The two-fold singularity of discontinuous vector fields

MR Jeffrey, A Colombo

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

68 Citations (Scopus)
295 Downloads (Pure)


When a vector field in 3 dimensions is discontinuous on a smooth codimension one surface, it may be simultaneously tangent to both sides of the surface at generic isolated points (singularities). For a piecewise-smooth dynamical system governed by the vector field, we show that the local dynamics depends on a single quantity: the jump in direction of the vector field through the singularity. This quantity controls a bifurcation, in which the initially repelling singularity becomes the apex of a pair of parabolic invariant surfaces. The surfaces are smooth except where they intersect the discontinuity surface, and divide local space into regions of attraction to, and repulsion from, the singularity.
Translated title of the contributionThe two-fold singularity of discontinuous vector fields
Original languageEnglish
Pages (from-to)624 - 640
JournalSIAM Journal on Applied Dynamical Systems
Publication statusPublished - 2009

Bibliographical note

Rose publication type: Journal article

Additional information: Postprint document accepted for publication in the SIAM Journal on Applied Dynamical Systems (SIADS).

Sponsorship: Engineering and Physical Sciences Research Council (EPSRC)

Terms of use: © 2009 SIAM


  • Filippov
  • singularity
  • sliding
  • fold
  • nonsmooth
  • discontinuous


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