Nonlinear Sliding of Discontinuous Vector Fields and Singular Perturbation

We consider piecewise smooth vector fields (PSVF) defined in open sets M subset of R-n with switching manifold being a smooth surface Sigma. We assume that M \ Sigma contains exactly two connected regions, namely Sigma(+) and Sigma(-). Then, the PSVF are given by pairs X = (X+ , X-), with X = X+ in Sigma(+) and X = X- in Sigma(-). A regularization of X is a 1-parameter family of smooth vector fields X-epsilon, epsilon > 0, satisfying that X-epsilon converges pointwise to X on M \ Sigma, when epsilon -> 0. Inspired by the Fenichel Theory, the sliding and sewing dynamics on the discontinuity locus Sigma can be defined as some sort of limit of the dynamics of a nearby smooth regularization X-epsilon. While the linear regularization requires that for every epsilon > 0 the regularized field X-epsilon is in the convex combination of X+ and X-, the nonlinear regularization requires only that X-epsilon is in a continuous combination of X+ and X-. We prove that, for both cases, the sliding dynamics on Sigma is determined by the reduced dynamics on the critical manifold of a singular perturbation problem. We apply our techniques in the description of the nonlinear regularization of normal forms of PSVF in R-2 and in R-3.