{strogb,
author = "S. H. Strogatz",
title = "Nonlinear Dynamics and Chaos",
series = "Studies in Nonlinearity",
publisher = "Perseus Publishing",
address = "Cambridge Massachusetts",
year = "2000"
}
@Article{SwinneyPhysD83,
author = "H. L. Swinney",
title = "Observations of order and chaos in nonlinear systems",
journal = "Physica D",
year = "1983",
volume = "7",
pages = "3--15"
}
@ARTICLE{Win92,
author = {D. Wintgen and K. Richter and G. Tanner},
title = {Quantization of chaotic systems},
journal = {CHAOS},
year = {1992},
volume = {2},
pages = {53}
}
@article{Ezr91,
title={Semiclassical cycle expansion for the helium atom},
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@BOOK{ Tennekes72,
author = "H. Tennekes and J.L. Lumley",
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author = "A. A. Townsend",
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author = "L. N. Trefethen",
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author = "L. N. Trefethen and D. Bau",
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@PhdThesis{ vanDoorne04,
author = "Casimir W. H. van Doorne",
title = "Stereoscopic {PIV} on transition in pipe flow",
school = "Delft",
year = "2004",
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author = "L. van Veen and G. Kawahara",
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author = "D. Viswanath",
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author = "D. Viswanath",
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author = "D. Viswanath",
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}
@Unpublished{ DVLindstedt,
author = "D. Viswanath",
title = "An extension of the {Lindstedt}-{Poincar\'e}
algorithm for computing periodic orbits",
note = "unpublished"
}
@Article{DV04,
author = "D. Viswanath",
title = "The fractal property of the {Lorenz} attractor",
journal = "Physica D",
year = "2004",
volume = "190",
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author = "D. Viswanath",
title = "The dynamics of transition to turbulence in plane {Couette} flow",
booktitle = "Mathematics and Computation, a Contemporary
View. The Abel Symposium 2006",
publisher = "Springer",
year = 2008,
volume = "3",
series = "Abel Symposia",
address = "Berlin",
note = "{\tt arXiv:physics/0701337}"
}
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author = {D. Viswanath and P. Cvitanovi\'{c}},
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year = "2009",
volume = "627",
pages = "215--233",
note = "{\tt arXiv:0801.1918}",
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% update
@Article{Visw08,
author = {D. Viswanath},
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author = "F. Waleffe",
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author = "F. Waleffe and J. Wang",
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%also jensen84
@article{JBB84,
author={Jensen, M. H. and Bak, P. and Bohr, T.},
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author={Jensen, M. H. and Bak, P. and Bohr, T.},
title={Complete devil's staircase fractal dimension and universality of
mode-locking structure in the circle map},
volume={50},
journal={Phys. Rev. Lett.},
year={1983},
pages={1637--1639}}
@Article{collet80,
author = "P. Collet and J.-P. Eckmann and O. E. Lanford",
title = "Universal Properties of Maps on an interval",
journal = "Commun.\ Math.\ Phys.",
volume = "76",
pages = "211",
year = "1980"
}
@Article{nhouse78,
author = "S. E. Newhouse and D. Ruelle and F. Takens",
title = "Occurrence of strange {A}xiom {A} attractors near quasi-periodic flows on ${T}^m$($m=3$ or more)",
journal = "Commun.\ Math.\ Phys.",
volume = "64",
pages = "35",
year = "1978"
}
@Article{transgol,
author = "R. Ramshankar and J. P. Gollub",
title = "Transport by capillary waves, {Part II}: {Scalar} dispersion and the structure of the concentration field",
journal = "Phys.\ Fluids A",
volume = "3",
pages = "1344",
year = "1991"
}
@Article{yuan00id,
author = "G.-C. Yuan and J. A. Yorke and T. L. Carroll and E. Ott and L. M. Pecora",
title = "Testing whether two chaotic one dimensional processes are dynamically identical",
journal = "Phys.\ Rev.\ Lett.",
volume = "85",
number = "20",
pages = "4265",
year = "2000",
abstract = "By using the natural measure to construct the canonical
