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 contribution||The two-fold singularity of discontinuous vector fields|
|Pages (from-to)||624 - 640|
|Journal||SIAM Journal on Applied Dynamical Systems|
|Publication status||Published - 2009|
Bibliographical noteRose 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)