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

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## Abstract

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).

## Keywords

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## Notes

### Acknowledgments

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.

## References

- Baer, S., Erneux, T.: Singular Hopf bifurcation to relaxation oscillations. SIAM J. Appl. Dyn. Syst.
**46**, 721–739 (1986)MathSciNetCrossRefzbMATHGoogle Scholar - Benes, G.N., Barry, A.M., Kaper, T.J., Kramer, M.A., Burke, J.: An elementary model of torus canards. Chaos
**21**, 023131 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - Benoît, E.: Canards et enlacements. Inst. Haut. Etud. Sci. Publ. Math.
**72**, 63–91 (1990)MathSciNetCrossRefGoogle Scholar - Benoit, E., Callot, J.-L., Diener, F., Diener, M.: Chasse au canard. Collectanea Mathematicae
**31–32**, 37–119 (1981)MathSciNetzbMATHGoogle Scholar - Bold, K., Edwards, C., Guckenheimer, J., Guharay, S., Hoffman, K., Hubbard, J., Oliva, R., Weckesser, W.: The forced van der Pol equation II: canards in the reduced system. SIAM J. Appl. Dyn. Syst.
**2**, 570–608 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - Braaksma, B.: Critical Phenomena in Dynamical Systems of van der Pol type, Ph.D. thesis, University of Utrecht (1993)Google Scholar
- Brøns, M., Krupa, M., Wechselberger, M.: Mixed mode oscillations due to the generalized canard phenomenon. In: “Bifurcation Theory and Spatio-Temporal Pattern Formation”, Fields Institute Communications, vol. 49, pp. 39–63. American Mathematical Society, Providence, RI (2006)Google Scholar
- Burke, J., Desroches, M., Barry, A.M., Kaper, T.J., Kramer, M.A.: A showcase of torus canards in neuronal bursters. J. Math. Neurosci.
**2**, 3 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - Cartwright, M.L.: Forced Oscillations in Nonlinear Systems Contrib. to Theory of Nonlinear Oscillations (Study 20), pp. 149–241. Princeton University Press, Princeton (1950)Google Scholar
- Cartwright, M.L., Littlewood, J.E.: On non-linear differential equations of the second order: I. The equation \(\ddot{y} - k(1-y^2)\dot{y}+y =b \lambda k \cos (\lambda t+a)\); \(k\) large. J. Lond. Math. Soc.
**20**, 180–189 (1945)MathSciNetCrossRefzbMATHGoogle Scholar - Delshams, A., Seara, T.M.: An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum. Comm. Math. Phys.
**150**(3), 443–463 (1992)MathSciNetCrossRefzbMATHGoogle Scholar - Delshams, A., Seara, T.M.: Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom. Math. Phys. Electron. J.
**3**, 4 (1997)MathSciNetzbMATHGoogle Scholar - Desroches, M., Krauskopf, B., Osinga, H.M.: The geometry of slow manifolds near a folded node. SIAM J. Appl. Dyn. Syst.
**7**, 1131–1162 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - Desroches, M., Krauskopf, B., Osinga, H.M.: Numerical continuation of canard orbits in slow-fast dynamical systems. Nonlinearity
**23**, 739–765 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - Desroches, M., Guckenheimer, J., Kuehn, C., Krauskopf, B., Osinga, H.M., Wechselberger, M.: Mixed-mode oscillations with multiple time scales. SIAM Rev.
**54**, 211–288 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - Desroches, M., Krupa, M., Rodrigues, S.: Inflection, canards and excitability threshold in neuronal models. J. Math. Biol.
**67**, 989–1017 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - Diener, M.: The canard unchained or how fast–slow systems bifurcate. Math. Intell.
**6**, 38–49 (1984)MathSciNetCrossRefzbMATHGoogle Scholar - Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Oldeman, K.E., Paffenroth, R.C., Sanstede, B., Wang, X.J., Zhang, C.: AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations. http://cmvl.cs.concordia.ca/ (2007)
- Dumortier, F., Roussarie, R.: Canard cycles and center manifolds. Mem. Am. Math. Soc.
**577**(1996)Google Scholar - Dumortier, F., Roussarie, R.: Geometric singular perturbation theory beyond normal hyperbolicity. In: Jones, C.K.R.T., Khibnik, A.I. (ed.) Multiple Time Scales Dynamical Systems, IMA Volumes in Mathematics and its Applications, vol. 122, pp. 29–64 (2001)Google Scholar
- Eckhaus, W.: Relaxation oscillations including a standard chase on French ducks. Lect. Notes Math.
**985**, 449–494 (1983)MathSciNetCrossRefzbMATHGoogle Scholar - Erchova, I., McGonigle, D.J.: Rhythms of the brain: an examination of mixed mode oscillation approaches to the analysis of neurophysiological data. Chaos
**18**, 015115 (2008)MathSciNetCrossRefGoogle Scholar - Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Eqs.
**31**, 53–98 (1979)MathSciNetCrossRefzbMATHGoogle Scholar - Flaherty, J.E., Hoppensteadt, F.C.: Frequency entrainment of a forced van der Pol oscillator. Stud. Appl. Math.
**58**, 5–15 (1978)MathSciNetCrossRefzbMATHGoogle Scholar - Gelfreich, V.G.: Melnikov method and exponentially small splitting of separatrices. Phys. D
**101**, 227–248 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory. Springer, Berlin (1988)CrossRefzbMATHGoogle Scholar
- Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, Berlin (1983)CrossRefzbMATHGoogle Scholar
