Regularization around a generic codimension one fold-fold singularity

TL;DR

This paper analyzes the fold-fold singularity in Filippov systems, exploring its bifurcations, regularizations, and the emergence of periodic orbits using geometric singular perturbation theory.

Contribution

It provides a comprehensive bifurcation analysis of the fold-fold singularity, including unfoldings and regularizations, and links regularized systems to known slow-fast systems.

Findings

Bifurcation diagrams of the fold-fold singularity are characterized.

Regularization reveals new periodic orbits not present in the Filippov system.

The regularized system is equivalent to a known slow-fast system by Krupa and Szmolyan.

Abstract

This paper is devoted to study the generic fold-fold singularity of Filippov systems on the plane, its unfoldings and its Sotomayor-Teixeira regularization. We work with general Filippov systems and provide the bifurcation diagrams of the fold-fold singularity and their unfoldings, proving that, under some generic conditions, is a codimension one embedded submanifold of the set of all Filippov systems. The regularization of this singularity is studied and its bifurcation diagram is shown. In the visible-invisible case, the use of geometric singular perturbation theory has been useful to give the complete diagram of the unfolding, specially the appearance and disappearance of periodic orbits that are not present in the Filippov vector field. In the case of a linear regularization, we prove that the regularized system is equivalent to a general slow-fast system studied by Krupa and Szmolyan [KrupaS01].