American Institute of Mathematical Sciences

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

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

Da-Peng Li 1,

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.
Keywords: Fitzhugh-Nagumo equation, orbit analysis, the van der Pol equation, Oscillators, relaxation systems, homoclinic bifurcation, travelling waves, the Winfree generic system..
Mathematics Subject Classification: Primary: 30D05, 32H50; Secondary: 34C23, 34K1.
Citation: Da-Peng Li. Phase transition of oscillators and travelling waves in a class of relaxation systems. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophysics, 17 (1955), 257-278. doi: 10.1007/BF02477753.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. Hodgkin, A quantitative description of membrane current and its applications to conduction and excitation in nerves, J. Physiol, 117 (1952), 500-544. Google Scholar

[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.  Google Scholar

[13]

M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J.Differential Equations, 174 (2001), 312-368. doi: 10.1006/jdeq.2000.3929.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

J. D. Murray, Mathematical Biology, Biomathematics, 19. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophysics, 17 (1955), 257-278. doi: 10.1007/BF02477753.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. Hodgkin, A quantitative description of membrane current and its applications to conduction and excitation in nerves, J. Physiol, 117 (1952), 500-544. Google Scholar

[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.  Google Scholar

[13]

M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J.Differential Equations, 174 (2001), 312-368. doi: 10.1006/jdeq.2000.3929.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

J. D. Murray, Mathematical Biology, Biomathematics, 19. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150
[10]

Zhaosheng Feng, Guangyue Gao, Jing Cui. Duffing--van der Pol--type oscillator system and its first integrals. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1377-1391. doi: 10.3934/cpaa.2011.10.1377
[11]

Vyacheslav Maksimov. Some problems of guaranteed control of the Schlögl and FitzHugh-Nagumo systems. Evolution Equations & Control Theory, 2017, 6 (4) : 559-586. doi: 10.3934/eect.2017028
[12]

Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1549-1563. doi: 10.3934/dcdsb.2020172
[13]

Anhui Gu, Bixiang Wang. Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1689-1720. doi: 10.3934/dcdsb.2018072
[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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (105)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences