Topological dynamics on finite directed graphs

TL;DR

This paper explores the topological dynamics of semiflows generated by finite directed graphs, analyzing Morse decompositions, recurrence, and attractors, and connecting graph theory with Markov chains for hybrid systems.

Contribution

It establishes a framework linking semiflows on graphs with dynamical systems concepts, extending analysis under weaker assumptions and exploring connections to Markov chains.

Findings

Characterization of Morse decompositions on directed graphs

Analysis of recurrence behavior in semiflows

Connections established between graph theory, Markov chains, and dynamical systems

Abstract

In this work we establish that finite directed graphs give rise to semiflows on the power set of their nodes. We analyze the topological dynamics for semiflows on finite directed graphs by characterizing Morse decompositions, recurrence behavior and attractor-repeller pairs under weaker assumptions. As is expected, the discrete metric plays an important role in our constructions and their consequences. The connections between the semiflow, graph theory and Markov chains are here explored. We lay the foundation for a dynamical systems approach to hybrid systems with Markov chain type perturbations.