Mixed-Mode Oscillations with Multiple Time Scales

• Published in: SIAM review Vol. 54; no. 2; pp. 211 - 288
• Main Authors: Desroches, Mathieu, Guckenheimer, John, Krauskopf, Bernd, Kuehn, Christian, Osinga, Hinke M., Wechselberger, Martin
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01-01-2012
• Subjects: Applied mathematics; Chaos theory; Differential equations; Dynamical Systems; Eigenvalues; Mathematical manifolds; Mathematics; Modeling; Oscillators; Periodic orbits; Scale models; Scholarships & fellowships; Signatures; Studies; SURVEY and REVIEW; System theory; Time series; Trajectories; Variables; United Kingdom > UK; Australia
• ISSN: 0036-1445; 1095-7200
• DOI: 10.1137/100791233

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.