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The Hopf bifurcation in the Shimizu–Morioka system

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Abstract

We study the local Hopf bifurcations of codimension one and two, which occur in the Shimizu–Morioka system. This system is a simplified model proposed for studying the dynamics of the well-known Lorenz system for large Rayleigh numbers. We present an analytic study and their bifurcation diagrams of these kinds of Hopf bifurcation, showing the qualitative changes in the dynamics of its solutions for different values of the parameters.

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Acknowledgments

The first author was partially supported by the Grants MINECO/FEDER MTM 2008-03437, AGAUR 2014SGR 568, ICREA Academia and FP7 PEOPLE-2012-IRSES-316338 and 318999. The second author was partially supported by Program CAPES/DGU Process 8333/13-0 and by FAPESP Project 2011/13152-8.

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Authors and Affiliations

  1. Departaments de Matematiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Catalonia, Spain

    Jaume Llibre

  2. Departamento de Matemática, Universidade Estadual Paulista, Campus de São José do Rio Preto - IBILCE, R. Cristóvão Colombo, 2265, São José do Rio Preto, SP, 15.054000, Brazil

    Claudio Pessoa

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  1. Jaume Llibre
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  2. Claudio Pessoa
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Correspondence to Claudio Pessoa.

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Llibre, J., Pessoa, C. The Hopf bifurcation in the Shimizu–Morioka system. Nonlinear Dyn 79, 2197–2205 (2015). https://doi.org/10.1007/s11071-014-1805-3

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