American Institute of Mathematical Sciences

October  2016, 21(8): 2601-2614. doi: 10.3934/dcdsb.2016063

Phase transition of oscillators and travelling waves in a class of relaxation systems

 1 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

Received  September 2015 Revised  April 2016 Published  September 2016

The main purpose of this article is to investigate the phase transition of oscillation solutions and travelling wave solutions in a class of relaxation systems as follows \begin{eqnarray} \left\{ \begin{array}{ll} \frac{\partial u}{\partial t}=\pm u(u-a)(u-b)-v+D \frac{\partial ^{2}u}{\partial^{2} x},~~~a\neq b, \\\frac{\partial v}{\partial t}=\varepsilon( mu + nv + p ), ~~~~0<\varepsilon\ll 1,\nonumber \end{array} \right. \end{eqnarray} where $a,b,m,n,p$ are parameters in this system. By using the orbit analysis method of planar dynamical system and the homoclinic bifurcation theory, the phase transitions of the solitary oscillators, kink oscillators, periodic oscillators and travelling waves in the relaxation system above are studied. Various critical parameters of the phase transition are obtained under different parametric conditions, while various sufficient conditions to guarantee the existence of the above oscillation solutions and travelling waves are given. As some applications, this paper studied the FitzHugh-Nagumo equation, the van der Pol-equation and the Winfree generic system.
Citation: Da-Peng Li. Phase transition of oscillators and travelling waves in a class of relaxation systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2601-2614. doi: 10.3934/dcdsb.2016063
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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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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