conjugate of the observed signal, the authors can identify the
dynamical equivalence of two signals from one process." }
@Article{frisch86,
author = "U. Frisch and B. Hasslacher and Y. Pomeau",
title = "Lattice-gas automata for the navier-stokes equation",
journal = "Phys.\ Rev.\ Lett.",
volume = "56",
pages = "1505--1508",
year = "1986"
}
@Book{bohr98tur,
author = "T. Bohr and M. H. Jensen and G. Paladin and A. Vulpiani",
title = "Dynamical Systems Approach to Turbulence",
publisher = "Cambridge Univ.\ Press",
address = "Cambridge",
year = "1998"
}
@Unpublished{ OleMog07,
author = "Olesen, P. and Jensen, M.s H.",
title = "Exact periodic solutions of shells models of turbulence",
abstract = "derive exact analytical solutions of the GOY shell model
of turbulence. In the absence of forcing and viscosity obtain
closed form solutions in terms of Jacobi elliptic functions. With
three shells the model is integrable. Obtain a Kolmogorov solution
in the limit of infinitely many shells. Also identify cases where
the models exhibits transitions to chaotic states lying on strange
attractors or ergodic energy surfaces.",
year = "2007",
note = "{\tt arXiv:0705.3123}",
}
@InCollection{ ampatt,
author = "M. van Hecke and P. C. Hohenberg and W. van Saarloos",
title = "Amplitude equations for pattern forming systems",
booktitle = "Fundamental Problems in Statistical Mechanics",
volume = "VIII",
editor = "H. van Beijeren and M. H. Ernst",
publisher = "North-Holland",
address = "Amsterdam",
year = "1994"
}
@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"
}
@Article{egolf00,
author = "D. A. Egolf and I. V. Melnikov and W. Pesch and R. E. Ecke",
title = "Mechanisms of extensive spatiotemporal chaos in {R}ayleigh-{B}{\'e}nard convection",
journal = "Nature",
volume = "404",
pages = "733",
year = "2000",
abstract = "Numerical study on Boussinesq equation(pseudospectral
method with time splitting scheme). Demonstrate the extensivity of
Lyapunov dimension of the extended system and showed that the
generating mechanism is breaking or connecting rolls mediated by
defects at the domain boundary and exsisting for only a short time."
}
@Article{egolf98,
author = "D. A. Egolf and I. V. Melnikov and E. Bodenschatz",
title = "Importance of local pattern properties in spiral defect chaos",
journal = "Phys.\ Rev.\ Lett.",
volume = "80",
pages = "3228",
year = "1998"
}
@Article{egolf94,
author = "D. A. Egolf and H. S. Greenside",
title = "Relation between fractal dimension and spatial correlation length for extensive chaos",
journal = "Nature",
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."
}
@Article{tajima02,
author = "S. Tajima and H. S. Greenside",
title = "Microextensive chaos of a spatially extended system",
journal = "Phys.\ Rev.\ E",
volume = "66",
pages = "017205",
year = "2002"
}
@Article{roth99,
author = "D. Rothstein and E. Henry and J. P. Gollub",
title = "Persistent patterns in transient chaotic fluid mixing",
journal = "Nature",
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."
}
@Article{voth02,
author = "G. A. Voth and G. Haller and J. P. Gollub",
title = "Experimental measurements of streching fields in fluid mixing",
journal = "Phys.\ Rev.\ Lett.",
volume = "88",
number = "25",
pages = "254501",
year = "2002",
abstract = "Based on the experimentally generated velocity field, fixed points on poincar{\'e} section are identified as well as their stable and unstable manifolds. These information can be related to the gradient of the stretching field and the distribution of passive scalar quantities. It seems that on these manifolds the stretching field gains its maximum."
}
@Article{golubsym84,
author = "M. Golubitsky and J. W. Swift and E. Knobloch",
title = "Symmetries and pattern selection in {R}ayleigh-{B}ernard convection",
journal = "Physica D",
volume = "10",
pages = "249",
year = "1984"
}
@book{golubitsky2002sp,
title={{The symmetry perspective}},
author={Golubitsky, M. and Stewart, I.},
year={2002},
publisher={Birkh{\"a}user},
address = "Boston"
}
@Article{golubord,
author = "J. P. Gollub",
title = "Order and disorder in fluid motion",
journal = "Proc.\ Natl.\ Acad.\ Sci.\ USA",
volume = "92",
pages = "6705",
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."
