Piecewise Implicit Differential Systems

In this article we deal with non-smooth dynamical systems expressed by a piecewise first order implicit differential equations of the form

$$begin{aligned} dot{x}=1,quad left( dot{y}right) ^2=left{ begin{array}{lll} g_1(x,y) quad text{ if }quad varphi (x,y)ge 0 g_2(x,y) quad text{ if }quad varphi (x,y)le 0 end{array},right. end{aligned}$$

where (g_1,g_2,varphi :Urightarrow mathbb {R}) are smooth functions and (Usubseteq mathbb {R}^2) is an open set. The main concern is to study sliding modes of such systems around some typical singularities. The novelty of our approach is that some singular perturbation problems of the form

$$begin{aligned} dot{x}= f(x,y,varepsilon ) ,quad (varepsilon dot{ y})^2=g ( x,y,varepsilon ) end{aligned}$$

arise when the Sotomayor–Teixeira regularization is applied with ((x, y) in U) , (varepsilon ge 0), and f, g smooth in all variables.