Conley Index Theory and the Attractor-Repeller Decomposition for Differential Inclusions

TL;DR

This paper extends Conley index theory and the attractor-repeller decomposition to continuous-time set-valued dynamical systems, including differential inclusions like Filippov systems, highlighting their stability under perturbation.

Contribution

It introduces a framework for applying attractor-repeller decomposition and stability analysis to differential inclusions, expanding the scope of Conley theory.

Findings

Extended attractor-repeller decomposition to differential inclusions.

Proved stability of the decomposition under perturbations.

Applicable to Filippov systems and similar set-valued dynamics.

Abstract

The Conley index theory is a powerful topological tool for describing the basic structure of dynamical systems. One important feature of this theory is the attractor-repeller decomposition of isolated invariant sets. In this decomposition, all points in the invariant set belong to the attractor, its associated dual repeller, or a connecting region. In this connecting region, points tend towards the attractor in forwards time and the repeller in backwards time. This decomposition is also, in a certain topological sense, stable under perturbation. Conley theory is well-developed for flows and homomorphisms, and has also been extended to some more abstract settings such as semiflows and relations. In this paper we aim to extend the attractor-repeller decomposition, including its stability under perturbation, to continuous time set-valued dynamical systems. The most common of these systems are differential inclusions such as Filippov systems.