Persistent topological features of dynamical systems

TL;DR

This paper introduces a method to analyze the topological features of dynamical systems' phase spaces using persistent homology, providing insights into chaotic behavior and embedding dimensions.

Contribution

It presents a novel approach to construct simplicial complexes from time series data that preserves topological features and enables advanced topological analysis.

Findings

Persistent homology reveals topological features related to chaos.

Topological properties are consistent across different regimes and embeddings.

The method offers advantages over complex network mappings.

Abstract

A general method for constructing simplicial complex from observed time series of dynamical systems based on the delay coordinate reconstruction procedure is presented. The obtained simplicial complex preserves all pertinent topological features of the reconstructed phase space and it may be analyzes from topological, combinatorial and algebraic aspects. In focus of this study is the computation of homology of the invariant set of some well known dynamical systems which display chaotic behavior. Persistent homology of simplicial complex and its relationship with the embedding dimensions are examined by studying the lifetime of topological features and topological noise. The consistency of topological properties for different dynamic regimes and embedding dimensions is examined. The obtained results shed new light on the topological properties of the reconstructed phase space and open up new possibilities for application of advanced topological methods. the method presented here may be used as a generic method for constructing simplicial complex from a scalar time series which has a number of advantages compared to the mapping of the time series to a complex network.