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.

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.

[1]

Boris Anicet Guimfack, Conrad Bertrand Tabi, Alidou Mohamadou, Timoléon Crépin Kofané. Stochastic dynamics of the FitzHugh-Nagumo neuron model through a modified Van der Pol equation with fractional-order term and Gaussian white noise excitation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2229-2243. doi: 10.3934/dcdss.2020397

[2]

Xiaoqin P. Wu, Liancheng Wang. Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 503-516. doi: 10.3934/dcdsb.2010.13.503

[3]

Matthieu Alfaro, Hiroshi Matano. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1639-1649. doi: 10.3934/dcdsb.2012.17.1639

[4]

John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 851-872. doi: 10.3934/dcdss.2009.2.851

[5]

Zhaosheng Feng. Duffing-van der Pol-type oscillator systems. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1231-1257. doi: 10.3934/dcdss.2014.7.1231

[6]

Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 775-794. doi: 10.3934/dcdsb.2020134

[7]

Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457

[8]

Willem M. Schouten-Straatman, Hermen Jan Hupkes. Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5017-5083. doi: 10.3934/dcds.2019205

[9]

Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150

[10]

Vyacheslav Maksimov. Some problems of guaranteed control of the Schlögl and FitzHugh-Nagumo systems. Evolution Equations and Control Theory, 2017, 6 (4) : 559-586. doi: 10.3934/eect.2017028

[11]

Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1549-1563. doi: 10.3934/dcdsb.2020172

[12]

Anhui Gu, Bixiang Wang. Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1689-1720. doi: 10.3934/dcdsb.2018072

[13]

Zhaosheng Feng, Guangyue Gao, Jing Cui. Duffing--van der Pol--type oscillator system and its first integrals. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1377-1391. doi: 10.3934/cpaa.2011.10.1377

[14]

Arnold Dikansky. Fitzhugh-Nagumo equations in a nonhomogeneous medium. Conference Publications, 2005, 2005 (Special) : 216-224. doi: 10.3934/proc.2005.2005.216

[15]

Anna Cattani. FitzHugh-Nagumo equations with generalized diffusive coupling. Mathematical Biosciences & Engineering, 2014, 11 (2) : 203-215. doi: 10.3934/mbe.2014.11.203

[16]

Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106

[17]

Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101

[18]

Wenqiang Zhao. Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3453-3474. doi: 10.3934/dcdsb.2018251

[19]

Bao Quoc Tang. Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 441-466. doi: 10.3934/dcds.2015.35.441

[20]

B. Ambrosio, M. A. Aziz-Alaoui, V. L. E. Phan. Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (155)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]