Piecewise-Linear (PWL) Canard Dynamics

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


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.


Piecewise-linear systems Singularly perturbed systems Canard solution Slow manifolds 



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.


  1. 1.
    Arima, N., Okazaki, H., Nakano, H.: A generation mechanism of canards in a piecewise linear system. IEICE T. Fundam. Electr. 80, 447–453 (1997)Google Scholar
  2. 2.
    Benoît, E.: Canards et enlacements. Publications Mathématiques de l’IHÉS 72, 63–91 (1990)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Benoît, E., Callot, J.L., Diener, F., Diener, M.: Chasse au canard. Collect. Math. 32, 37–119 (1981)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Brøns, M., Krupa, M., Wechselberger, M.: Mixed mode oscillations due to the generalized canard phenomenon. Fields Inst. Commun. 49, 39–63 (2006)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Desroches, M., Fernández-García, S., Krupa, M.: Canards in a minimal piecewise-linear square-wave burster. Chaos 26(7), 073,111 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Desroches, M., Guillamon, A., Ponce, E., Prohens, R., Rodrigues, S., Teruel, A.E.: Canards, folded nodes, and mixed-mode oscillations in piecewise-linear slow-fast systems. SIAM Rev. 58(4), 653–691 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Desroches, M., Kaper, T.J., Krupa, M.: Mixed-mode bursting oscillations: dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster. Chaos 23(4), 046,106 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Desroches, M., Guckenheimer, J.M., Krauskopf, B., Kuehn, C., Osinga, H.M., Wechselberger, M.: Mixed-mode oscillations with multiple time scales. SIAM Rev. 54, 211–288 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Diener, M.: The canard unchained or how fast/slow dynamical systems bifurcate. The Math. Intell. 6(3), 38–49 (1984)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Dumortier, F., Roussarie, R.: Canards cycles and center manifolds. Mem. Am. Math. Soc. 557, 1131–1162 (1996)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31, 53–98 (1979)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Fernández-García, S., Desroches, M., Krupa, M., Clément, F.: A multiple time scale coupling of piecewise linear oscillators. application to a neuroendocrine system. SIAM J. Appl. Dyn. Syst. 14(2), 643–673 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fernández-García, S., Desroches, M., Krupa, M., Teruel, A.E.: Canard solutions in planar piecewise linear systems with three zones. Dyn. Syst. A.I.J. 31, 173–197 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fernández-García, S., Krupa, M., Clément, F.: Mixed-mode oscillations in a piecewise linear system with multiple time scale coupling. Phys. D 332, 9–22 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1(6), 445–466 (1961)ADSCrossRefGoogle Scholar
  16. 16.
    Freire, E., Ponce E., Rodrigo. F., Torres, F.: Bifurcation sets of continuous piecewise linear systems with two zones. J. Bifur. Chaos Appl. Sci. Eng. 8, 2073–2097 (1998)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Freire, E., Ponce, E., Torres, F.: Hopf-like bifurcations in planar piecewise linear systems. Publ. Mat. 41, 135–148 (1997)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Jones, C.K.R.T.: Geometric Singular Perturbation Theory. Springer, Berlin, Heidelberg (1995)CrossRefGoogle Scholar
  19. 19.
    Kaper, T.: Systems theory for singular perturbation problems. In: O’Malley, R.E. Jr., Cronin, J. (eds.) Analyzing Multiscale Phenomena Using Singular Perturbation Methods; Proceedings of Symposia in Applied Mathematics, vol. 56, pp. 8–132; Am. Math. Soc. (1999)Google Scholar
  20. 20.
    Kramer, M.A., Traub, R.D., Kopell, N.J.: New dynamics in cerebellar purkinje cells: torus canards. Phys. Rev. Lett. 101(6), 068,103 (2008)Google Scholar
  21. 21.
    Krupa, M., Szmolyan, P.: Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions. SIAM J. Math. Anal. 33, 286–314 (2001)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Krupa, M., Szmolyan, P.: Relaxation oscillations and canard explosion. J. Differ. Equ. 174, 312–368 (2001)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    McKean, H.P.: Nagumo’s equation. Adv. Math. 4(3), 209–223 (1970)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mitry, J., Wechselberger, M.: Folded saddles and faux canards. SIAM J. Appl. Dyn. Syst. 16, 546–596 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50(10), 2061–2070 (1962)CrossRefGoogle Scholar
  26. 26.
    Prohens, R., Teruel, A.E.: Canard trajectories in 3d piecewise linear systems. Discret. Contin. Dyn. Syst. 33(3), 4595–4611 (2013)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Prohens, R., Teruel, A.E., Vich, C.: Slow-fast n-dimensional piecewise-linear differential systems. J. Differ. Equ. 260, 1865–1892 (2016)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Wechselberger, M.: Existence and bifurcation of canards in \({\mathbb{R}}^3\) in the case of a folded node. SIAM J. Appl. Dyn. Syst. 4, 101–139 (2005)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Mathieu Desroches
    • 1
    Email author
  • 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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