An Update on that Singularity
It took nearly 30 years from the translation of Filippov’s seminal book to be able to say that the two-fold singularity is understood. We now know that its structural stability requires nonlinear switching or hidden terms, and that it comes in three main flavours, with numerous subclasses between which bifurcations can occur. We know that it is neither an attractor nor a repellor, but a bridge between attracting and repelling sliding and, in certain cases, is a source of determinacy-breaking.
KeywordsSwitching Surface Deep Role Multiple Switch Switching Layer Simple Geometrical Interpretation
- 2.M. Buchanan, Generating chaos in a new way. Phys. Rev. Focus 28, 1 (2011)Google Scholar
- 6.A.F. Filippov, Differential Equations with Discontinuous Righthand Sides (Kluwer Academic Publishers, Dortrecht, 1988). (Russian 1985)Google Scholar
- 11.K.U. Kristiansen, S.J. Hogan, Regularizations of two-fold bifurcations in planar piecewise smooth systems using blow up, Submitted (2015)Google Scholar
- 13.D.J.W. Simpson, On resolving singularities of piecewise-smooth discontinuous vector fields via small perturbations, Discrete Continuous Dyn. Syst. (to appear)Google Scholar
- 14.D.J.W. Simpson, M.R. Jeffrey, Fast phase randomisation via two-folds, Submitted (2015)Google Scholar