Journal of Nonlinear Science

, Volume 26, Issue 2, pp 405–451 | Cite as

From Canards of Folded Singularities to Torus Canards in a Forced van der Pol Equation

  • John Burke
  • Mathieu Desroches
  • Albert Granados
  • Tasso J. KaperEmail author
  • Martin Krupa
  • Theodore Vo


In this article, we study canard solutions of the forced van der Pol equation in the relaxation limit for low-, intermediate-, and high-frequency periodic forcing. A central numerical observation made herein is that there are two branches of canards in parameter space which extend across all positive forcing frequencies. In the low-frequency forcing regime, we demonstrate the existence of primary maximal canards induced by folded saddle nodes of type I and establish explicit formulas for the parameter values at which the primary maximal canards and their folds exist. Then, we turn to the intermediate- and high-frequency forcing regimes and show that the forced van der Pol possesses torus canards instead. These torus canards consist of long segments near families of attracting and repelling limit cycles of the fast system, in alternation. We also derive explicit formulas for the parameter values at which the maximal torus canards and their folds exist. Primary maximal canards and maximal torus canards correspond geometrically to the situation in which the persistent manifolds near the family of attracting limit cycles coincide to all orders with the persistent manifolds that lie near the family of repelling limit cycles. The formulas derived for the folds of maximal canards in all three frequency regimes turn out to be representations of a single formula in the appropriate parameter regimes, and this unification confirms the central numerical observation that the folds of the maximal canards created in the low-frequency regime continue directly into the folds of the maximal torus canards that exist in the intermediate- and high-frequency regimes. In addition, we study the secondary canards induced by the folded singularities in the low-frequency regime and find that the fold curves of the secondary canards turn around in the intermediate-frequency regime, instead of continuing into the high-frequency regime. Also, we identify the mechanism responsible for this turning. Finally, we show that the forced van der Pol equation is a normal form-type equation for a class of single-frequency periodically driven slow/fast systems with two fast variables and one slow variable which possess a non-degenerate fold of limit cycles. The analytic techniques used herein rely on geometric desingularisation, invariant manifold theory, Melnikov theory, and normal form methods. The numerical methods used herein were developed in Desroches et al. (SIAM J Appl Dyn Syst 7:1131–1162, 2008, Nonlinearity 23:739–765 2010).


Folded singularities Canards Torus canards Torus bifurcation Mixed-mode oscillations 

Mathematics Subject Classification

Primary: 34E17 34E15 34A26 70K70 Secondary: 34E13 34D15 34C29 34C45 37G15 92C20 70K43 



The research of J.B., T.J.K., and T.V. was partially supported by NSF-DMS 1109587. T.J.K. thanks INRIA for their hospitality and for providing a climate highly conducive to research and collaboration during a visit. The authors thank Nick Benes, Mark Kramer, Christian Kuehn, John Mitry, and Martin Wechselberger for useful conversations.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • John Burke
    • 1
    • 3
  • Mathieu Desroches
    • 4
  • Albert Granados
    • 2
  • Tasso J. Kaper
    • 1
    Email author
  • Martin Krupa
    • 4
  • Theodore Vo
    • 1
  1. 1.Department of Mathematics and StatisticsBoston UniversityBostonUSA
  2. 2.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkKongens LyngbyDenmark
  3. 3.MSCI, Inc.BerkeleyUSA
  4. 4.NeuroMathComp Project-TeamInria Sophia-Antipolis Research CentreSophia Antipolis cedexFrance

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