Mixed-Mode Oscillations with Multiple Time Scales
Mathieu Desroches,John Guckenheimer,Bernd Krauskopf,Christian Kuehn,Hinke M. Osinga,Martin Wechselberger +5 more
TLDR
This survey of different types of MMOs is given, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale “mechanism.”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.read more
Citations
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A mathematical framework for critical transitions: Bifurcations, fast–slow systems and stochastic dynamics
TL;DR: In this article, the authors provide an overview of early warning signs for critical transitions in dynamical systems and their application in a wide range of applications from ecosystems and climate change to medicine and finance.
Journal ArticleDOI
Multi-timescale systems and fast-slow analysis.
TL;DR: In this review, the fast-slow analysis technique is described and applied to relaxation oscillations, neuronal bursting oscillation, canard oscillated systems, and mixed-mode oscillations.
Journal ArticleDOI
Long transients in ecology: Theory and applications
Andrew Morozov,Karen C. Abbott,Kim Cuddington,Tessa B. Francis,Gabriel Gellner,Alan Hastings,Ying-Cheng Lai,Sergei Petrovskii,Sergei Petrovskii,Katherine Scranton,Mary Lou Zeeman +10 more
TL;DR: Several main mechanisms leading to the emergence of long transients are revealed and hence lays the basis for a unifying theory in ecology.
Journal ArticleDOI
Mixed-mode bursting oscillations: Dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster
TL;DR: This article shows how MMBOs are created naturally in systems that have a spike-adding bifurcation or spike- adding mechanism, and in which the dynamics of one (or more) of the slow variables causes the system to pass slowly through that biforcation.
Journal ArticleDOI
À propos de canards (Apropos canards)
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