LAUSR

We consider piecewise smooth vector fields (PSVF) defined in open sets $$M\subseteq \mathbb {R}^n$$M⊆Rn with switching manifold being a smooth surface $$\Sigma $$Σ. We assume that $$M{\setminus }\Sigma $$M\Σ contains exactly two connected regions, namely $$\Sigma _+$$Σ+ and $$\Sigma _-$$Σ-. Then, the PSVF are given by pairs $$X = (X_+, X_-)$$X=(X+,X-), with $$X = X_+$$X=X+ in $$\Sigma _+$$Σ+ and $$X = X_-$$X=X- in $$\Sigma _-.$$Σ-. A regularization of X is a 1-parameter family of smooth vector fields $$X^{\varepsilon }$$Xε, $$\varepsilon >0,$$ε>0, satisfying that $$X^{\varepsilon }$$Xε converges pointwise to X on $$M{\setminus }\Sigma $$M\Σ, when $$\varepsilon \rightarrow 0$$ε→0. Inspired by the Fenichel Theory, the sliding and sewing dynamics on the discontinuity locus $$\Sigma $$Σ can be defined as some sort of limit of the dynamics of a nearby smooth regularization $$X^{\varepsilon }$$Xε. While the linear regularization requires that for every $$\varepsilon >0$$ε>0 the regularized field $$X^{\varepsilon }$$Xε is in the convex combination of $$X_+ $$X+ and $$X_- $$X-, the nonlinear regularization requires only that $$X^{\varepsilon }$$Xε is in a continuous combination of $$X_+ $$X+ and $$X_- $$X-. We prove that, for both cases, the sliding dynamics on $$\Sigma $$Σ 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 $$\mathbb {R}^2$$R2 and in $$\mathbb {R}^3$$R3.