Symmetry-driven layered dynamics in the Kuramoto-Sivashinsky equation

TL;DR

This paper reveals a layered organization of the state space in the Kuramoto-Sivashinsky equation, showing how invariant sets, chaotic attractors, and periodic orbits coexist and depend on initial energy and system parameters.

Contribution

It uncovers the layered structure of the state space and the systematic generation of invariant sets, linking symmetry to the dynamics in the Kuramoto-Sivashinsky equation.

Findings

Invariant sets coexist at fixed parameters

Periodic orbits decrease in period with increasing energy

Chaotic and periodic dynamics coexist in transitional regions

Abstract

In this work, we uncover a layered organization of the state space in the Kuramoto-Sivashinsky equation with periodic boundary conditions, in which multiple invariant sets coexist at fixed system parameters and are selected by the initial condition. Within this framework, both chaotic attractors and periodic orbits (traveling waves) can be systematically generated by amplifying a single initial condition and parameterized by the initial energy. As the energy increases, the period of the periodic orbits decreases according to an inverse scaling law. In transitional parameter regions, periodic dynamics at low initial energy is found to coexist with strange attractors at higher energy levels, revealing a unique layered landscape governed by the viscosity and the initial condition. We conjecture that this behavior is linked to continuous spatial translational symmetry, which is reflected in the degeneracy of the neutral part of the Lyapunov spectrum.