Local bifurcations in a class of piecewise-smooth Filippov systems with a nonregular switching curve via a nonlinear double regularization process

TL;DR

This paper investigates how bifurcations in certain piecewise-smooth systems are affected when these systems are smoothed, focusing on the preservation or alteration of bifurcation properties during the regularization process.

Contribution

It introduces a nonlinear double regularization method to analyze bifurcations in piecewise-smooth Filippov systems with nonregular switching curves, highlighting how bifurcation codimension and genericity are preserved or changed.

Findings

Bifurcations can be preserved or altered depending on the regularization.

The regularization process affects the codimension of bifurcations.

The generic nature of bifurcations may be maintained or lost during smoothing.

Abstract

We are interested in analyzing the preservation of bifurcations in a class of piecewise smooth vector fields with a nonregular switching set under a smoothing process that approximates them by smooth vector fields. We examine cases in which the codimension is either preserved or altered, as well as whether the generic nature of the bifurcation is maintained.