Equilibrium-Like Solutions of Asymptotically Autonomous Differential Equations

TL;DR

This paper extends Morse-Smale dynamical system concepts to asymptotically autonomous differential equations using a nonautonomous Conley index, revealing structured solutions and equilibrium connections.

Contribution

It introduces a nonautonomous Conley index framework to analyze equilibrium-like solutions in asymptotically autonomous systems, generalizing Morse-Smale properties.

Findings

Identifies equilibrium-like solutions in asymptotically autonomous equations.

Shows the structure of chain recurrent sets is close to Morse-Smale systems.

Provides tools to understand connections between generalized equilibria.

Abstract

We analyze the chain recurrent set of skew product semiflows obtained from nonautonomous differential equations -- ordinary differential equations or semilinear parabolic differential equations. For many gradient-like dynamical systems, Morse-Smale dynamical systems e.g., the chain recurrent set contains only isolated equilibria. The structure in the asymptotically autonomous setting is richer but still close to to the structure of a Morse-Smale dynamical system. The main tool used in this paper is a nonautonomous flavour of Conley index theory developed by the author. We will see that for a class of good equations, the Conley index can be understood in terms of equilibria (in a generalized meaning) and their connections. This allows us to find specific solutions of asymptotically autonomous equations and generalizes properties of Morse-Smale dynamical systems to the asymptotically autonomous setting.