An asymptotically stable cusp-fold singularity in 3D piecewise smooth vector fields

TL;DR

This paper analyzes a specific cusp-fold singularity in 3D piecewise smooth vector fields, demonstrating asymptotic stability through a normal form, advancing understanding of stability in such systems.

Contribution

It introduces a normal form for the cusp-fold singularity in 3D piecewise smooth vector fields and proves its asymptotic stability, a key property for these systems.

Findings

Normal form for cusp-fold singularity in 3D

Proof of asymptotic stability for the normal form

Enhanced understanding of stability in piecewise smooth systems

Abstract

This paper is concerned with the analysis of a typical singularity of piecewise smooth vector fields on $R^3$ composed by two zones. In our object of study, the cusp-fold singularity, we consider the simultaneous occurrence of a cusp singularity for one vector field and a fold singularity for the other one. We exhibit a normal form that presents one of the most important property searched for in piecewise smooth vector fields: the asymptotical stability.