What is the graph of a dynamical system?

TL;DR

This paper introduces a unified graph-theoretic framework for representing various dynamical systems using chain graphs based on epsilon-chains, capturing their fundamental properties and connectivity.

Contribution

It defines the concept of a chain graph for any dynamical system, linking nodes as limit sets and edges via trajectories, and proves their connectedness under mild conditions.

Findings

Chain graphs effectively represent dynamical systems.

Proved connectedness of chain graphs under mild hypotheses.

Applicable to diverse systems from maps to PDEs.

Abstract

Some of the basic properties of any dynamical system can be summarized by a graph. The dynamical systems in our theory run from maps like the logistic map to ordinary differential equations to dissipative partial differential equations. Our goal has been to define a meaningful concept of graph of any dynamical system. As a result, we base our definition of ``chain graph'' on ``epsilon-chains'', defining both nodes and edges of the graph in terms of chains. In particular, nodes are often maximal limit sets and there is an edge between two nodes if there is a trajectory whose forward limit set is in one node and its backward limit set is in the other. Our initial goal was to prove that every ``chain graph'' of a dynamical system is, in some sense, connected, and we prove connectedness under mild hypotheses.