Tracking Dynamical Features via Continuation and Persistence

TL;DR

This paper introduces a method to track isolated invariant sets in combinatorial dynamical systems across multivector fields using continuation and persistence, enabling analysis of dynamical features over sequences.

Contribution

It develops a novel 'Tracking Protocol' for continuation of invariant sets and connects continuation with persistence in a combinatorial setting.

Findings

Introduces a 'Tracking Protocol' for invariant set continuation.

Uses zigzag persistence to track homological features.

Shows continuation as a special case of persistence.

Abstract

Multivector fields and combinatorial dynamical systems have recently become a subject of interest due to their potential for use in computational methods. In this paper, we develop a method to track an isolated invariant set -- a salient feature of a combinatorial dynamical system -- across a sequence of multivector fields. This goal is attained by placing the classical notion of the "continuation" of an isolated invariant set in the combinatorial setting. In particular, we give a "Tracking Protocol" that, when given a seed isolated invariant set, finds a canonical continuation of the seed across a sequence of multivector fields. In cases where it is not possible to continue, we show how to use zigzag persistence to track homological features associated with the isolated invariant sets. This construction permits viewing continuation as a special case of persistence.