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Classification and unfoldings of 1:2 resonant Hopf Bifurcation

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Abstract

In this paper, we study the bifurcations of periodic solutions from an equilibrium point of a differential equation whose linearization has two pairs of simple pure imaginary complex conjugate eigenvalues which are in 1:2 ratio. This corresponds to a Hopf-Hopf mode interaction with 1:2 resonance, as occurs in the context of dissipative mechanical systems. Using an approach based on Liapunov-Schmidt reduction and singularity theory, we give a framework in which to study these problems and their perturbations in two cases: no distinguished parameter, and one distinguished (bifurcation) parameter. We give a complete classification of the generic cases and their unfoldings.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Ottawa, KIN 6N5, Ottawa, Ontario

    Victor G. LeBlanc & William F. Langford

  2. Department of Mathematics and Statistics, University of Guelph, N1G 2W1, Guelph, Ontario

    Victor G. LeBlanc & William F. Langford

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  1. Victor G. LeBlanc
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  2. William F. Langford
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Communicated byM. Golubitsky

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LeBlanc, V.G., Langford, W.F. Classification and unfoldings of 1:2 resonant Hopf Bifurcation. Arch. Rational Mech. Anal. 136, 305–357 (1996). https://doi.org/10.1007/BF02206623

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