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

Abstract

<p style="text-indent:20px;">In this paper, we first prove that the property of being a gradient-like 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 semi-continuous 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 <inline-formula><tex-math id="M1">\begin{document}$ X $\end{document}</tex-math></inline-formula>, 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.