The singular limit of a Hopf bifurcation
Pages: 2805 - 2823, Issue 8, August 2012

doi:10.3934/dcds.2012.32.2805      Abstract        References        Full text (2069.3K)           Related Articles

John Guckenheimer - Mathematics Department, Cornell University, Ithaca, NY 14853, United States (email)
Hinke M. Osinga - Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand (email)

1	D. Barkley, Slow manifolds and mixed-mode oscillations in the Belousov-Zhabotinskii reaction, J. Chem. Phys., 89 (1998), 5547-5559.
2	B. Braaksma, Singular Hopf bifurcation in systems with fast and slow variables, J. Nonlin. Sci., 8 (1998), 457-490.
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.
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.
5	J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'' 2nd 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," 2nd 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.
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," 2nd 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.
14	Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory,'' 3rd 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.
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.
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.
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.
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.

Go to top