Regularization of hidden dynamics in piecewise smooth flows

TL;DR

This paper explores the relationship between differentiable and non-differentiable dynamics in piecewise smooth flows, demonstrating how regularization and pinching methods relate these systems through slow-fast dynamics and differential inclusions.

Contribution

It extends the understanding of nonlinear combinations in Filippov systems and links sliding phenomena to equivalent smooth slow-fast systems, advancing the theory of discontinuous dynamical systems.

Findings

Existence of differentiable slow-fast systems with equivalent sliding dynamics.

Extension of regularization and pinching methods to nonlinear combinations.

Demonstration of the equivalence between discontinuous and smooth systems in specific conditions.

Abstract

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.