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

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 approximate embeddings of the chaotic attractor into three dimensions. By finding the projection 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 mirror each other. Hence, the organization of the periodic orbits is identical (up to a global change of sign) and consistent symbolic names for low-period UPOs are derived. This 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.