Regularization of hidden dynamics in piecewise smooth flows

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.

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• Ageing of an oscillator due to frequency switching
• Sliding mode on tangential sets of Filippov systems
• Poincaré-Hopf Theorem for Filippov vector fields on 2-dimensional compact manifolds
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• On the Hausdorff dimension and Cantor set structure of sliding Shilnikov invariant sets
• Smoothing tautologies, hidden dynamics, and sigmoid asymptotics for piecewise smooth systems
• The ghosts of departed quantities in switches and transitions