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1. | Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China |
References:
[1] |
D. Ambrosi, G. Arioli and H. Koch, A homoclinic solution for excitation waves on a contractile substratum, SIAM J. Appl. Dyn. Syst., 11 (2012), 1533-1542.
doi: 10.1137/12087654X. |
[2] |
A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, Theory of Bifurcations of Dynamical Systems on a Plane, Israel Program for Scientific Translations, Jerusalem-London, 1973. |
[3] |
G. Ariolia and H. Kochb, Existence and stability of traveling pulse solutions of the FitzHugh-Nagumo equation, Nonlinear Anal, 113 (2015), 51-70.
doi: 10.1016/j.na.2014.09.023. |
[4] |
G. A. Carpenter, A geometric approach to singular perturbation problems with applications to nerve impulse equations, J.Differential Equations, 23 (1977), 335-367.
doi: 10.1016/0022-0396(77)90116-4. |
[5] |
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, J.Biophys., 1 (1961), 445-466.
doi: 10.1016/S0006-3495(61)86902-6. |
[6] |
R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophysics, 17 (1955), 257-278.
doi: 10.1007/BF02477753. |
[7] |
P. Gaspard, R. Kapral and G. Nicolis, Bifurcation phenomena near homoclinic systems: A two-parameter analysis, J. Stat. Phys, 35 (1984), 697-727.
doi: 10.1007/BF01010829. |
[8] |
J. Gukenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[9] |
J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh-Nagumo equation: The singular limit, Discrete Contin. Dyn. Syst. 2 (2009), 851-872.
doi: 10.3934/dcdss.2009.2.851. |
[10] |
J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh-Nagumo equation: Bifurcations in the full system, SIAM J Applied Dynamical Systems, 9 (2009), 138-153.
doi: 10.1137/090758404. |
[11] |
H. Hodgkin, A quantitative description of membrane current and its applications to conduction and excitation in nerves, J. Physiol, 117 (1952), 500-544. |
[12] |
M. Krupa, B. Sandstede and P. Szmolyan, Fast and slow waves in the FitzHugh-Nagumo equation, J.Differential Equations, 133 (1997), 49-97.
doi: 10.1006/jdeq.1996.3198. |
[13] |
M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J.Differential Equations, 174 (2001), 312-368.
doi: 10.1006/jdeq.2000.3929. |
[14] |
Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-3978-7. |
[15] |
W. Liu and E. Van Vleck, Turning points and traveling waves in FitzHugh-Nagumo type equations, J. Differential Equations , 225 (2006), 381-410.
doi: 10.1016/j.jde.2005.10.006. |
[16] |
V. K. Melnikov, On the stability of the center for time periodic perturbations, (Russian) Trudy Moskov. Mat. Obu'su'c, 12 (1963), 3-52. |
[17] |
J. D. Murray, Mathematical Biology, Biomathematics, 19. Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-662-08539-4. |
[18] |
J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070.
doi: 10.1109/JRPROC.1962.288235. |
[19] |
L. Zhang and J. Li, Bifurcations of traveling wave solutions in a coupled non-linear wave equation, Chaos, Solitons and Fractals, 17 (2003), 941-950.
doi: 10.1016/S0960-0779(02)00442-3. |
show all references
References:
[1] |
D. Ambrosi, G. Arioli and H. Koch, A homoclinic solution for excitation waves on a contractile substratum, SIAM J. Appl. Dyn. Syst., 11 (2012), 1533-1542.
doi: 10.1137/12087654X. |
[2] |
A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, Theory of Bifurcations of Dynamical Systems on a Plane, Israel Program for Scientific Translations, Jerusalem-London, 1973. |
[3] |
G. Ariolia and H. Kochb, Existence and stability of traveling pulse solutions of the FitzHugh-Nagumo equation, Nonlinear Anal, 113 (2015), 51-70.
doi: 10.1016/j.na.2014.09.023. |
[4] |
G. A. Carpenter, A geometric approach to singular perturbation problems with applications to nerve impulse equations, J.Differential Equations, 23 (1977), 335-367.
doi: 10.1016/0022-0396(77)90116-4. |
[5] |
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, J.Biophys., 1 (1961), 445-466.
doi: 10.1016/S0006-3495(61)86902-6. |
[6] |
R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophysics, 17 (1955), 257-278.
doi: 10.1007/BF02477753. |
[7] |
P. Gaspard, R. Kapral and G. Nicolis, Bifurcation phenomena near homoclinic systems: A two-parameter analysis, J. Stat. Phys, 35 (1984), 697-727.
doi: 10.1007/BF01010829. |
[8] |
J. Gukenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[9] |
J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh-Nagumo equation: The singular limit, Discrete Contin. Dyn. Syst. 2 (2009), 851-872.
doi: 10.3934/dcdss.2009.2.851. |
[10] |
J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh-Nagumo equation: Bifurcations in the full system, SIAM J Applied Dynamical Systems, 9 (2009), 138-153.
doi: 10.1137/090758404. |
[11] |
H. Hodgkin, A quantitative description of membrane current and its applications to conduction and excitation in nerves, J. Physiol, 117 (1952), 500-544. |
[12] |
M. Krupa, B. Sandstede and P. Szmolyan, Fast and slow waves in the FitzHugh-Nagumo equation, J.Differential Equations, 133 (1997), 49-97.
doi: 10.1006/jdeq.1996.3198. |
[13] |
M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J.Differential Equations, 174 (2001), 312-368.
doi: 10.1006/jdeq.2000.3929. |
[14] |
Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-3978-7. |
[15] |
W. Liu and E. Van Vleck, Turning points and traveling waves in FitzHugh-Nagumo type equations, J. Differential Equations , 225 (2006), 381-410.
doi: 10.1016/j.jde.2005.10.006. |
[16] |
V. K. Melnikov, On the stability of the center for time periodic perturbations, (Russian) Trudy Moskov. Mat. Obu'su'c, 12 (1963), 3-52. |
[17] |
J. D. Murray, Mathematical Biology, Biomathematics, 19. Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-662-08539-4. |
[18] |
J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070.
doi: 10.1109/JRPROC.1962.288235. |
[19] |
L. Zhang and J. Li, Bifurcations of traveling wave solutions in a coupled non-linear wave equation, Chaos, Solitons and Fractals, 17 (2003), 941-950.
doi: 10.1016/S0960-0779(02)00442-3. |
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