}
@Article{luce95,
author = "B. P. Luce",
title = "Homoclinic explosions in the complex {Ginzburg-Landau} equation",
journal = "Physica D",
volume = "84",
pages = "553--581",
year = "1995"
}
@Article{doelman91n,
author = "A. Doelman",
title = "Finite-dimensional models of the {Ginzburg-Landau} equation",
journal = "Nonlinearity",
volume = "4",
pages = "231--250",
year = "1991"
}
@Article{garnier,
author = "N. Garnier and A. Chiffaudel and F. Daviaud and A. Prigent",
title = "Nonlinear dynamics of waves and modulated waves in 1{D} thermocapillary flows",
journal = "Physica D",
volume = "174",
number = "1-4",
pages = "1--29",
year = "2003"
}
@Article{witt99hol,
author = "R. W. Wittenberg and P. Holmes",
title = "Scale and space localization in the {Kuramoto-Sivashinsky} 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 = "??",
volume = "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{ksham95,
author = "S. Bouquet",
title = "Hamiltonian structure and integrability of the stationary
{Kuramoto-Sivashinsky} equation",
journal = "J.\ Math.\ Phys.",
volume = "36",
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."
}
@Article{kshooper88,
author = "A. P. Hooper and R. Grimshaw",
title = "Travelling wave solutions of the {Kuramoto-Sivashinsky} equation",
journal = "Wave Motion",
volume = "10",
pages = "405--420",
year = "1988",
abstract = "Numerically calculated connections of the KSe"
}
@Article{ksyang97,
author = "T.-S. Yang",
title = "On travelling-wave solutions of the {Kuramoto-Sivashinsky} 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 {Kuramoto-Sivashinsky} 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."
}
@Article{kskent92,
author = "P. Kent and J. Elgin",
title = "Travelling-waves of the {Kuramoto-Sivashinsky} 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."
}
@Article{kstroy89,
author = "W. C. Troy",
title = "The existence of steady solutions of the {Kuramoto-Sivashinsky} equation",
journal = "J.\ Diff.\ Eqn.",
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."
}
@Article{kstroy92,
author = "J. Jones and W. C. Troy and A. D. MacGillivary",
title = "Steady solutions of the {Kuramoto-Sivashinsky} equation for small wave speed",
journal = "J.\ Diff.\ Eqn.",
volume = "96",
pages = "28--55",
year = "1992",
abstract = "For small c>0, at least one odd periodic solution exists.
For any c>0, there existes no monotone bounded travelling wave
solution. The normal form analysis may not be able to give correct
results to all orders."
}
@Article{chang94,
author = "H.-C. Chang",
title = "Wave evolution on a falling film",
journal = "Ann.\ Rev.\ Fluid Mech.",
volume = "26",
pages = "103--136",
year = "1994"
}
@Article{mcord86,
author = "C. K. McCord",
title = "Uniqueness of connection orbits in the equation $y^{(3)}=y^2-1$",
journal = "J.\ Math.\ Anal.\ Appl.",
volume = "114",
pages = "584--592",
year = "1986"
}
@Article{lau92,
author = "Y.-T. Lau",
title = "The {``}cocoon"
}
@Article{HSGsize98,
author = "M. C. Strain and H. S. Greenside",
title = "Size-dependent transition to high-dimensional chaotic dynamics in a two-dimensional excitable media",
journal = "Phys.\ Rev.\ Lett.",
volume = "80",
number = "11",
pages = "2306--2309",
year = "1998"
}
@Article{saka90br,
author = "H. Sakaguchi",
title = "Breakdown of the phase dynamics",
journal = "Prog.\ Ther.\ Phys.",
volume = "84",
number = "5",
pages = "792--800",
year = "1990"
}
@Article{sten94sol,
author = "L. Stenflo and O. M. Gradov",
title = "Solitary surface-wave deformation of a plasma boundary",
journal = "Phys.\ Plasmas",
volume = "1",
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."
