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.