Closing Lemma for piecewise smooth vector fields with a recurrent point

Abstract

In this paper we provide a positive answer for the C${}^0$-Closing Lemma in the context of $n$-dimensional piecewise smooth vector fields governed by the Filippov’s rules. So, given a model presenting a nontrivially recurrent point it is possible to consider a C${}^0$-close perturbation of it possessing a closed trajectory. Also, we conclude the paper proving the existence of a closed orbit around a T-singularity.