Multiscale analysis of defective multiple-Hopf bifurcations

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

Control of Dynamic Hopf Bifurcations

The slow passage through a Hopf bifurcation leads to the delayed appearance of large amplitude oscillations. We construct a smooth scalar feedback control which suppresses the delay and causes the system to follow a stable equilibrium branch. This feature can be used to detect in time the loss of stability of an ageing device. As a by-product, we obtain results on the slow passage through a bif...

Hopf bifurcations on cubic lattices

Group theoretic means are employed to analyse the Hopf bifurcation on pattern forming systems with the periodicity of the face-centred (FCC) and bodycentred (BCC) cubic lattices. We find all C-axial subgroups of the normal form symmetry group by first extending the symmetry to a larger group. There are 15 such solutions for the FCC lattice, of which at least 12 can be stable for appropriate par...

Spectral Analysis of Noisy Oscillators near Hopf Bifurcations ∗

We compare the dynamics of nonlinear noisy oscillators near the two types of the Hopf bifurcation. Prior to the bifurcation, in the regime of damped oscillations around the stable focus, noise serves as a bifurcation precursor: the power spectrum includes a peak at the frequency of the self-sustained oscillations. Super-and sub-critical Hopf bifurcations differ crucially in the noise dependence...