Templex: a bridge between homologies and templates for chaotic attractors

TL;DR

This paper introduces Templex, a novel framework combining cell complexes and directed graphs to better characterize and classify chaotic attractors by incorporating flow dynamics into topological analysis.

Contribution

It extends homology-based analysis by integrating flow action through directed graphs, creating a new tool called Templex for analyzing chaotic attractors.

Findings

Templex accurately classifies well-known chaotic attractors.

The approach links topological and flow-based descriptions of attractors.

It provides a refined characterization compared to previous methods.

Abstract

The theory of homologies introduces cell complexes to provide an algebraic description of spaces up to topological equivalence. Attractors in state space can be studied using Branched Manifold Analysis through Homologies: this strategy constructs a cell complex from a cloud of points in state space and uses homology groups to characterize its topology. The approach, however, does not consider the action of the flow on the cell complex. The procedure is here extended to take this fundamental property into account, as done with templates. The goal is achieved endowing the cell complex with a directed graph that prescribes the flow direction between its highest dimensional cells. The tandem of cell complex and directed graph, baptized templex, is shown to allow for a sophisticated characterization of chaotic attractors and for an accurate classification of them. The cases of a few well-known chaotic attractors are investigated -- namely the spiral and funnel R\"ossler attractors, the Lorenz attractor and the Burke and Shaw attractor. A link is established with their description in terms of templates.