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Discretizing dynamical systems with generalized Hopf bifurcations

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Abstract

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.

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  1. Instituto de Ciencias Matemáticas, Escuela Superior Politécnica del Litoral, Km. 30.5 Vía Perimetral, P.O. Box 09-01-5863, Guayaquil, Ecuador

    Joseph Páez Chávez

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  1. Joseph Páez Chávez
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Correspondence to Joseph Páez Chávez.

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J. Páez Chávez was supported by CRC 701 ‘Spectral Structures and Topological Methods in Mathematics’, Bielefeld University.

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Páez Chávez, J. Discretizing dynamical systems with generalized Hopf bifurcations. Numer. Math. 118, 229–246 (2011). https://doi.org/10.1007/s00211-010-0340-5

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  • DOI: https://doi.org/10.1007/s00211-010-0340-5

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