Equilibrium-Like Solutions of Asymptotically Autonomous Differential Equations

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 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.