Mixed-mode dynamics and the canard phenomenon: towards a classification

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Mixed-mode dynamics is a complex type of oscillatory behavior that is characterized by an alternation of small-amplitude oscillations and large-amplitude excursions. In this overview article, we focus on one particular mechanism that has been shown to generate mixed-mode oscillations (MMOs) in multiple-scale systems: the generalized canard mechanism. After a brief review of the classical canard phenomenon, we present a model problem that was proposed in [23] as a canonical form for a family of three-dimensional three time-scale systems, and we reiterate some of the results obtained there. In particular, we discuss how that canonical form can be placed in the context of the well-developed geometric theory of canards in three dimensions. Finally, we introduce two examples of problems from mathematical neuroscience that fit into the framework of our model problem, and we discuss the implications of our results for the mixed-mode dynamics observed in these two examples. Our results are intended as a first step towards a more general classification of the mixed-mode dynamics that can arise via the generalized canard mechanism, with the long-term goal of constructing a 'toolbox' of prototypical minimal models.

Original languageEnglish
Title of host publicationINTERNATIONAL WORKSHOP ON MULTI-RATE PROCESSES AND HYSTERESIS
EditorsMP Mortell, RE OMalley, A Pokrovskii, D Rachinskii, VA Sobolev
Place of PublicationBRISTOL
PublisherIOP Publishing Ltd.
Pages-
Number of pages20
ISBN (Print)*****************
DOIs
Publication statusPublished - 2008

Keywords

  • SINGULAR PERTURBATION-THEORY
  • RELAXATION OSCILLATIONS
  • SYSTEMS
  • CHAOS
  • EQUATIONS
  • PATTERNS
  • GEOMETRY
  • NEURON
  • R-3

Fingerprint

Dive into the research topics of 'Mixed-mode dynamics and the canard phenomenon: towards a classification'. Together they form a unique fingerprint.

Cite this