Structurally Stable Discontinuous Vector Fields in the Plane

Let $M$ be a plane domain, partitioned into sub domains $N$ and $S$, with common border $D$. In $N$ and $S$ are defined vector fields $X$ and $Y$, respectively, leading to a discontinuous vector field $Z=(X,Y)$. This work pursues the stability and transition analysis of solutions between $N$ and $S$, started by Filippov and Kozlova and reformulated by Sotomayor and Teixeira in terms of the regularization method. This method consists in defining a one parameter family of continuous vector fields $Z_{\epsilon}$, by averaging $X$ and $Y$. This family approaches $Z$ when the parameter goes to zero. The results of Sotomayor and Teixeira providing conditions for the regularized vector fields to be structurally stable are extended and shown to be generic.