Numerical Methods for the Generalized Hopf Bifurcation

The presence of a generalized Hopf (GH) point (Hopf point where the first Lyapunov coefficient vanishes) deeply influences the qualitative behavior of a dynamical system. Further information on this behavior can be obtained from the second Lyapunov coefficient which depends on derivatives of order up to five. We describe two computational procedures, implemented in the software package content, to compute and numerically continue GH points. This is applied to a biochemical model and two examples from neurobiology. We make some comparisons between the two methods for computing GH points and discuss briefly the merits of three methods for computing curves of Hopf points.