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Regularization of hidden dynamics in piecewise smooth flows

Research output: Contribution to journalArticle

Original languageEnglish
Pages (from-to)4615-4633
Number of pages19
JournalJournal of Differential Equations
Issue number9
Early online date16 Jun 2015
DateAccepted/In press - 2 Jun 2015
DateE-pub ahead of print - 16 Jun 2015
DatePublished (current) - 5 Nov 2015


This paper studies the equivalence between differentiable and non-differentiable dynamics in R^n. 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.


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