Classification and unfoldings of degenerate Hopf bifurcations

Classification and Unfoldings of Degenerate Hopf Bifurcations

This paper initiates the classification, up to symmetry-covariant contact equivalence, of perturbations of local Hopf bifurcation problems which do not satisfy the classical non-degeneracy conditions. The only remaining hypothesis is that +i should be simple eigenvalues of the linearized right-hand side at criticality. Then the Lyapunov-Schmidt method allows a reduction to a scalar equation G(x...

Strange Attractors in Periodically-kicked Degenerate Hopf Bifurcations

We prove that spiral sinks (stable foci of vector fields) can be transformed into strange attractors exhibiting sustained, observable chaos if subjected to periodic pulsatile forcing. We show that this phenomenon occurs in the context of periodically-kicked degenerate supercritical Hopf bifurcations. The results and their proofs make use of a new k-parameter version of the theory of rank one ma...

North-HoUand, Amsterdam CLASSIFICATION AND UNFOLDING OF DEGENERATE HOPF BIFURCATIONS WITH 0(2) SYMMETRY: NO DISTINGUISHED PARAMETER

The Hopf bifurcation in the presence of 0(2) symmetry is considered. When the bifurcation breaks the symmetry, the critical imaginary eigenvalues have multiplicity two and generically there are two primary branches of periodic orbits which bifurcate simultaneously. In applications these correspond to rotating (traveling) waves and standing waves. Using equivariant singularity theory a classific...

Discretizing Dynamical Systems with Hopf-Hopf Bifurcations

We consider parameter-dependent, continuous-time dynamical systems under discretizations. It is shown that Hopf-Hopf bifurcations are O(h)-shifted and turned into double Neimark-Sacker points by general one-step methods of order p. Then we discuss the effect of discretization methods on the emanating Hopf curves. The numerical approximation of the critical eigenvalues is analyzed too. The resul...

Double Hopf Bifurcations