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Piecewise-Linear (PWL) Canard Dynamics

Simplifying Singular Perturbation Theory in the Canard Regime Using Piecewise-linear Systems
  • Mathieu Desroches
  • Soledad Fernández-García
  • Martin Krupa
  • Rafel Prohens
  • Antonio E. Teruel
Chapter
Part of the Understanding Complex Systems book series (UCS)

Abstract

In this chapter we gather recent results on piecewise-linear (PWL) slow-fast dynamical systems in the canard regime. By focusing on minimal systems in \(\mathbb {R}^2\) (one slow and one fast variables) and \(\mathbb {R}^3\) (two slow and one fast variables), we prove the existence of (maximal) canard solutions and show that the main salient features from smooth systems is preserved. We also highlight how the PWL setup carries a level of simplification of singular perturbation theory in the canard regime, which makes it more amenable to present it to various audiences at an introductory level. Finally, we present a PWL version of Fenichel theorems about slow manifolds, which are valid in the normally hyperbolic regime and in any dimension, which also offers a simplified framework for such persistence results.

Keywords

Piecewise-linear systems Singularly perturbed systems Canard solution Slow manifolds 

Notes

Acknowledgements

SFG is supported by the University of Seville VPPI-US and partially supported by Proyectos de Excelencia de la Junta de Andalucía under Grant No. P12-FQM-1658 and Ministerio de Economía y Competitividad under Grant No. MTM2015-65608-P. RP and AET are supported by the Spanish Ministerio de Economía y Competitividad through project MTM2014-54275-P.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Mathieu Desroches
    • 1
  • Soledad Fernández-García
    • 2
  • Martin Krupa
    • 1
    • 3
    • 4
  • Rafel Prohens
    • 5
  • Antonio E. Teruel
    • 5
  1. 1.MathNeuro TeamInria Sophia Antipolis Research CentreSophia Antipolis CedexFrance
  2. 2.Departamento EDAN, Facultad de MatemáticasUniversity of SevillaSevillaSpain
  3. 3.Université Côte d’Azur (UCA)NiceFrance
  4. 4.Laboratoire J. A. DieudonnéUniversité de Nice Sophia AntipolisNice Cedex 02France
  5. 5.Departament de Matemàtiques i InformàticaUniversitat de les Illes BalearsPalma de MallorcaSpain

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