}
@Article{doeb94nls,
author = "H.-D. Doebner and G. A. Goldin",
title = {Properties of nonlinear Schr{\"o}dinger equations associated with diffeomorphism group representations},
journal = "J.\ Phys.\ A",
volume = "27",
pages = "1771--1780",
year = "1994"
}
@Article{yu94sur,
author = "S. V. Vladimirov and M. Y. Yu",
title = "Coupled Electron Surface Plasma Wave Solitons in Slabs",
journal = "Phys.\ Lett.\ A",
volume = "184",
pages = "459",
year = "1994"
}
@Article{yu93bd,
author = "S. V. Vladimirov and M. Y. Yu",
title = "Boundary Effects on the Nonlinear Interactions of Surface Waves",
journal = "Phys.\ Fluids B",
volume = "5",
pages = "2888",
year = "1993"
}
@Article{bro94sur,
author = "G. Brodin and J. Lundberg",
title = "Nonlinear surface waves in a plasma with a diffuse boundary",
journal = "Phys.\ Plasmas",
volume = "1",
pages = "96",
year = "1994"
}
@Article{chate94,
author = "H. Chat{\'e}",
title = "Spatiotemporal intermittency regimes of the one-dimensional complex {Ginzburg-Landau} 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 "
}
@Article{newcgl1,
author = "P. K. Newton and L. Sirovich",
title = "Instabilities of {Ginzburg-Landau} equation: periodic solutions",
journal = "Quart.\ Appl.\ Math.",
volume = "XLIV",
pages = "49--58",
year = "1986"
}
@Article{newcgl2,
author = "P. K. Newton and L. Sirovich",
title = "Instabilities of {Ginzburg}-{{L}andau} equation: secondary bifurcation",
journal = "Quart.\ Appl.\ Math.",
volume = "XLIV",
pages = "367--374",
year = "1986"
}
% sometimes cited as Article{cgl,
@Article{AKcgl02,
author = "I. S. Aranson and L. Kramer",
title = "The World of the Complex {Ginzburg-Landau} Equation",
journal = "Rev.\ Mod.\ Phys.",
volume = "74",
pages = "99--143",
year = "2002"
}
@Article{hoopks85,
author = "A. P. Hooper and R. Grimshaw",
title = "Nonlinear instability at the interface between two viscous fluids",
journal = "Phys.\ Fluids",
volume = "28",
pages = "37--45",
year = "1985"
}
@Article{cohen72,
author = "B. I. Cohen and J. A. Krommes and W. M. Tang and M. N. Rosenbluth",
title = "Non-linear saturation of the dissipative trapped-ion mode by mode coupling",
journal = "Nucl.\ Fusion",
volume = "16",
pages = "971--991",
year = "1972"
}
@Article{laquey74,
author = "R. E. La Quey and S. M. Mahajan and P. H. Rutherford and W. M. Tang",
title = "Nonlinear saturation of the trapped-ion mode",
journal = "Phys.\ Rev.\ Lett.",
volume = "34",
pages = "391--394",
year = "1974"
}
@Article{umb94pat,
author = "F. Melo and P. Umbanhowar and H. L. Swinney",
title = "Transition to parametric wave patterns in
a vertically oscillated granular layer",
journal = "Phys.\ Rev.\ Lett.",
volume = "72",
pages = "172--175",
year = "1994"
}
% aslo Article{sup,
@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 {Ginzburg-Landau} Equation for a Subcritical Bifurcation",
journal = "Phys.\ Rev.\ Lett.",
volume = "66",
number = "18",
pages = "2316",
year = "1991"
}
% also Article{saddks,
@Article{KNSks90,
author = "I. G. Kevrekidis and B. Nicolaenko and J. C. Scovel",
title = "Back in the saddle again: a computer assisted study of the {Kuramoto-Sivashinsky} equation",
journal = "SIAM J.\ Appl.\ Math.",
volume = "50",
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."
