# Nonlinear Sliding of Discontinuous Vector Fields and Singular Perturbation

## TL;DR

This paper studies the behavior of piecewise smooth vector fields near discontinuities, introducing nonlinear regularization and linking sliding dynamics to singular perturbation theory.

## Contribution

It extends the analysis of sliding dynamics to nonlinear regularizations, connecting them to singular perturbation problems and critical manifolds.

## Key findings

- Sliding dynamics are determined by reduced singular perturbation systems.
- Nonlinear regularization broadens the class of smoothings beyond convex combinations.
- The approach generalizes Fenichel theory to discontinuous vector fields.

## Abstract

We consider piecewise smooth vector fields (PSVF) defined in open sets $M\subseteq R^n$ with switching manifold being a smooth surface $\Sigma$. The PSVF are given by pairs $X = (X_+, X_-)$, with $X = X_+$ in $\Sigma_+$ and $X = X_-$ in $\Sigma_-$ where $\Sigma _+$ and $\Sigma _-$ are the regions on $M$ separated by $\Sigma.$ A regularization of $X$ is a 1-parameter family of smooth vector fields $X^{\epsilon},\epsilon>0,$ satisfying that $X^{\epsilon}$ converges pointwise to $X$ on $M\setminus\Sigma$, when $\epsilon\rightarrow 0$. Inspired by the Fenichel Theory , the sliding and sewing dynamics on the discontinuity locus $\Sigma$ can be defined as some sort of limit of the dynamics of a nearby smooth regularization $X^{\epsilon}$. While the linear regularization requires that for every $\epsilon>0$ the regularized field $X^{\epsilon}$ is in the convex combination of $X_+ $ and $X_- $ the nonlinear regularization requires only that $X^{\epsilon}$ is in a continuous combination of $X_+ $ and $X_- $. We prove that for both cases, the sliding dynamics on $\Sigma$ is determined by the reduced dynamics on the critical manifold of a singular perturbation problem. \end{abstract}