Discretizing Dynamical Systems with Generalized Hopf Bifurcations JosephP´aezCh´avez ∗ Instituto de Ciencias Matem´aticas, Escuela Superior Polit´ ecnica del Litoral, Km. 30.5 V´ ıa Perimetral, P.O. Box 09-01-5863 Guayaquil, Ecuador jpaez@espol.edu.ec April 12, 2010 Abstract 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. 1 Introduction Consider a continuous-time dynamical system depending on parameters ˙ x(t)= f (x(t),α), (1.1) where f ∈ C k (Ω × Λ, N ) with open sets 0 ∈ Ω ⊂ N ,0 ∈ Λ ⊂ 2 , k ≥ 1 sufficiently large, N ≥ 2. The first and commonly used tool for understanding the dynamics generated by the vector field (1.1) is numerical time-integration. For this purpose we can utilize one- step methods, which consists in approximating the evolution operator by a discrete-time system x → g (x,α), (1.2) ∗ Supported by CRC 701 ‘Spectral Structures and Topological Methods in Mathematics’, Bielefeld University. 1