Nonautonomous Conley Index Theory: The Connecting Homomorphism

TL;DR

This paper extends Conley index theory to nonautonomous dynamical systems by expressing the homology Conley index as a direct limit, revealing connections between attractor-repeller decompositions and system trajectories.

Contribution

It introduces a nonautonomous homology Conley index as a direct limit and links nontrivial connecting homomorphisms to uniform connectedness in nonautonomous systems.

Findings

Nonautonomous homology Conley index expressed as a direct limit.

Nontrivial connecting homomorphism indicates uniform connectedness.

Generalizes classical Conley index theory to nonautonomous settings.

Abstract

Attractor-repeller decompositions of isolated invariant sets give rise to so-called connecting homomorphisms. These homomorphisms reveal information on the existence and structure of connecting trajectories of the underlying dynamical system. To give a meaningful generalization of this general principle to nonautonomous problems, the nonautonomous homology Conley index is expressed as a direct limit. Moreover, it is shown that a nontrivial connecting homomorphism implies, on the dynamical systems level, a sort of uniform connectedness of the attractor-repeller decomposition.