Discretizing dynamical systems with generalized Hopf bifurcations

We consider parameter-dependent, continuous-time dynamical systems under discretizations. It is shown that generalized Hopf bifurcations are shifted and turned into generalized Neimark-Sacker points by general one-step methods. We analyze the effect of discretizations methods on the emanating Hopf curve. In particular, we obtain estimates of the discretized eigenvalues along this curve. A detailed analysis of the discretized first Lyapunov coefficient is also given. The results are illustrated by a numerical example.