Birth of limit cycles bifurcating from a nonsmooth center

Abstract

We will perform a codimension analysis of a two-fold singularity of piecewise smooth planar vector fields, when it behaves itself like a center of smooth vector fields  (also called nondegenerate $\Sigma$-center). Given a positive integer number k we explicitly construct families of piecewise smooth vector fields emerging from the nondegenerate $\Sigma$-center that have k hyperbolic limit cycles bifurcating from it. Moreover, we also exhibit families of piecewise smooth vector fields of codimension k also emerging from the nondegenerate $\Sigma$-center. As a consequence we prove that it has infinite codimension. It is a joint work with Tiago de Carvalho and Marco Antonio Teixeira.