Nonlinear Sliding of Discontinuous Vector Fields and Singular Perturbation

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}