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

详细信息    查看全文

• 作者：John Burke ; Mathieu Desroches ; Albert Granados…
• 关键词：Folded singularities ; Canards ; Torus canards ; Torus bifurcation ; Mixed ; mode oscillations
• 刊名：Journal of Nonlinear Science
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：26
• 期：2
• 页码：405-451
• 全文大小：2,425 KB
• 参考文献：Baer, S., Erneux, T.: Singular Hopf bifurcation to relaxation oscillations. SIAM J. Appl. Dyn. Syst. 46, 721–739 (1986)MathSciNet CrossRef MATH
  Benes, G.N., Barry, A.M., Kaper, T.J., Kramer, M.A., Burke, J.: An elementary model of torus canards. Chaos 21, 023131 (2011)MathSciNet CrossRef MATH
  Benoît, E.: Canards et enlacements. Inst. Haut. Etud. Sci. Publ. Math. 72, 63–91 (1990)MathSciNet CrossRef
  Benoit, E., Callot, J.-L., Diener, F., Diener, M.: Chasse au canard. Collectanea Mathematicae 31–32, 37–119 (1981)MathSciNet MATH
  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)MathSciNet CrossRef MATH
  Braaksma, B.: Critical Phenomena in Dynamical Systems of van der Pol type, Ph.D. thesis, University of Utrecht (1993)
  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)
  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)MathSciNet CrossRef MATH
  Cartwright, M.L.: Forced Oscillations in Nonlinear Systems Contrib. to Theory of Nonlinear Oscillations (Study 20), pp. 149–241. Princeton University Press, Princeton (1950)
  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)MathSciNet CrossRef MATH
  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)MathSciNet CrossRef MATH
  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)MathSciNet MATH
  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)MathSciNet CrossRef MATH
  Desroches, M., Krauskopf, B., Osinga, H.M.: Numerical continuation of canard orbits in slow-fast dynamical systems. Nonlinearity 23, 739–765 (2010)MathSciNet CrossRef MATH
  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)MathSciNet CrossRef MATH
  Desroches, M., Krupa, M., Rodrigues, S.: Inflection, canards and excitability threshold in neuronal models. J. Math. Biol. 67, 989–1017 (2013)MathSciNet CrossRef MATH
  Diener, M.: The canard unchained or how fast–slow systems bifurcate. Math. Intell. 6, 38–49 (1984)MathSciNet CrossRef MATH
  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)
  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)
  Eckhaus, W.: Relaxation oscillations including a standard chase on French ducks. Lect. Notes Math. 985, 449–494 (1983)MathSciNet CrossRef MATH
  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)MathSciNet CrossRef
  Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Eqs. 31, 53–98 (1979)MathSciNet CrossRef MATH
  Flaherty, J.E., Hoppensteadt, F.C.: Frequency entrainment of a forced van der Pol oscillator. Stud. Appl. Math. 58, 5–15 (1978)MathSciNet CrossRef MATH
  Gelfreich, V.G.: Melnikov method and exponentially small splitting of separatrices. Phys. D 101, 227–248 (1997)MathSciNet CrossRef MATH
  Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory. Springer, Berlin (1988)CrossRef MATH
  Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, Berlin (1983)CrossRef MATH
  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)MathSciNet CrossRef MATH
  Haiduc, R.: Horseshoes in the forced van der Pol system. Nonlinearity 22, 213–237 (2009)MathSciNet CrossRef MATH
  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)MathSciNet CrossRef MATH
  Izhikevich, E.M.: Neural excitability, spiking and bursting. Int. J. Bifurc. Chaos 10, 11711266 (2000)MathSciNet CrossRef MATH
  Izhikevich, E.: Synchronization of elliptic bursters. SIAM Rev. 43, 315–344 (2001)MathSciNet CrossRef MATH
  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)CrossRef
  Kramer, M.A., Traub, R.D., Kopell, N.J.: New dynamics in cerebellar Purkinje cells: torus canards. Phys. Rev. Lett. 101, 068103 (2008)CrossRef
  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)
  Krupa, M., Wechselberger, M.: Local analysis near a folded saddle-node singularity. J. Differ. Equ. 248, 2841–2888 (2010)MathSciNet CrossRef MATH
  Kuehn, C.: From first Lyapunov coefficients to maximal canards. Int. J. Bifurc. Chaos 20, 1467–1475 (2010)MathSciNet CrossRef MATH
  Kuehn, C.: Multiple Time Scale Dynamics. Springer, Berlin (2015)CrossRef MATH
  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)
  Levi, M.: Qualitative analysis of the periodically-forced relaxation oscillations. Mem. AMS 32, 244 (1981)MathSciNet MATH
  Levinson, N.: A second-order differential equation with singular solutions. Ann. Math. 50(1), 127–153 (1949)MathSciNet CrossRef MATH
  Mitry, J., McCarthy, M., Kopell, N., Wechselberger, M.: Excitable neurons, firing threshold manifolds and canards. J. Math. Neurosci. 3, 12 (2013)MathSciNet CrossRef MATH
  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)
  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)MathSciNet CrossRef MATH
  Rubin, J., Wechselberger, M.: The selection of mixed-mode oscillations in a Hodgkin–Huxley model with multiple timescales. Chaos 18, 015105 (2008)MathSciNet CrossRef MATH
  Sanders, J.A., Verhulst, F.: Averaging Methods in Nonlinear Dynamical Systems. Springer, Berlin (1985)CrossRef MATH
  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)
  Szmolyan, P., Wechselberger, M.: Canards in \(\mathbb{R}^3\) . J. Differ. Equ. 177, 419–453 (2001)MathSciNet CrossRef MATH
  Szmolyan, P., Wechselberger, M.: Relaxation oscillations in \(\mathbb{R}^3\) . J. Differ. Equ. 200, 69–104 (2004)MathSciNet CrossRef MATH
  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)MathSciNet CrossRef MATH
  van der Pol, B.: A theory of the amplitude of free and forced triode vibrations. Radio Rev. 1, 701–710, 754–762 (1920)
  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)
  Vo, T., Wechselberger, M.: Canards of folded saddle-node type I. SIAM J. Math. Anal. 47, 3235–3283 (2015)MathSciNet CrossRef MATH
  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)MathSciNet CrossRef MATH
  Wechselberger, M.: À propos de canards (apropos canards). Trans. Am. Math. Soc. 364, 3289–3309 (2012)MathSciNet CrossRef MATH
  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)
  
• 作者单位：John Burke (1) (3)
  Mathieu Desroches (4)
  Albert Granados (2)
  Tasso J. Kaper (1)
  Martin Krupa (4)
  Theodore Vo (1)
  
  1. Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, MA, 02215, USA
  3. MSCI, Inc., 2100 Milvia Street, Berkeley, CA, 94704, USA
  4. NeuroMathComp Project-Team, Inria Sophia-Antipolis Research Centre, 2004 route des Lucioles, BP93, 06902, Sophia Antipolis cedex, France
  2. Department of Applied Mathematics and Computer Science, Technical University of Denmark, Building 303B, 2800, Kongens Lyngby, Denmark
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Mathematical and Computational Physics
  Mechanics
  Applied Mathematics and Computational Methods of Engineering
  Economic Theory
  
• 出版者：Springer New York
• ISSN：1432-1467

文摘

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).