Piecewise smooth dynamical systems theory: the case of the missing boundary equilibrium bifurcations

John Hogan, Martin Homer*, Mike Jeffrey, Robert Szalai

*Corresponding author for this work

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

26 Citations (Scopus)
387 Downloads (Pure)


We present two codimension-one bifurcations that occur when an equilibrium collides with a discontinuity in a piecewise smooth dynamical system. These simple cases appear to have escaped recent classifications. We present them here to highlight some of the powerful results from Filippov's book Differential Equations with Discontinuous Righthand Sides (Kluwer, 1988). Filippov classified the so-called boundary equilibrium collisions without providing their unfolding. We show the complete unfolding here, for the first time, in the particularly interesting case of a node changing its stability as it collides with a discontinuity. We provide a prototypical model that can be used to generate all codimension-one boundary equilibrium collisions, and summarize the elements of Filippov's work that are important in achieving a full classification.
Original languageEnglish
Pages (from-to)1161-1173
Number of pages13
JournalJournal of Nonlinear Science
Issue number5
Early online date20 May 2016
Publication statusPublished - Oct 2016


  • Bifurcation
  • Boundary equilibrium collision
  • Dynamical systems
  • Piecewise smooth


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