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Bautin bifurcation analysis for synchronous solution of a coupled FHN neural system with delay. (English) Zbl 1221.34188
Summary: The Bautin bifurcation of synchronous solution of a coupled FHN neural system with delay is investigated. Firstly, the method of Lyapunov functional is used to obtain conditions for the synchronization of the neural system. Then, distributions of the roots of the characteristic equation associated with the linearization of the synchrosystem are discussed. Center manifold and normal form are employed to calculate its Lyapunov coefficients. A group of sufficient conditions are given to present Bautin bifurcation of the synchrosystem by applying the Bautin bifurcation theorem of delay differential equations developed by Anca-Veronica Ion. The Bautin bifurcation diagram in the physical parameter space is provided to illustrate the correctness of our theoretical analysis.

34K18Bifurcation theory of functional differential equations
34K20Stability theory of functional-differential equations
37N25Dynamical systems in biology
92B20General theory of neural networks (mathematical biology)
Full Text: DOI
[1] Fitzhugh, R.: Impulses and physillogical states in theoretical models of nerve membrane, Biophys J 1, 445-466 (1961)
[2] Nagumo, J.; Arimoto, S.; Yoshizawa, S.: An active pulse transmission line simulating nerve axon, Proc IRE 50, 2061-2070 (1962)
[3] Shepherd, G. M.: Neurobiology, (1983)
[4] Murray, J. D.: Mathematical biology, (1990) · Zbl 0704.92001
[5] Tetsushi, U.; Hisayo, M.; Takuji, K.; Hiroshi, K.: Bifurcation and chaos in coupled BVP oscillators, Int J bifur chaos 4, No. 14, 1305-1324 (2004) · Zbl 1086.37530
[6] Tetsushi, U.; Hiroshi, K.: Bifurcation in asymmetrically coupled BVP oscillators, Int J bifur chaos 5, No. 13, 1319-1327 (2003) · Zbl 1064.34513 · doi:10.1142/S0218127403007199
[7] Kunichika, T.; Kazuyuki, A.; Hiroshi, K.: Bifurcations in synaptically coupled BVP nerons, Int J bifur chaos 4, No. 11, 1053-1064 (2001)
[8] Nikola, B.; Dragana, T.: Dynamics of Fitzhugh -- Nagumo excitable systems with delayed coupling, Phys rev E 67, 0662215-0662221 (2003)
[9] Nikola, B.; Ines, G.; Nebojsa, V.: Type I vs. Type II excitable systems with delayed coupling, Chaos, solitons & fractals 23, No. 2, 1221-1233 (2005) · Zbl 1100.34060
[10] Wang, Qingyun; Lu, Qishao; Chen, Guanrong: Bifurcation and synchronization of synaptically coupled FHN models with time delay, Chaos, solitons & fractals 39, 918-925 (2009)
[11] Ion A-V. On the Bautin bifurcation for systems of delay differential equations. In: Proceedings of ICTAMI 2004. vol. 8, Thessaloniki, Breece: Acta Univ. Apulensis; 2004. p. 235 -- 46. · Zbl 1142.34371
[12] Ion, A. -V.: An example of bautin-type bifurcation in a delay differential equation, J math anal appl 329, 777-789 (2007) · Zbl 1153.34347 · doi:10.1016/j.jmaa.2006.06.083
[13] Hale, J.: Theory of functional differential equations, (1977) · Zbl 0352.34001
[14] Kuznetsov, Y.: Elements of applied bifurcation theory, Appl math sci 112 (1995) · Zbl 0829.58029
[15] Drover, J. D.; Ermentrout, B.: Nonlinear coupling near a degenerate Hopf (Bautin) bifurcation, SIAM J appl math 63, 1627-1647 (2003) · Zbl 1038.37064 · doi:10.1137/S0036139902412617
[16] Govaerts, W.; Sautois, W. B.: Computation of the phase response curve: a direct numerical approach, Neural comput 18, 817-847 (2006) · Zbl 1087.92001 · doi:10.1162/neco.2006.18.4.817
[17] Govaerts, W.; Kuznetsov, Y. A.; Sijnave, B.: Numerical methods for the generalized Hopf bifurcation, SIAM J numer anal 38, 329-346 (2000) · Zbl 0968.65109 · doi:10.1137/S0036142999352552
[18] Yang, Xiaofan; Yang, Maobin; Liu, Huaiyi; Liao, Xiaofeng: Bautin bifurcation in a class of two-neuron networks with resonant bilinear terms, Chaos, solitons & fractals 38, 575-589 (2008) · Zbl 1146.34315
[19] Burić, N.; Todorović, D.: Bifurcations due to small time-lag in coupled excitable systems, Int J bifur chaos 15, 1775-1785 (2005) · Zbl 1092.37538 · doi:10.1142/S0218127405012831
[20] Drover, Jonathan D.; Ermentrout, Bard: Nonlinear coupled near a degenerate Hopf (Bautin) bifurcatio, SIAM J appl math 63, No. 5, 1627-1647 (2003) · Zbl 1038.37064 · doi:10.1137/S0036139902412617