Bifurcations of piecewise smooth flows: perspectives, methodologies and open problems

Alessandro Colombo, M di Bernardo, S. J. Hogan, MR Jeffrey

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

78 Citations (Scopus)
425 Downloads (Pure)


In this paper the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study.
Original languageEnglish
Pages (from-to)1845-1860
Number of pages16
JournalPhysica D: Nonlinear Phenomena
Issue number22
Publication statusPublished - 15 Nov 2012

Bibliographical note

Sponsorship: Preprint document submitted for publication in the journal Physica D


  • Filippov
  • bifurcation


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