On the state space geometry of the Kuramoto-Sivashinsky flow in a periodic domain

TL;DR

This paper explores the geometric structure of the state space in the Kuramoto-Sivashinsky system with periodic boundary conditions, highlighting how symmetries shape invariant sets and the organization of chaotic dynamics.

Contribution

It provides a detailed analysis of how continuous and discrete symmetries influence the invariant structures and heteroclinic connections in the state space of the Kuramoto-Sivashinsky system.

Findings

Heteroclinic connections form a 'cage' organizing the state space.

Unstable relative periodic orbits are prevalent and form a dynamical skeleton.

Novel low-dimensional visualizations reveal the geometry of the high-dimensional flow.

Abstract

The continuous and discrete symmetries of the Kuramoto-Sivashinsky system restricted to a spatially periodic domain play a prominent role in shaping the invariant sets of its chaotic dynamics. The continuous spatial translation symmetry leads to relative equilibrium (traveling wave) and relative periodic orbit (modulated traveling wave) solutions. The discrete symmetries lead to existence of equilibrium and periodic orbit solutions, induce decomposition of state space into invariant subspaces, and enforce certain structurally stable heteroclinic connections between equilibria. We show, on the example of a particular small-cell Kuramoto-Sivashinsky system, how the geometry of its dynamical state space is organized by a rigid `cage' built by heteroclinic connections between equilibria, and demonstrate the preponderance of unstable relative periodic orbits and their likely role as the skeleton underpinning spatiotemporal turbulence in systems with continuous symmetries. We also offer novel visualizations of the high-dimensional Kuramoto-Sivashinsky state space flow through projections onto low-dimensional, PDE representation independent, dynamically invariant intrinsic coordinate frames, as well as in terms of the physical, symmetry invariant energy transfer rates.