}
@InProceedings{BrKevr96,
author = "Brown, H. S. and Kevrekidis, I. G.",
title = "Modulated traveling waves for the {Kuramoto-Sivashinsky}
equation",
booktitle = "Pattern Formation: Symmetry Methods and Applications",
editor = "D. Benest and C. Froeschl\'{e}",
series = "Fields Inst.\ Commun.",
volume = "5",
publisher = "AMS",
address = "Providence, RI",
pages = "45--66",
year = "1996"
}
@Article{pattfm,
author = "M. C. Cross and P. C. Hohenberg",
title = "Pattern formation outside of equilibrium",
journal = "Rev.\ Mod.\ Phys.",
volume = "65",
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 {Ginzburg-Landau} 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 {Ginzburg-Landau} Equation",
journal = "J.\ Phys.\ Soc.\ Japan",
volume = "53",
pages = "1581--1582",
year = "1984"
}
@Article{bknk83,
author = "K. Nozaki and N. Bekki",
title = "Pattern Selection and Spatiotemporal Transition to Chaos in the {Ginzburg-Landau} Equation",
journal = "Phys.\ Rev.\ Lett.",
volume = "51",
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",
pages = "152--167",
year = "2003"
}
% also cites as Article{defectcgl,
@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 {Ginzburg-Landau} equation",
journal = "Physica D",
volume = "160",
pages = "127--148",
year = "2001"
}
% also cites as Article{mawcgl,
@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",
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 {Ginzburg-Landau} Equation near a Subcritical Bifurcation",
journal = "Phys.\ Rev.\ Lett.",
volume = "64",
pages = "749",
year = "1990"
}
% also {shka,
@ARTICLE{ShenKad82,
author = {S. J. Shenker and L. P. Kadanoff},
title = {Critical behavior of a {KAM} surface: I. {Empirical} results},
journal = {J. Stat. Phys.},
year = {1982},
volume = {27},
pages = {631}
}
@article{SeOsRaSi83,
author={Sethna, J. and Ostlund, S. and Rand, D. A. and Siggia, E.},
title={Universal properties of the transition from quasi-periodicity
to chaos in dissipative systems}, volume={8}, journal={Physica D},
year={1983}, pages={303}}
@article{RaOsSeSi82,
author = {D. Rand and S. Ostlund and J. Sethna and E. Siggia},
title = {Universal transition from quasiperiodicity to chaos in dissipative systems},
journal = {Phys. Rev. Lett.},
volume = {49},
year = {1982},
pages = {132--135},
doi = {10.1103/PhysRevLett.49.132},
}
@article{FeKaSh82,
author = {M.J. Feigenbaum and L.P. Kadanoff and S.J. Shenker},
title = {Quasiperiodicity in dissipative systems:
{A} renormalization group analysis},
journal = {Physica D},
volume = {5},
year = {1982},
pages = {370--386},
doi = {10.1016/0167-2789(82)90030-6},
}
@article{Shenker82,
author = {S.J. Shenker},
title = {Scaling behavior in a map of a circle onto itself: {Empirical} results},
journal = {Physica D},
volume = {5},
year = {1982},
pages = {405--411},
doi = {10.1016/0167-2789(82)90033-1},
}
@article{CSS85,
author = {P. Cvitanovi\'c and B. Shraiman and B. S\"oderberg},
title = {Scaling laws for mode lockings in circle maps},
journal = {Phys. Scr.},
volume = {32},
year = {1985},
pages = {263--270},
doi = {10.1088/0031-8949/32/4/003},
}
@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 {Ginzburg-Landau} equation",
journal = "Physica D",
volume = "57",
pages = "241--248",
year = "1992"
}
% also Article{buildcgl,
@Article{MvH98,
author = "M. van Hecke",
title = "Building Blocks of Spatiotemporal Intermittency",
journal = "Phys.\ Rev.\ Lett.",
volume = "80",
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",
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 {Ginzburg-Landau} equation",
journal = "Phys.\ Rev.\ Lett.",
volume = "86",
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",
pages = "112",
year = "1987"
}
@Article{MC96pha,
author = "P. Manneville and H. Chat{\'e}",
title = "Phase turbulence in the two-dimensional complex {Ginzburg-Landau} 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",
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 {Ginzburg-Landau} equation",
journal = "Phys.\ Rev.\ Lett.",
volume = "77",
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 1{D} {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."
