Morse decomposition for gradient-like multi-valued autonomous and nonautonomous dynamical systems

Abstract

In this paper, we first prove that the property of being a gradientlike general dynamical system and the existence of a Morse decomposition are equivalent. Next, the stability of gradient-like general dynamical systems is analyzed. In particular, we show that a gradient-like general dynamical system is stable under perturbations, and that Morse sets are upper semicontinuous with respect to perturbations. Moreover, we prove that any solution of perturbed general dynamical systems should be close to some Morse set of the unperturbed gradient-like general dynamical system. We do not assume local compactness for the metric phase space X, unlike previous results in the literature. Finally, we extend the Morse decomposition theory of single-valued nonautonomous dynamical systems to the multi-valued case, without imposing any compactness of the parameter spaces.