Birth of limit cycles bifurcating from a nonsmooth center

This paper is concerned with 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). We prove that any nondegenerate Sigma-center is Sigma-equivalent to a particular normal form Z(0). Given a positive integer number k we explicitly construct families of piecewise smooth vector fields emerging from Z(0) that have k hyperbolic limit cycles bifurcating from the nondegenerate Sigma-center of Z(0) (the same holds for k = infinity). Moreover, we also exhibit families of piecewise smooth vector fields of codimension k emerging from Z(0). As a consequence we prove that Z(0) has infinite codimension. (c) 2013 Elsevier Masson SAS. All rights reserved.