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Piecewise-Linear (PWL) Canard Dynamics

Simplifying Singular Perturbation Theory in the Canard Regime Using Piecewise-linear Systems

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Book cover Nonlinear Systems, Vol. 1

Part of the book series: Understanding Complex Systems ((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.

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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.

Author information

Authors and Affiliations

  1. MathNeuro Team, Inria Sophia Antipolis Research Centre, 2004 route des Lucioles BP93, 06902, Sophia Antipolis Cedex, France

    Mathieu Desroches & Martin Krupa

  2. Departamento EDAN, Facultad de Matemáticas, University of Sevilla, C/ Tarfia s/n., 41012, Sevilla, Spain

    Soledad Fernández-García

  3. Université Côte d’Azur (UCA), Nice, France

    Martin Krupa

  4. Laboratoire J. A. Dieudonné, Université de Nice Sophia Antipolis, 06108, Nice Cedex 02, France

    Martin Krupa

  5. Departament de Matemàtiques i Informàtica, Universitat de les Illes Balears, Carretera de Valldemossa km. 7.5., 07122, Palma de Mallorca, Spain

    Rafel Prohens & Antonio E. Teruel

Authors
  1. Mathieu Desroches
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  2. Soledad Fernández-García
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  3. Martin Krupa
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  4. Rafel Prohens
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  5. Antonio E. Teruel
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Corresponding author

Correspondence to Mathieu Desroches .

Editor information

Editors and Affiliations

  1. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Victoriano Carmona

  2. Departamento de Física Aplicada I, Universidad de Sevilla, Sevilla, Spain

    Jesús Cuevas-Maraver

  3. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Fernando Fernández-Sánchez

  4. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Elisabeth García- Medina

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Desroches, M., Fernández-García, S., Krupa, M., Prohens, R., Teruel, A.E. (2018). Piecewise-Linear (PWL) Canard Dynamics. In: Carmona, V., Cuevas-Maraver, J., Fernández-Sánchez, F., García- Medina, E. (eds) Nonlinear Systems, Vol. 1. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-66766-9_3

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