Persistence of Conley-Morse Graphs in Combinatorial Dynamical Systems

TL;DR

This paper introduces a method to analyze changes in combinatorial dynamical systems by capturing the evolution of Conley-Morse graphs, providing a more detailed understanding than previous approaches focused solely on Conley index or Morse decomposition.

Contribution

The paper presents a novel approach to summarize dynamical system changes through Conley-Morse graphs, integrating both Morse decomposition structure and Conley index information.

Findings

Captures finer details of dynamical changes

Provides a more comprehensive summary than previous methods

Enhances understanding of combinatorial dynamical systems

Abstract

Multivector fields provide an avenue for studying continuous dynamical systems in a combinatorial framework. There are currently two approaches in the literature which use persistent homology to capture changes in combinatorial dynamical systems. The first captures changes in the Conley index, while the second captures changes in the Morse decomposition. However, such approaches have limitations. The former approach only describes how the Conley index changes across a selected isolated invariant set though the dynamics can be much more complicated than the behavior of a single isolated invariant set. Likewise, considering a Morse decomposition omits much information about the individual Morse sets. In this paper, we propose a method to summarize changes in combinatorial dynamical systems by capturing changes in the so-called Conley-Morse graphs. A Conley-Morse graph contains information about both the structure of a selected Morse decomposition and about the Conley index at each Morse set in the decomposition. Hence, our method summarizes the changing structure of a sequence of dynamical systems at a finer granularity than previous approaches.