Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics.

We undertake an exploration of recurrent patterns in the antisymmetric subspace of the one-dimensional Kuramoto-Sivashinsky system. For a small but already rather "turbulent" system, the long-time dynamics takes place on a low-dimensional invariant manifold. A set of equilibria offers a coarse geometrical partition of this manifold. The Newton descent method enables us to determine numerically a large number of unstable spatiotemporally periodic solutions. The attracting set appears surprisingly thin-its backbone consists of several Smale horseshoe repellers, well approximated by intrinsic local one-dimensional return maps, each with an approximate symbolic dynamics. The dynamics appears decomposable into chaotic dynamics within such local repellers, interspersed by rapid jumps between them.

[1]  P. Cvitanović,et al.  Spatiotemporal chaos in terms of unstable recurrent patterns , 1996, chao-dyn/9606016.

[2]  P. Holmes,et al.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 1996 .

[3]  John Guckenheimer,et al.  Kuramoto-Sivashinsky dynamics on the center-unstable manifold , 1989 .

[4]  Stéphane Zaleski,et al.  Order and complexity in the Kuramoto-Sivashinsky model of weakly turbulent interfaces , 1986 .

[5]  Hsueh-Chia Chang,et al.  Wave evolution on a falling film , 1994 .

[6]  S. Bouquet.  Hamiltonian structure and integrability of the stationary Kuramoto–Sivashinsky equation , 1995 .

[7]  I. Kevrekidis,et al.  Back in the saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation , 1990 .

[8]  Yueheng Lan,et al.  Variational method for finding periodic orbits in a general flow. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Hsueh-Chia Chang,et al.  Traveling waves on fluid interfaces: Normal form analysis of the Kuramoto–Sivashinsky equation , 1986 .

[10]  B. M. Fulk.  MATH , 1992 .

[11]  James Shipman,et al.  Wave Motion , 2006 .

[12]  G. Sivashinsky.  Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations , 1977 .

[13]  P. Cvitanović,et al.  BIFURCATION STRUCTURES IN MAPS OF HENON TYPE , 1998 .

[14]  D Michelson,et al.  Steady solutions of the Kuramoto-Sivashinsky equation , 1986 .

[15]  On the geometric structure of non-hyperbolic attractors , 1988 .

[16]  Ioannis G. Kevrekidis,et al.  The Oseberg Transition: Visualization of Global bifurcations for the Kuramoto-Sivashinsky equation , 2001, Int. J. Bifurc. Chaos.

[17]  Michael T. Heath,et al.  Relative Periodic Solutions of the Complex Ginzburg-Landau Equation , 2004, SIAM J. Appl. Dyn. Syst..

[18]  Auerbach,et al.  Exploring chaotic motion through periodic orbits. , 1987, Physical review letters.

[19]  Genta Kawahara,et al.  Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst , 2001, Journal of Fluid Mechanics.

[20]  E. Hopf.  A mathematical example displaying features of turbulence , 1948 .

[21]  Yoshiki Kuramoto,et al.  Diffusion-Induced Chaos in Reaction Systems , 1978 .

[22]  On kinetic Ising models in one dimension , 1988 .

[23]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[24]  Christiansen,et al.  Generating partition for the standard map. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[25]  Henry S Greenside,et al.  Microextensive chaos of a spatially extended system. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[27]  R. Grimshaw,et al.  Travelling wave solutions of the Kuramoto-Sivashinsky equation , 1988 .

[28]  James A. Yorke,et al.  A procedure for finding numerical trajectories on chaotic saddles , 1989 .

[29]  Cvitanovic,et al.  Topological and metric properties of Hénon-type strange attractors. , 1988, Physical review. A, General physics.

[30]  Christophe Letellier,et al.  The Symmetry of Chaos , 2007 .

[31]  J. Greene,et al.  The steady states of the Kuramoto-Sivashinsky equation , 1988 .

[32]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[33]  J. Hyman,et al.  The Kuramoto-Sivashinsky equation: a bridge between PDE's and dynamical systems , 1986 .

[34]  Silvina Ponce Dawson,et al.  The PIM-simplex method: an extension of the PIM-triple method to saddles with an arbitrary number of expanding directions , 1999 .