The topology of a chaotic attractor in the Kuramoto-Sivashinsky equation

TL;DR

This paper investigates the topological structure of a chaotic attractor in the Kuramoto-Sivashinsky PDE by using dimensionality reduction and topological analysis of periodic orbits, providing a robust characterization of its topology.

Contribution

It demonstrates that different dimensionality reduction methods yield consistent topological templates for the attractor, advancing understanding of PDE chaos topology.

Findings

Templates from two reduction methods are equivalent

Symbolic sequences for low-period UPOs are derived

Dimensional reduction is shown to be robust for topological analysis

Abstract

The Birman-Williams theorem gives a connection between the collection of unstable periodic orbits (UPOs) contained within a chaotic attractor and the topology of that attractor, for three-dimensional systems. In certain cases, the fractal dimension of a chaotic attractor in a partial differential equation (PDE) is less than three, even though that attractor is embedded within an infinite-dimensional space. Here we study the Kuramoto-Sivashinsky PDE at the onset of chaos. We use two different dimensionality-reduction techniques -- proper orthogonal decomposition and an autoencoder neural network -- to find two different mappings of the chaotic attractor into three dimensions. By finding the image of the attractor's UPOs in these reduced spaces and examining their linking numbers, we construct templates for the branched manifold which encodes the topological properties of the attractor. The templates obtained using two different dimensionality reduction methods are equivalent. The organization of the periodic orbits is identical and consistent symbolic sequences for low-period UPOs are derived. While this is not a formal mathematical proof, this agreement is strong evidence that the dimensional reduction is robust, in this case, and that an accurate topological characterization of the chaotic attractor of the chaotic PDE has been achieved.