The singular limit of a Hopf bifurcation

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August 2012, 32(8): 2805-2823. doi: 10.3934/dcds.2012.32.2805

The singular limit of a Hopf bifurcation

John Guckenheimer ^1, and Hinke M. Osinga ^2,

1.	Mathematics Department, Cornell University, Ithaca, NY 14853
2.	Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142

Received June 2011 Revised November 2011 Published March 2012

Abstract
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Hopf bifurcation in systems with multiple time scales takes several forms, depending upon whether the bifurcation occurs in fast directions, slow directions or a mixture of these two. Hopf bifurcation in fast directions is influenced by the singular limit of the fast time scale, that is, when the ratio $\epsilon$ of the slowest and fastest time scales goes to zero. The bifurcations of the full slow-fast system persist in the layer equations obtained from this singular limit. However, the Hopf bifurcation of the layer equations does not necessarily have the same criticality as the corresponding Hopf bifurcation of the full slow-fast system, even in the limit $\epsilon \to 0$ when the two bifurcations occur at the same point. We investigate this situation by presenting a simple slow-fast system that is amenable to a complete analysis of its bifurcation diagram. In this model, the family of periodic orbits that emanates from the Hopf bifurcation accumulates onto the corresponding family of the layer equations in the limit as $\epsilon \to 0$; furthermore, the stability of the orbits is dictated by that of the layer equation. We prove that a torus bifurcation occurs $O(\epsilon)$ near the Hopf bifurcation of the full system when the criticality of the two Hopf bifurcations is different.

Keywords: torus bifurcation, Lyapunov coefficient, slow-fast system, Hopf bifurcation.

Mathematics Subject Classification: 34C20, 34D15, 37G1.

Citation: John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805

References:

[1]	D. Barkley, Slow manifolds and mixed-mode oscillations in the Belousov-Zhabotinskii reaction, J. Chem. Phys., 89 (1998), 5547-5559. doi: 10.1063/1.455561.
[2]	B. Braaksma, Singular Hopf bifurcation in systems with fast and slow variables, J. Nonlin. Sci., 8 (1998), 457-490. doi: 10.1007/s003329900058.
[3]	M. Desroches, B. Krauskopf and H. M. Osinga, The geometry of slow manifolds near a folded node, SIAM J. Appl. Dyn. Syst., 7 (2008), 1131-1162. doi: 10.1137/070708810.
[4]	A. Dhooge, W. Govaerts and Yu. A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Software, 29 (2003), 141-164. doi: 10.1145/779359.779362.
[5]	J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'' 2^nd edition, Applied Mathematical Sciences, 42, Springer-Verlag, New York/Berlin, 1986.
[6]	E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I. Nonstiff Problems," 2^nd edition, Springer Series in Computational Mathematics, 8, Springer-Verlag, Berlin, 1993.
[7]	J. Hindmarsh and M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. R. Soc. London B, 221 (1984), 87-102. doi: 10.1098/rspb.1984.0024.
[8]	A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (London), 117 (1952), 205-249.
[9]	F. C. Hoppensteadt and E. M. Izhikevich, "Weakly Connected Neural Networks,'' Applied Mathematical Sciences, 126, Springer-Verlag, New York, 1997.
[10]	E. M. Izhikevich, "Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting,'' Computational Neuroscience, MIT Press, Cambridge, Mass., 2007.
[11]	C. K. R. T. Jones, Geometric singular perturbation theory, in "Dynamical Systems" (Montecatini Terme, 1994), Lecture Notes in Math., 1609, Springer, Berlin, (1995), 44-118.
[12]	J. Keener and J. Sneyd, "Mathematical Physiology," 2^nd edition, Interdisciplinary Applied Mathematics, 8, Springer-Verlag, New York, 2008.
[13]	B. Krauskopf, K. R. Schneider, J. Sieber, S. M. Wieczorek and M. Wolfrum, Excitability and self-pulsations near homoclinic bifurcations in semiconductor laser systems, Optics Communications, 215 (2003), 367-379. doi: 10.1016/S0030-4018(02)02239-3.
[14]	Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory,'' 3^rd edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 2004.
[15]	H. M. Osinga, A. Sherman and K. T. Tsaneva-Atanasova, Cross-currents between biology and mathematics: the codimension of pseudo-plateau bursting, Discrete and Continuous Dynamical Systems - Series A, 32 (2012), 2853-2878.
[16]	H. M. Osinga and K. T. Tsaneva-Atanasova, Dynamics of plateau bursting depending on the location of its equilibrium, J. Neuroendocrinology, 22 (2010), 1301-1314. doi: 10.1111/j.1365-2826.2010.02083.x.
[17]	B. van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Review, 1 (1920), 701-710.
[18]	B. van der Pol, On relaxation oscillations, Philosophical Magazine, 7 (1926), 978-992.
[19]	J. Rinzel, A formal classification of bursting mechanisms in excitable systems, in "Proceedings of the International Congress of Mathematicians," Vol. 1, 2 (Berkeley, Calif., 1986) (ed. A. M. Gleason), American Mathematical Society, Providence, RI, (1987), 1578-1593.
[20]	H. G. Rotstein, T. Oppermann, J. A. White and N. Kopell, The dynamic structure underlying subthreshold oscillatory activity and the onset of spikes in a model of medial entorhinal cortex stellate cells, J. Comput. Neurosci., 21 (2006), 271-292. doi: 10.1007/s10827-006-8096-8.
[21]	A. Shilnikov and M. Kolomiets, Methods of the qualitative theory for the Hindmarsh-Rose model: A case study. A tutorial, Int. J. Bifurcat. Chaos Appl. Sci. Engrg., 18 (2008), 2141-2168. doi: 10.1142/S0218127408021634.
[22]	K. T. Tsaneva-Atanasova, H. M. Osinga, T. Rieß, and A. Sherman, Full system bifurcation analysis of endocrine bursting models, J. Theoretical Biology, 264 (2010), 1133-1146. doi: 10.1016/j.jtbi.2010.03.030.
[23]	W. Zhang, V. Kirk, J. Sneyd and M. Wechselberger, Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales, The Journal of Mathematical Neuroscience, 1 (2011), 9. doi: 10.1186/2190-8567-1-9.

