Threshold dynamics of a delayed multi-group heroin epidemic model in heterogeneous populations
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,
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
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. doi: 10.1137/12087654X. Google Scholar
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, (1973). Google Scholar
G. Ariolia and H. Kochb, Existence and stability of traveling pulse solutions of the FitzHugh-Nagumo equation,, Nonlinear Anal, 113 (2015), 51. doi: 10.1016/j.na.2014.09.023. Google Scholar
G. A. Carpenter, A geometric approach to singular perturbation problems with applications to nerve impulse equations,, J.Differential Equations, 23 (1977), 335. doi: 10.1016/0022-0396(77)90116-4. Google Scholar
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, J.Biophys., 1 (1961), 445. doi: 10.1016/S0006-3495(61)86902-6. Google Scholar
R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane,, Bull. Math. Biophysics, 17 (1955), 257. doi: 10.1007/BF02477753. Google Scholar
P. Gaspard, R. Kapral and G. Nicolis, Bifurcation phenomena near homoclinic systems: A two-parameter analysis,, J. Stat. Phys, 35 (1984), 697. doi: 10.1007/BF01010829. Google Scholar
J. Gukenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-1140-2. Google Scholar
J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh-Nagumo equation: The singular limit,, Discrete Contin. Dyn. Syst. 2 (2009), 2 (2009), 851. doi: 10.3934/dcdss.2009.2.851. Google Scholar
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. doi: 10.1137/090758404. Google Scholar
H. Hodgkin, A quantitative description of membrane current and its applications to conduction and excitation in nerves,, J. Physiol, 117 (1952), 500. Google Scholar
M. Krupa, B. Sandstede and P. Szmolyan, Fast and slow waves in the FitzHugh-Nagumo equation,, J.Differential Equations, 133 (1997), 49. doi: 10.1006/jdeq.1996.3198. Google Scholar
M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion,, J.Differential Equations, 174 (2001), 312. doi: 10.1006/jdeq.2000.3929. Google Scholar
Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory,, Third edition, (2004). doi: 10.1007/978-1-4757-3978-7. Google Scholar
W. Liu and E. Van Vleck, Turning points and traveling waves in FitzHugh-Nagumo type equations,, J. Differential Equations , 225 (2006), 381. doi: 10.1016/j.jde.2005.10.006. Google Scholar
V. K. Melnikov, On the stability of the center for time periodic perturbations,, (Russian) Trudy Moskov. Mat. Obu'su'c, 12 (1963), 3. Google Scholar
J. D. Murray, Mathematical Biology,, Biomathematics, (1989). doi: 10.1007/978-3-662-08539-4. Google Scholar
J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon,, Proc. IRE, 50 (1962), 2061. doi: 10.1109/JRPROC.1962.288235. Google Scholar
L. Zhang and J. Li, Bifurcations of traveling wave solutions in a coupled non-linear wave equation,, Chaos, 17 (2003), 941. doi: 10.1016/S0960-0779(02)00442-3. Google Scholar
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
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
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
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
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
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 - A, 2019, 39 (9) : 5017-5083. doi: 10.3934/dcds.2019205
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
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
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
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
Arnold Dikansky. Fitzhugh-Nagumo equations in a nonhomogeneous medium. Conference Publications, 2005, 2005 (Special) : 216-224. doi: 10.3934/proc.2005.2005.216
Anna Cattani. FitzHugh-Nagumo equations with generalized diffusive coupling. Mathematical Biosciences & Engineering, 2014, 11 (2) : 203-215. doi: 10.3934/mbe.2014.11.203
Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101
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
Bao Quoc Tang. Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 441-466. doi: 10.3934/dcds.2015.35.441
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
Abiti Adili, Bixiang Wang. Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 643-666. doi: 10.3934/dcdsb.2013.18.643
Abiti Adili, Bixiang Wang. Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise. Conference Publications, 2013, 2013 (special) : 1-10. doi: 10.3934/proc.2013.2013.1
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
Stefan Siegmund. Normal form of Duffing-van der Pol oscillator under nonautonomous parametric perturbations. Conference Publications, 2001, 2001 (Special) : 357-361. doi: 10.3934/proc.2001.2001.357
PDF downloads (5)
HTML views (0)
Da-Peng Li