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The simplest parametrized normal forms of Hopf and generalized Hopf bifurcations

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Abstract

This paper considers the computation of the simplest parameterized normal forms (SPNF) of Hopf and generalized Hopf bifurcations. Although the notion of the simplest normal form has been studied for more than two decades, most of the efforts have been spent on the systems that do not involve perturbation parameters due to the restriction of the computational complexity. Very recently, two singularities – single zero and Hopf bifurcation – have been investigated, and the SPNFs for these two cases have been obtained. This paper extends a recently developed method for Hopf bifurcation to compute the SPNF of generalized Hopf bifurcations. The attention is focused on a codimension-2 generalized Hopf bifurcation. It is shown that the SPNF cannot be obtained by using only a near-identity transformation. Additional transformations such as time and parameter rescaling are further introduced. Moreover, an efficient recursive formula is presented for computing the SPNF. Examples are given to demonstrate the applicability of the new method.

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Authors and Affiliations

  1. Department of Applied Mathematics, The University of Western Ontario, London, Ontario, N6A 5B7, Canada

    Pei Yu

  2. Department of Electronic Engineering, City University of Hong Kong, Hong Kong, SAR, P.R. China

    Pei Yu & Guanrong Chen

Authors
  1. Pei Yu
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  2. Guanrong Chen
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Correspondence to Pei Yu.

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Yu, P., Chen, G. The simplest parametrized normal forms of Hopf and generalized Hopf bifurcations. Nonlinear Dyn 50, 297–313 (2007). https://doi.org/10.1007/s11071-006-9158-1

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  • DOI: https://doi.org/10.1007/s11071-006-9158-1

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