Unstable manifolds of relative periodic orbits in the symmetry-reduced state space of the Kuramoto-Sivashinsky system

TL;DR

This paper introduces a new symmetry reduction method for analyzing systems with continuous and discrete symmetries, enabling detailed visualization of unstable manifolds and bifurcations of relative periodic orbits in the Kuramoto-Sivashinsky system.

Contribution

The paper develops a novel symmetry reduction scheme combining slices and invariant polynomial methods, applied to the Kuramoto-Sivashinsky system, facilitating analysis of complex bifurcations and unstable manifolds.

Findings

Computed and visualized unstable manifolds of relative periodic orbits

Identified bifurcations leading to chaos via torus breakdown

Mapped heteroclinic connections between relative periodic orbits

Abstract

Systems such as fluid flows in channels and pipes or the complex Ginzburg-Landau system, defined over periodic domains, exhibit both continuous symmetries, translational and rotational, as well as discrete symmetries under spatial reflections or complex conjugation. The simplest, and very common symmetry of this type is the equivariance of the defining equations under the orthogonal group O(2). We formulate a novel symmetry reduction scheme for such systems by combining the method of slices with invariant polynomial methods, and show how it works by applying it to the Kuramoto-Sivashinsky system in one spatial dimension. As an example, we track a relative periodic orbit through a sequence of bifurcations to the onset of chaos. Within the symmetry-reduced state space we are able to compute and visualize the unstable manifolds of relative periodic orbits, their torus bifurcations, a transition to chaos via torus breakdown, and heteroclinic connections between various relative periodic orbits. It would be very hard to carry through such analysis in the full state space, without a symmetry reduction such as the one we present here.