}
% also Article{hohencgl,
@Article{SHcoh92,
author = "W. van Saarloos and P. C. Hohenberg",
title = "Fronts, pulses, sources and sinks in generalized complex {Ginzburg-Landau} 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
{Ginzburg-Landau} 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{SMcgl02,
author = "J. Scheuer and B. A. Malomed",
title = "Stable and chaotic solutions of the complex {Ginzburg-Landau} 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{CEEks93,
author = "P. Collet and J.-P. Eckmann and H. Epstein and J. Stubbe",
title = "Analyticity for the {Kuramoto-Sivashinsky} 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 {Kuramoto-Sivashinsky} 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 complex
Ginzburg Landau 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 {Kuramoto-Sivashinsky} 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 {Kuramoto-Sivashinsky}
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{FSTks86,
author = "U. Frisch and Z. S. She and O. Thual",
title = "Viscoelastic behavior of cellular solutions to
the {Kuramoto-Sivashinsky} 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 "
}
@Article{FNSTks85,
author = "C. Foias and B. Nicolaenko and G. R. Sell and R. Temam",
title = "Inertial manifold for the {Kuramoto-Sivashinsky} equation",
journal = "C.\ R.\ Acad.\ Sci.\ I-Math",
volume = "301",
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 {Kuramoto-Sivashinsky}
equation and an estimate of their lowest dimension",
journal = "J.\ Math.\ Pure Appl.",
volume = "67",
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",
pages = "2822",
year = "1999"
}
@Book{ns,
author = "J. W. Thomas",
title = "Numerical Partial Differential Equations",
publisher = "Spring-Verlag",
address = "New York",
year = "1995"
}
@Book{bl,
author = "J. Stoer and R. Bulirsch",
title = "Introduction to Numerical Analysis",
publisher = "Springer",
address = "New York",
year = "1983"
}
@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\~nez",
note = "{\tt arXiv:astro-ph/0109497}",
publisher = "Taylor \& Francis",
address = "London",
pages = "269--344",
year = "2003"
}
@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",
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",
address = "Amsterdam",
year = "1989"
}
% also {carsim
@INPROCEEDINGS{simo89,
author = {C. Simo},
title = {On the analytical and numerical approximation of invariant
manifolds},
booktitle = {Les M\'{e}thodes Modernes de la M\'{e}canique C\'{e}leste
(Goutelas '89)},
editor = {D. Benest and C. Froeschl\'{e}},
publisher = {Editions Frontieres},
address = {Gif-sur-Yvette},
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",
pages = "951--985",
year = "2000"
}
@Article{pimyk,
author = "H. E. Nusse and J. A. Yorke",
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",
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."
}
@MISC{franzosi-2004,
author = {R. Franzosi and P. Poggi and M. Cerruti-Sola},
title = {Lyapunov exponents from unstable periodic orbits
in the {FPU}-beta model},
note = "{\tt arXiv:nlin/0408056}",
year = {2004}
}
@ARTICLE{ginelli-2007-99,
author = {F. Ginelli and P. Poggi and A. Turchi and H. Chate
and R. Livi and A. Politi},
title = {Characterizing dynamics with covariant {Lyapunov} vectors},
journal = {Phys.\ Rev.\ Lett.},
volume = {99},
pages = {130601},
note = "{\tt arXiv:0706.0510}",
year = {2007}
}
@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 {O}seberg transition: visualization
of global bifurcations for the {Kuramoto-Sivashinsky} equation",
journal = "Int.\ J.\ Bifurc.\ Chaos",
volume = "11",
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",
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 "
}
@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{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 arXiv:chao-dyn/9810022}"
}
@Article{PSDB00,
author = "D. 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 arXiv:nlin.CD/0006011}"
}
@Article{afind,
author = "O. Biham and W. Wenzel",
title = "Characterization of unstable periodic orbits in chaotic attractors and repellers",
journal = "Phys.\ Rev.\ Lett.",
volume = "63",
pages = "819",
year = "1989"
}
@Article{mfind,
author = "R. L. Davidchack and Y.-C. Lai",
title = "Efficient Algorithm for Detecting Unstable Periodic Orbits in Chaotic Systems",
journal = "Phys.\ Rev.\ E",
volume = "60",
pages = "6172",
year = "1999"
}
@Article{PSD01,
author = "D. Pingel and P. Schmelcher and F. K. Diakonos",
title = "Detecting unstable periodic orbits in chaotic continuous-time dynamical systems",
journal = "Phys.\ Rev.\ E",
volume = "64",
pages = "026214",
year = "2001",
abstract = "Change the stability of the fixed points on the
Poincar{\'e} section to find periodic orbits."
