An iterative method for the canard explosion in general planar systems

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    Abstract

    The canard explosion is the change of amplitude and period of a limit cycle born in a Hopf bifurcation in a very narrow parameter interval. The phenomenon is well understood in singular perturbation problems where a small parameter controls the slow/fast dynamics. However, canard explosions are also observed in systems where no such parameter is present. Here we show how the iterative method of Roussel and Fraser, devised to construct regular slow manifolds, can be used to determine a canard point in a general planar system of nonlinear ODEs. We demonstrate the method on the van der Pol equation, showing that the asymptotics of the method is correct, and on a templator model for a self-replicating system.
    Original languageEnglish
    Title of host publicationProceedings of the The 9th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    Number of pages9
    PublisherAmerican Institute of Mathematical Sciences
    Publication date2012
    Publication statusPublished - 2012
    EventThe 9th AIMS Conference on Dynamical Systems, Differential Equations and Applications - Orlando, FL, United States
    Duration: 1 Jul 20125 Jul 2012

    Conference

    ConferenceThe 9th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    CountryUnited States
    CityOrlando, FL
    Period01/07/201205/07/2012

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