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

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