The two-fold singularity of discontinuous vector fields

Mike R Jeffrey, A Colombo

Research output: Contribution to journalArticlepeer-review

69 Citations (SciVal)
212 Downloads (Pure)


When a vector field in three 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 they divide local space into regions of attraction to, and repulsion from, the singularity.
Original languageEnglish
Pages (from-to)624-640
Number of pages17
JournalSIAM Journal on Applied Dynamical Systems
Issue number2
Publication statusPublished - 2009


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