The discontinuous Hopf-transversal system and its geometric regularization

PhD ceremony: Ms. X. Liu, 14.30 uur, Academiegebouw, Broerstraat 5, Groningen

Dissertation: The discontinuous Hopf-transversal system and its geometric regularization

Promotor(s): prof. H.W. Broer

Faculty: Mathematics and Natural Sciences

This thesis studies a particular class of planar Filippov systems, called Hopf-transversal (HT).

A HT system is composed of two families of planar smooth vector fields that are separated by a smooth discontinuity boundary. The vector field on one side undergoes a Hopf bifurcation, while the vector field on the other side intersects the boundary transversally.

Five families of codimension-1 bifurcations generically occur in a HT system. The point where all these five bifurcations meet is called the boundary-Hopf-fold (BHF) bifurcation, which has codimension 3. The main contribution of this work is deriving the generic unfoldings of this BHF bifurcation, which turn out to be suitable HT families. We prove that these unfoldings are structurally stable in a sense of contact equivalence, which amounts to dffeomorphic persistence of the bifurcation set and topological stability of the dynamics.

A promising line of research for Filippov systems is to relate the discontinuous dynamics to a regularized system, that is, to a smooth or continuous approximation of the original system. Then the dynamical properties of the Filippov system can be associated to corresponding properties of the regularized, in particular, slow-fast system for which the theory is better developed. In our approach the Filippov system is approximated by a piecewise smooth, continuous system, where the invariant sets are `preserved'. Eventually, we apply our regularization to the generic codimension-1 bifurcations of planar Filippov systems and then to the HT system.