A torus bifurcation theorem with symmetry

S.A. van Gils, M. Golubitsky

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    Abstract

    A general theory for the study of degenerate Hopf bifurcation in the presence of symmetry has been carried out only in situations where the normal form equations decouple into phase/amplitude equations. In this paper we prove a theorem showing that in general we expect such degeneracies to lead to secondary torus bifurcations. We then apply this theorem to the case of degenerate Hopf bifurcation with triangular (D3) symmetry, proving that in codimension two there exist regions of parameter space where two branches of asymptotically stable 2-tori coexist but where no stable periodic solutions are present. Although this study does not lead to a theory for degenerate Hopf bifurcations in the presence of symmetry, it does present examples that would have to be accounted for by any such general theory.
    Original languageEnglish
    Pages (from-to)133-162
    Number of pages30
    JournalJournal of dynamics and differential equations
    Volume2
    Issue number2
    DOIs
    Publication statusPublished - 1990

    Keywords

    • METIS-140642
    • IR-96534

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