Birth of limit cycles bifurcating from a nonsmooth center

Abstract

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 Σ-center). We prove that any nondegenerate Σ-center is Σ-equivalent to a particular normal form Z0. Given a positive integer number k we explicitly construct families of piecewise smooth vector fields emerging from Z0 that have k hyperbolic limit cycles bifurcating from the nondegenerate Σ-center of Z0 (the same holds for k=∞). Moreover, we also exhibit families of piecewise smooth vector fields of codimension k emerging from Z0. As a consequence we prove that Z0 has infinite codimension.