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

Abstract

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