The generic unfolding of a codimension-two connection to a two-fold singularity of planar Filippov systems

TL;DR

This paper develops a bifurcation theory for 2-parameter families of planar Filippov systems with a codimension-two minimal set, analyzing the unfolding of a two-fold singularity and its bifurcation diagram.

Contribution

It introduces a qualitative analysis of bifurcations near a codimension-two two-fold singularity in Filippov systems, extending classical smooth bifurcation results to nonsmooth systems.

Findings

Bifurcation diagram of the elementary simple two-fold cycle

Characterization of the codimension-two scenario

Analysis of the regular trajectory connecting a two-fold singularity to itself

Abstract

Generic bifurcation theory was classically well developed for smooth differential systems, establishing results for $k$-parameter families of planar vector fields. In the present study we focus on a qualitative analysis of $2$-parameter families, $Z_{\alpha,\beta}$, of planar Filippov systems assuming that $Z_{0,0}$ presents a codimension-two minimal set. Such object, named elementary simple two-fold cycle, is characterized by a regular trajectory connecting a visible two-fold singularity to itself, for which the second derivative of the first return map is nonvanishing. We analyzed the codimension-two scenario through the exhibition of its bifurcation diagram.