Persistence of equilibria as periodic solutions of forced systems

Given an autonomous system with an isolated equilibrium, we consider general periodic perturbations. We say that the equilibrium persists if it can be continued as a periodic solution. The question of persistence is very classical and we find that the search of sharp conditions is linked with Topology. Besides the topological degree, the notion of diffeotopy and Hopfʼs Theorem of homotopy classes play a role. For dimension two we find a complete characterization of persistence.

► Isolated equilibria of autonomous systems are perturbed. ► An equilibrium persists when it is continued as periodic solution. ► We find a characterization of persistence in two dimensions. ► Diffeotopies and homotopy classes play a role.