- Guckenheimer, J., Hoffman, K., Weckesser, W.: The forced van der Pol equation I: the slow flow and its bifurcations. SIAM J. Appl. Dyn. Syst.
**2**, 1–35 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - Haiduc, R.: Horseshoes in the forced van der Pol system. Nonlinearity
**22**, 213–237 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - Han, X., Bi, Q.: Slow passage through canard explosion and mixed-mode oscillations in the forced Van der Pol’s equation. Nonlinear Dyn.
**68**, 275–283 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - Izhikevich, E.M.: Neural excitability, spiking and bursting. Int. J. Bifurc. Chaos
**10**, 11711266 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - Izhikevich, E.: Synchronization of elliptic bursters. SIAM Rev.
**43**, 315–344 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - Jones, C.K.R.T.: Geometric singular perturbation theory. In: Johnson, R. (ed.) Dynamical Systems. Lecture Notes in Mathematics, pp. 44–120. Springer, New York (1995)CrossRefGoogle Scholar
- Kramer, M.A., Traub, R.D., Kopell, N.J.: New dynamics in cerebellar Purkinje cells: torus canards. Phys. Rev. Lett.
**101**, 068103 (2008)CrossRefGoogle Scholar - 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)Google Scholar - Krupa, M., Wechselberger, M.: Local analysis near a folded saddle-node singularity. J. Differ. Equ.
**248**, 2841–2888 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - Kuehn, C.: From first Lyapunov coefficients to maximal canards. Int. J. Bifurc. Chaos
**20**, 1467–1475 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - Kuehn, C.: Multiple Time Scale Dynamics. Springer, Berlin (2015)CrossRefzbMATHGoogle Scholar
- Lanford, O.E., III: Bifurcation of periodic solutions into invariant tori: the work of Ruelle and Takens. In: Nonlinear Problems in the Physical Sciences and Biology, pp. 159–192. Springer, Berlin (1973)Google Scholar
- Levi, M.: Qualitative analysis of the periodically-forced relaxation oscillations. Mem. AMS
**32**, 244 (1981)MathSciNetzbMATHGoogle Scholar - Levinson, N.: A second-order differential equation with singular solutions. Ann. Math.
**50**(1), 127–153 (1949)MathSciNetCrossRefzbMATHGoogle Scholar - Mitry, J., McCarthy, M., Kopell, N., Wechselberger, M.: Excitable neurons, firing threshold manifolds and canards. J. Math. Neurosci.
**3**, 12 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - Roberts, K.-L., Rubin, J., Wechselberger, M.: Averaging, Folded Singularities, and Torus Canards: Explaining Transitions Between Bursting and Spiking in a Coupled Neuron Model. SIAM J. Appl. Dyn. Syst.
**14**, 1808–1844 (2015)Google Scholar - Rotstein, H., Wechselberger, M., Kopell, N.: Canard induced mixed-mode oscillations in a medial entorhinal cortex layer II stellate cell model. SIAM J. Appl. Dyn. Syst.
**7**, 1582–1611 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - Rubin, J., Wechselberger, M.: The selection of mixed-mode oscillations in a Hodgkin–Huxley model with multiple timescales. Chaos
**18**, 015105 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - Sanders, J.A., Verhulst, F.: Averaging Methods in Nonlinear Dynamical Systems. Springer, Berlin (1985)CrossRefzbMATHGoogle Scholar
- Sekikawa, M., Inaba, N., Yoshinaga, T., Kawakami, H.: Collapse of duck solution in a circuit driven by an extremely small periodic force. Electron. Comm. Jpn. Part 3
**88**(4), 199–207 (2005)Google Scholar - Szmolyan, P., Wechselberger, M.: Canards in \(\mathbb{R}^3\). J. Differ. Equ.
**177**, 419–453 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - Szmolyan, P., Wechselberger, M.: Relaxation oscillations in \(\mathbb{R}^3\). J. Differ. Equ.
**200**, 69–104 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - Teka, W., Tabak, J., Vo, T., Wechselberger, M., Bertram, R.: The dynamics underlying pseudo-plateau bursting in a pituitary cell model. J. Math. Neurosci.
**1**, 12 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - van der Pol, B.: A theory of the amplitude of free and forced triode vibrations. Radio Rev.
**1**, 701–710, 754–762 (1920)Google Scholar - van der Pol, B.: Forced oscillations in a circuit with non-linear resistance (reception with reactive triode). Lond. Edinb. Dublin Phil. Mag. J. Sci. Ser. 7,
**3**, 65–80 (1927)Google Scholar - Vo, T., Wechselberger, M.: Canards of folded saddle-node type I. SIAM J. Math. Anal.
**47**, 3235–3283 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 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)MathSciNetCrossRefzbMATHGoogle Scholar - Wechselberger, M.: À propos de canards (apropos canards). Trans. Am. Math. Soc.
**364**, 3289–3309 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - Wechselberger, M., Mitry, J., Rinzel, J.: Canard theory and excitability. In: Nonautonomous Dynamical Systems in the Life Sciences, Lecture Notes in Mathematics, vol. 2102 (Mathematical Biosciences Subseries) (2014)Google Scholar