Slow-fast normal forms arising from piecewise smooth vector fields

Abstract

We studied piecewise smooth differential systems of the form $$\dot{z} = Z(z) = \dfrac{1 + \operatorname{sgn}(F)}{2}X(z) + \dfrac{1 - \operatorname{sgn}(F)}{2}Y(z),$$ where $F: \mathbb{R}^{n}\rightarrow \mathbb{R}$ is a smooth map having 0 as a regular value. We consider linear regularizations of the vector field $Z$ given by $$\dot{z}= Z_{\varepsilon}(z) = \dfrac{1 + \varphi(F/\varepsilon)}{2}X(z) +\dfrac{1 - \varphi(F /\varepsilon)}{2}Y(z),$$where $\varphi$ is a transition function (not necessarily monotonic) and nonlinear regularizations of the vector field $Z$ whose transition function is monotonic. It is a well-known fact that the regularized system is a slow-fast system. The main contribution of this paper is the study of typical singularities of slow-fast systems that arise from (linear or nonlinear) regularizations. We developed an algorithm to construct suitable transition functions, and we apply these ideas in order to create slow-fast singularities from normal forms of piecewise smooth vector fields. We present examples of transition functions that, after regularization of a PSVF normal form, generate normally hyperbolic, fold, transcritical, and pitchfork singularities.