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Mixed-mode oscillations with multiple time scales

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  • MF Desroches
  • J Guckenheimer
  • B Krauskopf
  • Christian Kuehn
  • HM Osinga
  • Martin Wechselberger
Original languageEnglish
Publication dateApr 2010
StatePublished

Abstract

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.

Additional information

Sponsorship: M.D. was supported by grants EP/C54403X/1 and EP/E032249/1 from the Engineering and Physical Sciences Research Council (EPSRC); J.G. by the National Science Foundation, the Department of Energy and by an International Visiting Fellowship from Sydney University; B.K. by a visitor grant from the Center for Applied Mathematics (CAM) at Cornell University; C.K. by the National Science Foundation; H.M.O. by an EPSRC Advanced Research Fellowship grant and by the IGERT programme of CAM at Cornell University; and M.W. by sabbatical support from the Institute for Analysis and Scientific Computing at the Vienna University of Technology and from the Mathematical Biosciences Institute at Ohio State University.

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  1. Mixed-mode oscillations with multiple time scales

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