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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 linearregularizations 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 (notnecessarily monotonic) and nonlinear regularizations of the vector field $Z$whose transition function is monotonic. It is a well-known fact that theregularized system is a slow-fast system. The main contribution of this paperis the study of typical singularities of slow-fast systems that arise from(linear or nonlinear) regularizations. We developed an algorithm to constructsuitable transition functions, and we apply these ideas in order to createslow-fast singularities from normal forms of piecewise smooth vector fields. Wepresent examples of transition functions that, after regularization of a PSVFnormal form, generate normally hyperbolic, fold, transcritical, and pitchforksingularities.

Slow-fast normal forms arising from piecewise smooth vector fields