On non-smooth slow-fast systems

Abstract

We deal with non-smooth differential systems $\dot{z}=X(z),z\in R^n,$ with discontinuity occurring in a codimension one smooth surface $\Sigma$. A regularization of $X$ is a 1-parameter family of smooth vector fields $X^{\delta },\delta >0$, satisfying that $X^{\delta }$ converges pointwise to $X$ in $R^n\setminus \Sigma$, when $\delta \to 0$. We work with two known regularizations: the classical one proposed by Sotomayor and Teixeira and its generalization, using non-monotonic transition functions. Using the techniques of geometric singular perturbation theory we study minimal sets of regularized systems. Moreover, non-smooth slow-fast systems are studied and the persistence of the sliding region by singular perturbations is analyzed.