Conley-Morse-Forman theory for combinatorial multivector fields

TL;DR

This paper introduces combinatorial multivector fields, extending Forman's combinatorial vector fields, to analyze dynamical systems topologically and algorithmically, with applications in differential equations and sampled dynamics.

Contribution

It develops a new framework for combinatorial multivector fields, defining key dynamical concepts and providing a prototype algorithm for practical analysis.

Findings

Defined isolated invariant sets, Conley index, attractors, repellers, Morse decompositions.

Proved Morse inequalities for the new framework.

Provided a prototype algorithm for analyzing dynamical systems.

Abstract

We introduce combinatorial multivector fields, associate with them multivalued dynamics and study their topological features. Our combinatorial multivector fields generalize combinatorial vector fields of Forman. We define isolated invariant sets, Conley index, attractors, repellers and Morse decompositions. We provide a topological characterization of attractors and repellers and prove Morse inequalities. The generalization aims at algorithmic analysis of dynamical systems through combinatorialization of flows given by differential equations and through sampling dynamics in physical and numerical experiments. We provide a prototype algorithm for such applications.