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September  2011, 16(2): 457-474. doi: 10.3934/dcdsb.2011.16.457

Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme

Fang Han 1, , Bin Zhen 2, , Ying Du 3, , Yanhong Zheng 4, and  Marian Wiercigroch 5,

1. 

School of Information Science and Technology, Donghua University, Shanghai 201620, China
2. 

School of Civil and Architecture Engineering, Wuhan University of Technology, Wuhan 430070, China
3. 

Institute for Cognitive Neurodynamics,School of Science, East China University of Science and Technology, Shanghai, 200237, China
4. 

School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, China
5. 

Centre for Applied Dynamics Research, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom

Received  March 2010 Revised  December 2010 Published  June 2011

Global Hopf bifurcation analysis is carried out on a six-dimensional FitzHugh-Nagumo (FHN) neural network with a time delay. First, the existence of local Hopf bifurcations of the system is investigated and the explicit formulae which can determine the direction of the bifurcations and the stability of the periodic solutions are derived using the normal form method and the center manifold theory. Then the sufficient conditions for the system to have multiple periodic solutions when the delay is far away from the critical values of Hopf bifurcations are obtained by using the Wu's global Hopf bifurcation theory and the Bendixson's criterion. Especially, a synchronized scheme is used during the analysis to reduce the dimension of the system. Finally, example numerical simulations are given to support the theoretical analysis.
Keywords: delay, FitzHugh-Nagumo model, synchronize., Global Hopf bifurcation, neural network.
Mathematics Subject Classification: Primary: 34C23; Secondary: 34C2.
Citation: 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
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show all references

References:
[1]

Proc. Natl. Acad. Sci. USA, 81 (1984), 3088-3092. doi: 10.1073/pnas.81.10.3088.  Google Scholar

[2]

Chaos, Solitons & Fractals, 40 (2009), 377-397. doi: 10.1016/j.chaos.2007.08.040.  Google Scholar

[3]

SIAM J. Applied Dynamical Systems, 6 (2007), 29-60. doi: 10.1137/040614207.  Google Scholar

[4]

IEIC Technical Report, 183 (2003), 33-36. Google Scholar

[5]

Phys. Rev. E, 67 (2003), 066222. doi: 10.1103/PhysRevE.67.066222.  Google Scholar

[6]

Chaos, Solitons & Fractals, 4 (2005), 1221-1233.  Google Scholar

[7]

Chaos, Solitons & Fractals, 39 (2009), 918-925. doi: 10.1016/j.chaos.2007.01.061.  Google Scholar

[8]

Physica D, 198 (2004), 106-119. doi: 10.1016/j.physd.2004.08.023.  Google Scholar

[9]

Physics Letters A, 360 (2007), 689-695. doi: 10.1016/j.physleta.2006.08.078.  Google Scholar

[10]

J. Comp. Appl. Math., 196 (2006), 579-595. doi: 10.1016/j.cam.2005.10.012.  Google Scholar

[11]

Cambridge University Press, Cambridge, 1981.  Google Scholar

[12]

Trans. Amer. Math. Soc., 350 (1998), 4799-4838. doi: 10.1090/S0002-9947-98-02083-2.  Google Scholar

[13]

J. Differential Equations, 106 (1994), 27-39. doi: 10.1006/jdeq.1993.1097.  Google Scholar

[14]

Biophys. J., 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

[15]

Proc. IRE, 50 (1962), 2061-2070. Google Scholar

[16]

Academic Press, INC, 1993.  Google Scholar

[17]

Springer, New York, 1977.  Google Scholar

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