}
@Article{SD98,
author = "P. Schmelcher and F. K. Diakonos",
title = "General appoach to the localization of unstable periodic
orbits in chaotic dynamical systems",
journal = "Phys.\ Rev.\ E",
volume = "57",
pages = "2739",
year = "1998",
abstract = "Change stability of maps to locate periodic orbits."
}
@Article{biham90c,
author = "O. Biham and W. Wenzel",
title = "Unstable periodic orbits and the symbolic dynamics of the complex {H}{\'e}non map",
journal = "Phys.\ Rev.\ E",
volume = "42",
pages = "4639",
year = "1990",
abstract = "Previous work is extended to the complex plane so that
2^n complex orbits are found for symbol sequences of length n."
}
@Article{ks04com,
author = "A.-K. Kassam and L. N. Trefethen",
title = "Fourth-order time stepping for stiff {PDE}s",
journal = "SIAM J. Sci. Comp.",
volume = "26",
pages = "1214--1233",
year = "2005",
abstract = "A contour integral method is presented to evaluate
accurately the matrix functions with removable singularities."
}
@Article{cpmani,
author = "A. J. Homburg and H. M. Osinga and G. Vegter",
title = "On the computation of invariant manifolds of fixed points",
journal = "Z.\ Angew.\ Math.\ Phys.",
volume = "46",
pages = "171",
year = "1995"
}
@Article{DDF00,
author = {J.W. Demmel and L. Dieci and M.J. Friedman},
title = {Computing connecting orbits via an improved algorithm
for continuing invariant spaces},
journal = {SIAM J.\ Sci.\ Comput.},
year = {2000},
volume = {22},
pages = {81-94}
}
@Article{foias88,
author = "C. Foias and M. S. Jolly and I. G. Kevrekidis and G. R. Sell and E. S. Titi",
title = "On the computation of inertial manifolds",
journal = "Phys.\ Lett.\ A",
volume = "131",
pages = "433",
year = "1988"
}
@Article{broer97,
author = "H. W. Broer and H. M. Osinga and G. Vegter",
title = "Algorithms for computing {N}ormally {H}yperbolic {I}nvariant {M}anifolds",
journal = "Z.\ Angew.\ Math.\ Phys.",
volume = "48",
pages = "480",
year = "1997"
}
@Article{misdis,
author = "K. Mischaikow and M. Mrozek and J. Reiss and A. Szymczak",
title = "Construction of Symbolic Dynamics from Experimental Time Series",
journal = "Phys.\ Rev.\ Lett.",
volume = "82",
pages = "1144",
year = "1999"
}
@Article{farmer87,
author = "J. D. Farmer and J. J. Sidorowich",
title = "Predicting Chaotic Time Series",
journal = "Phys.\ Rev.\ Lett.",
volume = "59",
pages = "845",
year = "1987"
}
@Article{lattref,
author = "B. I. Henry and S. D. Watt and S. L. Wearne",
title = "A lattice refinement scheme for finding periodic orbits",
journal = "Anziam J.",
volume = "42E",
pages = "C735--C751",
year = "2000",
abstract = "A lattice refinement scheme is proposed based on the
linear approximation of a given map. Jacobian matrix is not
calculated directly but inferred from the neighboring points."
