Hopf-Takens bifurcations and centres

Interactions of Andronov-Hopf and Bogdanov-Takens Bifurcations

A codimension-three bifurcation, characterized by a pair of purely imaginary eigenvalues and a nonsemisimple double zero eigenvalue, arises in the study of a pair of weakly coupled nonlinear oscillators with Z2 Z2 symmetry. The methodology is based on Arnold's ideas of versal deformations of matrices for the linear analysis, and Poincar e normal forms for the nonlinear analysis of the system. B...

Generalized Liénard Equations, Cyclicity and Hopf–Takens Bifurcations

We investigate the bifurcation of small–amplitude limit cycles in generalized Liénard equations. We use the simplicity of the Liénard family, to illustrate the advantages of the approach based on Bautin ideals. Essentially, this Bautin ideal is generated by the so–called Lyapunov quantities, that are computed for generalized Liénard equations and used to detect the presence of a Hopf– Takens bi...

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

Takens-Bogdanov bifurcations without parameters, and oscillatory shock profiles