Piecewise Implicit Differential Systems - Lopes, Bruno D. [UNESP]; da Silva, Paulo R. [UNESP]; Teixeira, Marco A. (Springer Nature, 2017)
In this article we deal with non-smooth dynamical systems expressed by a piecewise first order implicit differential equations of the form x˙=1,(y˙)2={g1(x,y)ifφ(x,y)≥0g2(x,y)ifφ(x,y)≤0,where g1, g2, φ: U→ R are smooth functions and U⊆ R2 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 x˙=f(x,y,ε),(εy˙)2=g(x,y,ε)arise when the Sotomayor–Teixeira regularization is applied with (x, y) ∈ U , ε≥ 0 , and f, g smooth in all variables.
Keywords: Implicit differential equation; Non-smooth dynamical system; Singular perturbation; Sliding vector fields