Limit sets near a cusp-fold singularity in a class of 3D non-smooth vector fields

Abstract

We consider a class of three-dimensional non-smooth vector fields presenting a {\it cusp-fold singularity}. We develop a local approach to classify the limit sets of points positioned near the cusp-fold. The results state the existence of center manifolds connecting to the cusp-fold, the absence of isolated periodic orbits of any kind and sufficient conditions for the asymptotic stability of the cusp-fold. The results bring new, relevant material to the subject of structural stability and bifurcations of co-dimension one in three-dimensional non-smooth vector fields. Mathematics Subject Classification (2020) MSC 34A36 · 34C05 · 37C27