Nonautonomous Conley Index Theory: The Homology Index and Attractor-Repeller decompositions

TL;DR

This paper extends Conley index theory to nonautonomous systems, introducing homology indices and attractor-repeller decompositions to analyze the qualitative behavior of such dynamical systems.

Contribution

It develops a refined nonautonomous Conley index framework, including homology indices and attractor-repeller decompositions, for families of nonautonomous evolution operators.

Findings

Introduces a homology Conley index for nonautonomous systems

Establishes attractor-repeller decompositions in the nonautonomous setting

Defines connecting homomorphisms for nonautonomous dynamics

Abstract

In a previous work, the author established a nonautonomous Conley index based on the interplay between a nonautonomous evolution operator and its skew-product formulation. This index is refined to obtain a Conley index for families of nonautonomous evolution operators. Different variants such as a categorial index, a homotopy index and a homology index are obtained. Furthermore, attractor-repeller decompositions and conecting homomorphisms are introduced for the nonautonomous setting.