Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics

Details Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics

2008-04-15

arxiv.org/abs/0804.2474v1

Nonlinear Sciences

Pattern Formation and Solitons

11 pages, 11 figures

Scientific paper

10.1103/PhysRevE.78.026208

We undertake a systematic exploration of recurrent patterns in a 1-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. A variational method enables us to determine numerically a large number of unstable spatiotemporally periodic solutions. The attracting set appears surprisingly thin - its backbone are several Smale horseshoe repellers, well approximated by intrinsic local 1-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.

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