Bifurcações genéricas em sistemas planares de Filippov

Consider a planar Filippov system Z=(X,Y), where X and Y are smooth vector fields defined in a neighborhood of the origin and whose discontinuity curve is given by the set of zeros of f(x,y)=y. In this work we present a rigorous study of the fold-fold singularities of a planar Filippov system. We show that, under some generic assumptions, the set of the planar Filippov systems having a fold-fold singularity at the origin is an embedded codimension one submanifold contained in the set of all planar Filippov systems. In addition, we show that all the unfoldings of Z=(X,Y) belonging to this submanifold are equivalent. We also consider piecewise smooth systems of the kind Z=(X,Y) satisfying Z(x,y)=X(x,y) if xy> 0 and Z(x,y)=Y(x,y) if xy<0 . In this case, the discontinuity set is the set of zeros of f(x,y)=xy. For these systems, we present a classification of structurally stable and generic codimension one singularities. In addition, we present the bifurcation diagram of each codimension one singularity and we show that they are, in fact, universal unfoldings. In the sequel we study the Teixeira-Sotomayor regularization of planar Filippov systems having a fold-fold singularity and whose regularization has a critical point around the origin. In this context, we study the nature of this critical point and when the critical point presents a bifurcation, we study the relations between the bifurcation for the planar Filippov system and for the regularized system (AU)