Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics

TL;DR

This paper explores the complex recurrent patterns in the turbulent Kuramoto-Sivashinsky system, revealing a low-dimensional structure with unstable periodic solutions and chaotic dynamics characterized by Smale horseshoe repellers.

Contribution

It introduces a variational method to numerically identify numerous unstable spatiotemporally periodic solutions in the Kuramoto-Sivashinsky system, advancing understanding of its invariant manifold structure.

Findings

Long-time dynamics confined to a low-dimensional manifold

Presence of multiple Smale horseshoe repellers

Dynamics decomposed into local chaotic regions with rapid transitions

Abstract

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.