On non-smooth slow-fast systems

TL;DR

This paper investigates non-smooth differential systems with discontinuities, analyzing their regularizations and slow-fast dynamics using geometric singular perturbation theory to understand minimal sets and sliding region persistence.

Contribution

It introduces a detailed analysis of regularizations of non-smooth systems, including classical and generalized methods, and studies their slow-fast behavior and minimal sets.

Findings

Regularizations converge to the original non-smooth system outside the discontinuity surface.

Persistence of sliding regions under singular perturbations is established.

Minimal sets of regularized systems are characterized using geometric singular perturbation theory.

Abstract

We deal with non-smooth differential systems $\dot{z}=X(z), z\in R^{n},$ with discontinuity occurring in a codimension one smooth surface $\Sigma$. A regularization of $X$ is a 1-parameter family of smooth vector fields $X^{\delta},\delta>0$, satisfying that $X^{\delta}$ converges pointwise to $X$ in $R^{n}\setminus\Sigma$, when $\delta\rightarrow 0$. We work with two known regularizations: the classical one proposed by Sotomayor and Teixeira and its generalization, using non-monotonic transition functions. Using the techniques of geometric singular perturbation theory we study minimal sets of regularized systems. Moreover, non-smooth slow-fast systems are studied and the persistence of the sliding region by singular perturbations is analyzed.