Regularization of piecewise smooth singularly perturbed systems

Abstract

In this work we deal with piecewise smooth singularly perturbed systems

{ F (x, y, ε)   if   h(x, y, ε) ≤ 0, 

x˙ =

εy˙ = H(x, y, ε).                     (1)

G(x, y, ε)   if   h(x, y, ε) ≥ 0,

In system (1), ε ∈ R is a non-negative small parameter, x ∈ Rn and y ∈ R denote the slow and fast variables, respectively, and F , G, h and H are Cr maps which vary differentially with respect to ε, with r ≥ 1. We study the Sotomayor–Teixeira regularization of periodic orbits of system (1) with ε = 0 and with ε > 0 sufficiently small. More specifically, we establish the persistence of periodic orbits with sewing or with sliding of system (1) with ε = 0 and with ε > 0 for their respective regularized systems.