Computer-Assisted Proofs of Hopf Bubbles and 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...

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...

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...

Simplest Normal Forms of Hopf and Generalized Hopf Bifurcations

The normal forms of Hopf and generalized Hopf bifurcations have been extensively studied, and obtained using the method of normal form theory and many other different approaches. It is well known that if the normal forms of Hopf and generalized Hopf bifurcations are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal form. In this paper, three theorem...

Double Hopf Bifurcations