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The European Physical Journal Special Topics

, Volume 227, Issue 5–6, pp 591–601 | Cite as

Relaxation oscillations and canards in the Jirsa–Kelso excitator model: global flow perspective

  • Piotr Słowiński
  • Sohaib Al-Ramadhani
  • Krasimira Tsaneva-Atanasova
Open Access
Regular Article
Part of the following topical collections:
  1. Nonlinear Phenomena in Physics: New Techniques and Applications

Abstract

Fenichel’s geometric singular perturbation theory and the blow-up method have been very successful in describing and explaining global non-linear phenomena in systems with multiple time-scales, such as relaxation oscillations and canards. Recently, the blow-up method has been extended to systems with flat, unbounded slow manifolds that lose normal hyperbolicity at infinity. Here, we show that transition between discrete and periodic movement captured by the Jirsa–Kelso excitator is a new example of such phenomena. We, first, derive equations of the Jirsa–Kelso excitator with explicit time scale separation and demonstrate existence of canards in the systems. Then, we combine the slow-fast analysis, blow-up method and projection onto the Poincaré sphere to understand the return mechanism of the periodic orbits in the singular case, ϵ = 0.

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Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of MathematicsCollege of Engineering, Mathematics and Physical Sciences, University of ExeterExeter, DevonUK
  2. 2.Living Systems Institute, University of ExeterExeterUK
  3. 3.EPSRC Centre for Predictive Modelling in Healthcare, University of ExeterExeterUK

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