Slow--fast systems and sliding on codimension 2 switching manifolds

TL;DR

This paper studies piecewise smooth vector fields with complex switching manifolds using a double regularization approach, transforming the problem into a slow-fast system analyzed via geometric singular perturbation theory.

Contribution

It introduces a double regularization method for vector fields with self-intersecting switching manifolds, enabling analysis through slow-fast system techniques.

Findings

Defined sliding regions as limits of invariant manifolds.

Applied geometric singular perturbation theory to analyze the regularized systems.

Provided a framework for understanding complex switching dynamics.

Abstract

In this work we consider piecewise smooth vector fields $X$ defined in $\R^n\setminus \Sigma$, where $\Sigma$ is a self-intersecting switching manifold. A double regularization of $X$ is a 2-parameter family of smooth vector fields $X_{\e.\eta}$, $\e,\eta>0,$ satisfying that $X_{\e,\eta}$ converges pointwise to $X$ on $\R^n\setminus\Sigma$, when $\e,\eta\rightarrow 0$. We define the sliding region on the non regular part of $\Sigma$ as a limit of invariant manifolds of $X_{\e.\eta}$. Since the double regularization provides a slow--fast system, the GSP-theory (geometric singular perturbation theory) is our main tool.