Simplicial Multivalued Maps and the Witness Complex for Dynamical Analysis of Time Series

TL;DR

This paper introduces a novel simplicial complex-based discretization method for topological analysis of time series from dynamical systems, aiming to improve the connection with invariant densities and facilitate Conley index computations.

Contribution

It proposes a witness complex approach for discretizing dynamics, providing conditions for the witness map to approximate the true dynamics and enabling topological invariants computation.

Findings

The witness map can approximate the dynamics under certain conditions.

The method is demonstrated on data from the Hénon map.

It offers a potentially more natural discretization aligned with invariant densities.

Abstract

Topology based analysis of time-series data from dynamical systems is powerful: it potentially allows for computer-based proofs of the existence of various classes of regular and chaotic invariant sets for high-dimensional dynamics. Standard methods are based on a cubical discretization of the dynamics and use the time series to construct an outer approximation of the underlying dynamical system. The resulting multivalued map can be used to compute the Conley index of isolated invariant sets of cubes. In this paper we introduce a discretization that uses instead a simplicial complex constructed from a witness-landmark relationship. The goal is to obtain a natural discretization that is more tightly connected with the invariant density of the time series itself. The time-ordering of the data also directly leads to a map on this simplicial complex that we call the witness map. We obtain conditions under which this witness map gives an outer approximation of the dynamics, and thus can be used to compute the Conley index of isolated invariant sets. The method is illustrated by a simple example using data from the classical H\'enon map.