Analysis and Applications of Nonsmooth Bifurcations

Applied and Computational Mathematics Seminar
Monday, October 28, 2019 - 1:55pm for 1 hour (actually 50 minutes)
Skiles 05
Oleg Makarenkov – Univ Texas at Dallas –
Luca Dieci
In this talk I will first give a brief overview of how nonsmooth bifurcations (border-splitting, grazing, and fold-fold bifurcations) help to rigorously explain the existence of nonsmooth limit cycles in the models of anti-lock braking systems, power converters, integrate-and-fire neurons, and climate dynamics. I will then focus on one particular application that deals with nonsmooth bifurcations in dispersing billiards. In [Nonlinearity 11 (1998)] Turaev and Rom-Kedar discovered that every periodic orbit that is tangent to the boundary of the billiard produces an island of stability upon smoothening the boundary of the billiard. The result to be presented in the talk (joint work with Turaev) proves that any dispersing billiard admits such an arbitrary small perturbation that ensures the occurrence of a tangent periodic orbit.