Mixed Mode Oscillations due to the Generalized Canard Phenomenon

Morten Brøns, Martin Krupa, Martin Wechselberger

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    Abstract

    Mixed mode oscillations combine features of small oscillations and large oscillations of relaxation type. We describe a mechanism for mixed mode oscillations based on the presence of canard solutions, which are trajectories passing from a stable to an unstable slow manifold. An important ingredient of this mechanism are singularities known as folded nodes. The main focus of this article is to show how the local dynamics near a folded node can combine with global features, leading to mixed mode oscillations. We review and extend the results of [26] on the dynamics near a folded node and state some results on mixed mode periodic orbits with Farey sequences of the form 1s. We also show how to generalize the context of one fast variable to an arbitrary number of fast variables.
    Original languageEnglish
    Title of host publicationBifurcation Theory and Spatio-Temporal Pattern Formation
    EditorsW. Nagata
    PublisherAmerican Mathematical Society
    Publication date2006
    Pages39-64
    ISBN (Print)0-8218-3725-7
    Publication statusPublished - 2006
    SeriesFields Insititute Communications
    Number49

    Cite this

    Brøns, M., Krupa, M., & Wechselberger, M. (2006). Mixed Mode Oscillations due to the Generalized Canard Phenomenon. In W. Nagata (Ed.), Bifurcation Theory and Spatio-Temporal Pattern Formation (pp. 39-64). American Mathematical Society. Fields Insititute Communications, No. 49