Geometric singular perturbation on a positive measure minimal set of a planar piecewise smooth vector field

Abstract

In this paper, we employ the geometric theory of singular perturbations to obtain detailed insights concerning a class of piecewise smooth vector fields exhibiting a positive measure minimal set. The canonical form used in our analysis represents a larger class of piecewise smooth systems, encompassing models of discontinuous harmonic oscillators. Through a desingularization process, which entails the application of a $C^n$-regularization function along with successive weighted blow-ups (directional, spherical and polar), we obtain an attractor for the trajectories of the desingularized vector field $Z_{\omega }$.