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Stability and stabilization of the constrained runs schemes for equation-free projection to a slow manifold
The singular limit of a Hopf bifurcation
1. | Mathematics Department, Cornell University, Ithaca, NY 14853 |
2. | Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142 |
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,'' 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.
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," 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.
doi: 10.1016/S0030-4018(02)02239-3. |
[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.
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,'' 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.
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," 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.
doi: 10.1016/S0030-4018(02)02239-3. |
[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.
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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