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Mixed-mode oscillations and slow manifolds in the self-coupled FitzHugh-Nagumo system

Chaos 18, 015107 (2008); doi:10.1063/1.2799471

Published 27 March 2008

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Mathieu Desroches, Bernd Krauskopf, and Hinke M. Osinga
Bristol Centre for Applied Nonlinear Mathematics, Department of Engineering Mathematics, University of Bristol, Queen's Building, Bristol BS8 1TR, United Kingdom
We investigate the organization of mixed-mode oscillations in the self-coupled FitzHugh-Nagumo system. These types of oscillations can be explained as a combination of relaxation oscillations and small-amplitude oscillations controlled by canard solutions that are associated with a folded singularity on a critical manifold. The self-coupled FitzHugh-Nagumo system has a cubic critical manifold for a range of parameters, and an associated folded singularity of node-type. Hence, there exist corresponding attracting and repelling slow manifolds that intersect in canard solutions. We present a general technique for the computation of two-dimensional slow manifolds (smooth surfaces). It is based on a boundary value problem approach where the manifolds are computed as one-parameter families of orbit segments. Visualization of the computed surfaces gives unprecedented insight into the geometry of the system. In particular, our techniques allow us to find and visualize canard solutions as the intersection curves of the attracting and repelling slow manifolds. ©2008 American Institute of Physics
History: Received 17 August 2007; accepted 24 September 2007; published 27 March 2008
Permalink: http://link.aip.org/link/?CHAOEH/18/015107/1
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KEYWORDS and PACS

Keywords
PACS
  • 05.45.-a
    Nonlinear dynamics and chaos
  • 02.30.-f
    Function theory, analysis
  • YEAR: 2008

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ISSN:
1054-1500 (print)   1089-7682 (online)
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