show all references

References:

[1]	D. Barkley, Slow manifolds and mixed-mode oscillations in the Belousov-Zhabotinskii reaction, J. Chem. Phys., 89 (1998), 5547-5559. doi: 10.1063/1.455561.
[2]	B. Braaksma, Singular Hopf bifurcation in systems with fast and slow variables, J. Nonlin. Sci., 8 (1998), 457-490. doi: 10.1007/s003329900058.
[3]	M. Desroches, B. Krauskopf and H. M. Osinga, The geometry of slow manifolds near a folded node, SIAM J. Appl. Dyn. Syst., 7 (2008), 1131-1162. doi: 10.1137/070708810.
[4]	A. Dhooge, W. Govaerts and Yu. A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Software, 29 (2003), 141-164. doi: 10.1145/779359.779362.
[5]	J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'' 2^nd edition, Applied Mathematical Sciences, 42, Springer-Verlag, New York/Berlin, 1986.
[6]	E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I. Nonstiff Problems," 2^nd edition, Springer Series in Computational Mathematics, 8, Springer-Verlag, Berlin, 1993.
[7]	J. Hindmarsh and M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. R. Soc. London B, 221 (1984), 87-102. doi: 10.1098/rspb.1984.0024.
[8]	A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (London), 117 (1952), 205-249.
[9]	F. C. Hoppensteadt and E. M. Izhikevich, "Weakly Connected Neural Networks,'' Applied Mathematical Sciences, 126, Springer-Verlag, New York, 1997.
[10]	E. M. Izhikevich, "Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting,'' Computational Neuroscience, MIT Press, Cambridge, Mass., 2007.
[11]	C. K. R. T. Jones, Geometric singular perturbation theory, in "Dynamical Systems" (Montecatini Terme, 1994), Lecture Notes in Math., 1609, Springer, Berlin, (1995), 44-118.
[12]	J. Keener and J. Sneyd, "Mathematical Physiology," 2^nd edition, Interdisciplinary Applied Mathematics, 8, Springer-Verlag, New York, 2008.
[13]	B. Krauskopf, K. R. Schneider, J. Sieber, S. M. Wieczorek and M. Wolfrum, Excitability and self-pulsations near homoclinic bifurcations in semiconductor laser systems, Optics Communications, 215 (2003), 367-379. doi: 10.1016/S0030-4018(02)02239-3.
[14]	Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory,'' 3^rd edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 2004.
[15]	H. M. Osinga, A. Sherman and K. T. Tsaneva-Atanasova, Cross-currents between biology and mathematics: the codimension of pseudo-plateau bursting, Discrete and Continuous Dynamical Systems - Series A, 32 (2012), 2853-2878.
[16]	H. M. Osinga and K. T. Tsaneva-Atanasova, Dynamics of plateau bursting depending on the location of its equilibrium, J. Neuroendocrinology, 22 (2010), 1301-1314. doi: 10.1111/j.1365-2826.2010.02083.x.
[17]	B. van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Review, 1 (1920), 701-710.
[18]	B. van der Pol, On relaxation oscillations, Philosophical Magazine, 7 (1926), 978-992.
[19]	J. Rinzel, A formal classification of bursting mechanisms in excitable systems, in "Proceedings of the International Congress of Mathematicians," Vol. 1, 2 (Berkeley, Calif., 1986) (ed. A. M. Gleason), American Mathematical Society, Providence, RI, (1987), 1578-1593.
[20]	H. G. Rotstein, T. Oppermann, J. A. White and N. Kopell, The dynamic structure underlying subthreshold oscillatory activity and the onset of spikes in a model of medial entorhinal cortex stellate cells, J. Comput. Neurosci., 21 (2006), 271-292. doi: 10.1007/s10827-006-8096-8.
[21]	A. Shilnikov and M. Kolomiets, Methods of the qualitative theory for the Hindmarsh-Rose model: A case study. A tutorial, Int. J. Bifurcat. Chaos Appl. Sci. Engrg., 18 (2008), 2141-2168. doi: 10.1142/S0218127408021634.
[22]	K. T. Tsaneva-Atanasova, H. M. Osinga, T. Rieß, and A. Sherman, Full system bifurcation analysis of endocrine bursting models, J. Theoretical Biology, 264 (2010), 1133-1146. doi: 10.1016/j.jtbi.2010.03.030.
[23]	W. Zhang, V. Kirk, J. Sneyd and M. Wechselberger, Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales, The Journal of Mathematical Neuroscience, 1 (2011), 9. doi: 10.1186/2190-8567-1-9.

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