Abstract:

Our aim is to study the dynamics of a discontinuous system when its discontinuity set belongs to a class of algebraic sets. In order to do this we first consider $F:mathcal{U} ightarrowmathbb R$ a polynomial function defined on the open subset $mathcal{U}subset mathbb R^n$. The set $F^{-1}(0)$ divides $mathcal U$ into subdomains $mathcal U_1,mathcal U_2,...,mathcal U_k,$ with border $F^{-1}(0)$. These subdomains provide a Whitney stratification on $mathcal U$. We consider $Z_i:mathcal U_i ightarrowmathbb R^n$ smooth vector fields and we get $Z = (Z_1,...,Z_k )$ a discontinuous vector field with discontinuities in $F^{-1}(0)$. Our approach combines several techniques such as $ m e$-regularization process, blowing-up method and singular perturbation theory.