We consider piecewise smooth vector fields (PSVF) defined in open sets M ⊆ R n with switching manifold being a smooth surface Σ. The PSVF are given by pairs X = (X+, X−), with X = X+ in Σ+ and X = X− in Σ− where Σ+ and Σ− are the regions on M separated by Σ. A regularization of X is a 1-parameter family of smooth vector fields X ε , ε > 0, satisfying that X ε converges pointwise to X on M \ Σ, when ε → 0. Inspired by the Fenichel Theory [6], the sliding and sewing dynamics on the discontinuity locus Σ can be defined as some sort of limit of the dynamics of a nearby smooth regularization X ε . While the linear regularization requires that for every ε > 0 the regularized field X ε is in the convex combination of X+ and X− the nonlinear regularization requires only that X ε is in a continuous combination of X+ and X−. We prove that for both cases, the sliding dynamics on Σ is determined by the reduced dynamics on the critical manifold of a singular perturbation problem.2010 Mathematics Subject Classification. Primary 34C20, 34C26, 34D15, 34H05.

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