Improved homoclinic predictor for Bogdanov-Takens bifurcation

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    Abstract

    An improved homoclinic predictor at a generic codim 2 Bogdanov-Takens (BT) bifucation is derived. We use the classical "blow-up" technique to reduce the canonical smooth normal form near a generic BT bifurcation to a perturbed Hamiltonian system. With a simple perturbation method, we derive explicit rst- and second-order corrections of the unperturbed homoclinic orbit and parameter value. To obtain the normal form on the center manifold, we apply the standard parameter-dependent center manifold reduction combined with the normalization, that is based on the Fredholm solvability of the homological equation. By systematically solving all linear systems appearing from the homological equation, we remove an ambiguity in the parameter transformation existing in the literature. The actual implementation of the improved predictor in MatCont and numerical examples illustrating its eciency are discussed.
    Original languageUndefined
    Pages (from-to)1450057
    Number of pages12
    JournalInternational journal of bifurcation and chaos in applied sciences and engineering
    Volume24
    Issue number4
    DOIs
    Publication statusPublished - 24 Apr 2014

    Keywords

    • EWI-24077
    • Homoclinic orbit
    • Normal form
    • MatCont
    • IR-90569
    • Center manifold
    • Bogdanov-Takens bifurcation
    • METIS-303975

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