Slow-Fast Normal Forms Arising from Piecewise Smooth Vector Fields

Abstract

We study planar piecewise smooth differential systems of the form z˙=Z(z)=1+sgn(F)2X(z)+1-sgn(F)2Y(z), where F: R2→ R is a smooth map having 0 as a regular value. We consider linear regularizations Zεφ of Z by replacing sgn (F) by φ(F/ ε) in the last equation, with ε> 0 small and φ 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 well-known 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.