On Non-Smooth Perturbations of Degenerate or Non-Degenerate Planar Centers

We provide sufficient conditions for the existence of limit cycles of non–smooth perturbed planar centers, when the set of discontinuity is an algebraic variety. It is introduced a mechanism which allows us to deal with such system, even in higher dimension. The main tool used in this paper is the averaging method. Two applications are given in careful detail.