  This paper studies the equivalence between differentiable and
non-differentiable dynamics in Rn. Filippov's theory of discontinuous
differential equations allows us to find flow solutions of dynamical systems
whose vector fields undergo switches at thresholds in phase space. The
canonical convex combination at the discontinuity is only the linear part of a
nonlinear combination that more fully explores Filippov's most general problem:
the differential inclusion. Here we show how recent work relating discontinuous
systems to singular limits of continuous (or regularized) systems extends to
nonlinear combinations. We show that if sliding occurs in a discontinuous
systems, there exists a differentiable slow-fast system with equivalent slow
invariant dynamics. We also show the corresponding result for the pinching
method, a converse to regularization which approximates a smooth system by a
discontinuous one.
