Recent results on the stability of time dependent sets and their application to bifurcation problems

Abstract

In the first part of the paper we give a short review of our recent results concerning the relationship between conditional and unconditional stability properties of time dependent sets, under smooth differential systems in ${\mathbf{R}}^n$. More precisely, let $M$ be an ‘‘$s$-compact’’ invariant set in $\mathbf{R}\times {\mathbf{R}}^n$ and let $\Phi$ be a smooth invariant set in $\mathbf{R}\times {\mathbf{R}}^n$ containing $M$. It is assumed that $M$ is uniformly asymptotically stable with respect to the perturbations lying on $\Phi$. The unconditional stability properties of $M$ depend on the stability properties of $\Phi$ ‘‘near $M$’’. This dependence has been analyzed in general, and, in the periodic case, complete characterizations are obtained. In the second part, the above results have been applied to bifurcation problems for periodic differential systems. Some our previous statements on the matter are revisited and enriched.