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Mixed-mode oscillations with multiple time scales. (English) Zbl 1250.34001

Summary: Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale “mechanism”. Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.

MSC:

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34E13 Multiple scale methods for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
34E17 Canard solutions to ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C45 Invariant manifolds for ordinary differential equations
65L11 Numerical solution of singularly perturbed problems involving ordinary differential equations

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