Hidden Degeneracies in Piecewise Smooth Dynamical Systems

Mike R. Jeffrey*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

13 Citations (Scopus)
383 Downloads (Pure)


When a flow suffers a discontinuity in its vector field at some switching surface, the flow can cross through or slide along the surface. Sliding along the switching surface can be understood as the flow along an invariant manifold inside a switching layer. It turns out that the usual method for finding sliding modes - the Filippov convex combination or Utkin equivalent control - results in a degeneracy in the switching layer whenever the flow is tangent to the switching surface from both sides. We derive the general result and analyze the simplest case here, where the flow curves parabolically on either side of the switching surface (the so-called fold-fold or two-fold singularities). The result is a set of zeros of the fast switching flow inside the layer, which is structurally unstable to perturbation by terms nonlinear in the switching parameter, terms such as (signx)2 [where the superscript does mean "squared"]. We provide structurally stable forms, and show that in this form the layer system is equivalent to a generic singularity of a two timescale system. Finally we show that the same degeneracy arises when a discontinuity is smoothed using standard regularization methods.

Original languageEnglish
Article number1650087
JournalInternational Journal of Bifurcation and Chaos
Issue number5
Publication statusPublished - 1 May 2016

Structured keywords

  • Engineering Mathematics Research Group


  • degeneracy
  • Discontinuity
  • Filippov
  • hidden
  • layer
  • sliding
  • structural stability


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