Slow-Fast Normal Forms Arising from Piecewise Smooth Vector Fields

We study planar piecewise smooth differential systems of the form [GRAPHICS] . where F : R-2 -> R is a smooth map having 0 as a regular value. We consider linear regularizations Z(epsilon)(phi) of Z by replacing sgn(F) by phi(F/epsilon) in the last equation, with epsilon > 0 small and phi being a transition function (not necessarily monotonic). Nonlinear regularizations of the vector field Z whose transition function is monotonic are considered too. It is a wellknown fact that the regularized system is a slow-fast system. In this paper, we study typical singularities of slow-fast systems that arise from (linear or nonlinear) regularizations, namely, fold, transcritical and pitchfork singularities. Furthermore, the dependence of the slow-fast system on the graphical properties of the transition function is investigated.