}
@Article{grass89,
author = "P. Grassberger and H. Kantz and U. Moenig",
title = "On the symbolic dynamics of {H\'enon} map",
journal = "J.\ Phys.\ A",
volume = "22",
pages = "5217--5230",
year = "1989",
abstract = "Discuss Biham-Wenzel method in {H}{\'e}non map and point
out its defficiency: 1. It converges to a limit cycle sometimes. 2.
Two symbol sequence converges to the same cycle."
}
@Article{hansen92,
author = "K. T. Hansen",
title = "Remarks on the symbolic dynamics for the {H}{\'e}non map",
journal = "Phys.\ Lett.\ A",
volume = "165",
pages = "100--104",
year = "1992",
abstract = "Discuss the change of the symbol sequence of one
particular periodic orbit in the {H}{\'e}non map."
}
@Article{mislor1,
author = "K. Mischaikow and M. Mrozek",
title = "Chaos in the {Lorenz} equations: {A} computer assisted proof {Part II}: {Details}",
journal = "Math.\ Comp.",
volume = "67",
pages = "1023--1046",
year = "1998"
}
@Article{postgal89,
author = "B. Garc{\'i}a-Archilla and J. Novo and E. S. Titi",
title = {Postprocessing the {G \"alerkin} method: a novel approach to
approximate inertial manifolds},
journal = "SIAM J.\ Numer.\ Anal.",
volume = "35",
pages = "941--972",
year = "1998",
}
@Article{nongal89,
author = "M. Martine and R. Temam",
title = {Nonlinear {G}{\"a}lerkin Methods},
journal = "SIAM J.\ Numer.\ Anal.",
volume = "26",
pages = "1139--1157",
year = "1989",
}
@Article{condisfam1,
author = "J.-A. Sepulchre and R. Mackay",
title = "Localized oscillations in conservative or dissipative
networks of weakly coupled autonomous oscillators",
journal = "Nonlinearity",
volume = "10",
pages = "679--713",
year = "1997"
}
@Article{hell78,
author = "R. Helleman and T. Bountis",
title = "Periodic solutions of arbitrary period, variational methods",
journal = "LNP",
volume = "93",
pages = "353",
year = "1978"
}
@Article{hhsys,
author = "M. H{\'e}non and C. Heiles",
title = "The applicability of the third integral of motion:
{Some} numerical experiments",
journal = "Astron.\ J.",
volume = "69",
pages = "73",
year = "1964"
}
@Article{Mather91,
author = "J. N. Mather",
title = "Variational construction of orbits of twist difeomorphisms",
journal = "J.\ Am.\ Math.\ Soc.",
volume = "4",
pages = "207--263",
year = "1991"
}
@Article{angen88,
author = "S. B. Angenent",
title = "The periodic orbits of an area preserving twist-map",
journal = "Commun.\ Math.\ Phys.",
volume = "115",
pages = "353--374",
year = "1988"
}
@Article{cgole92,
author = "C. Gol{\'e}",
title = "A new proof of the {A}ubry-{M}ather's theorem",
journal = "Math.\ Z.",
volume = "210",
pages = "441--448",
year = "1992",
abstract = "A auxiliary infinite gradient-like flow is constructed
for the area-preserving map such that orbits of the original system
are rest points of the new system. In this way, existence of orbits
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resolution merely increases the number of isolated modes.}
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of physical modes hyperbolically isolated from a set of
residual degrees of freedom, themselves individually
isolated from each other. In the context of dissipative
partial differential equations, our results imply that a
faithful numerical integration needs to incorporate at
least all physical modes and that increasing the resolution
merely increases the number of isolated modes.},
note = "{\tt arXiv:0807.5073}",
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Standard techniques for computing Lyapunov vectors produce
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for constructing the n most rapidly growing Lyapunov
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construction of the n most rapidly decaying Lyapunov
vectors from n decaying forward and n-1 decaying backward
singular vectors.}
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title = "{Frequency locking by external forcing in systems with
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note = "{\tt arXiv:1108.5990}",
year = {2011},
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