CHAOTIC REGIMES, POST-BIFURCATION DYNAMICS, AND COMPETITIVE MODES FOR A GENERALIZED DOUBLE HOPF NORMAL FORM

    Authors

    B. Reeves; R. A. Van Gorder;S. R. Choudhury

    Comments

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    Abbreviated Journal Title

    Int. J. Bifurcation Chaos

    Keywords

    Generalized double Hopf normal form; competitive modes; chaos; chaotic; regime; LORENZ; Mathematics, Interdisciplinary Applications; Multidisciplinary Sciences

    Abstract

    We discuss the post-bifurcation dynamics of the general double Hopf normal form, which allows us to study two intermittent routes to chaos (routes following either (i) subcritical or (ii) supercritical Hopf or double Hopf bifurcations). In particular, the route following supercritical bifurcations is somewhat subtle. Such behavior following repeated Hopf bifurcations is well-known and widely observed, including the classical Ruelle-Takens and quasiperiodic routes to chaos. However, it has not, to the best of our knowledge, been considered in the context of the double Hopf normal form, although it has been numerically observed and tracked in the post-double-Hopf regime. We then apply the method of competitive modes to verify parameter regimes for which the double Hopf normal form exhibits chaotic behavior. Such an analysis is useful, as it allows us to potentially identify specific parameter regimes for which the system may exhibit strange or irregular behavior, something which would be extremely difficult otherwise in a system with so many parameters. Indeed, it is conjectured that for parameter sets with two of the square-mode frequencies competitive or nearly competitive, chaotic behavior is likely to be observed in the system. We apply the method of competitive modes to two representative cases where intermittent chaos is found, and the competitive mode analysis there seems to verify the occurrence of chaos in each of the two types of regimes.

    Journal Title

    International Journal of Bifurcation and Chaos

    Volume

    22

    Issue/Number

    12

    Publication Date

    1-1-2012

    Document Type

    Article

    Language

    English

    First Page

    10

    WOS Identifier

    WOS:000313704900012

    ISSN

    0218-1274

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