Discretizing dynamical systems with generalized Hopf bifurcations

We consider the discretizations of parameter-dependent, continuous-time dynamical systems. We show that the general one-step methods shift a generalized Hopf bifurcation and turn it into a generalized Neimark–Sacker point. We analyze the effect of discretization methods on the emanating Hopf curve. In particular, we obtain estimates for the eigenvalues of the discretized system along this curve. A detailed analysis of the discretized first Lyapunov coefficient is also given. The results are illustrated by a numerical example. Dynamical consequences are discussed.