Two regularizations of the grazing-sliding bifurcation giving non equivalent dynamics

TL;DR

This paper compares two regularization methods for a specific bifurcation in piecewise smooth systems, revealing that one leads to saddle-node bifurcations while the other induces chaos, highlighting their different dynamical outcomes.

Contribution

It introduces and analyzes two regularization techniques for grazing-sliding bifurcations, demonstrating their distinct effects on system dynamics.

Findings

Sotomayor-Teixeira regularization results in saddle-node bifurcations.

Hysteretic regularization leads to chaotic dynamics.

Both regularizations agree in sliding modes but differ at tangencies.

Abstract

We present two ways of regularizing a one parameter family of piece-wise smooth dynamical systems undergoing a codimension one grazing-sliding global bifurcation of periodic orbits. First we use the Sotomayor-Teixeira regularization and prove that the regularized family has a saddle-node bifurcation of periodic orbits. Then we perform a hysteretic regularization and show that the regularized family has chaotic dynamics. Our result shows that, in spite that the two regularizations will give the same dynamics in the sliding modes, when a tangency appears the hysteretic process generates chaotic dynamics.