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Bifurcation of a Cohen-Grossberg neural network with discrete delays. (English) Zbl 1238.34075
Summary: A simple Cohen-Grossberg neural network with discrete delays is investigated in this paper. The existence of local Hopf bifurcations is first considered by choosing the appropriate bifurcation parameter, and then explicit formulas are given to determine the direction of Hopf bifurcation and stability of the periodic solutions. Moreover, a set of sufficient conditions are given to guarantee the global Hopf bifurcation. Numerical simulations are given to illustrate the obtained results.

34C23Bifurcation (ODE)
92B20General theory of neural networks (mathematical biology)
37N35Dynamical systems in control
Full Text: DOI
[1] M. A. Cohen and S. Grossberg, “Absolute stability of global pattern formation and parallel memory storage by competitive neural networks,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 13, no. 5, pp. 815-826, 1983. · Zbl 0553.92009 · doi:10.1109/TSMC.1983.6313075
[2] S. Townley, A. Ilchmann, M. G. Weiß et al., “Existence and learning of oscillations in recurrent neural networks,” IEEE Transactions on Neural Networks, vol. 11, no. 1, pp. 205-214, 2000. · doi:10.1109/72.822523
[3] Z. Huang and Y. Xia, “Exponential periodic attractor of impulsive BAM networks with finite distributed delays,” Chaos, Solitons and Fractals, vol. 39, no. 1, pp. 373-384, 2009. · Zbl 1197.34124 · doi:10.1016/j.chaos.2007.04.014
[4] J. Wei and S. Ruan, “Stability and bifurcation in a neural network model with two delays,” Physica D, vol. 130, no. 3-4, pp. 255-272, 1999. · Zbl 1066.34511 · doi:10.1016/S0167-2789(99)00009-3
[5] J. Cao and M. Xiao, “Stability and Hopf bifurcation in a simplified BAM neural network with two time delays,” IEEE Transactions on Neural Networks, vol. 18, no. 2, pp. 416-430, 2007. · doi:10.1109/TNN.2006.886358
[6] S. Guo, L. Huang, and L. Wang, “Linear stability and Hopf bifurcation in a two-neuron network with three delays,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 14, no. 8, pp. 2799-2810, 2004. · Zbl 1062.34078 · doi:10.1142/S0218127404011016
[7] C. Huang, L. Huang, J. Feng, M. Nai, and Y. He, “Hopf bifurcation analysis of a two-neuron network with four delays,” Chaos, Solitons and Fractals, vol. 34, no. 3, pp. 795-812, 2007. · Zbl 1149.34047 · doi:10.1016/j.chaos.2006.03.089
[8] X. Zhou, Y. Wu, Y. Li, and X. Yao, “Stability and Hopf bifurcation analysis on a two-neuron network with discrete and distributed delays,” Chaos, Solitons and Fractals, vol. 40, no. 3, pp. 1493-1505, 2009. · Zbl 1197.37132 · doi:10.1016/j.chaos.2007.09.034
[9] Y. Yang and J. Ye, “Stability and bifurcation in a simplified five-neuron BAM neural network with delays,” Chaos, Solitons and Fractals, vol. 42, no. 4, pp. 2357-2363, 2009. · Zbl 1198.34170 · doi:10.1016/j.chaos.2009.03.123
[10] Y. Song, M. Han, and J. Wei, “Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays,” Physica D, vol. 200, no. 3-4, pp. 185-204, 2005. · Zbl 1062.34079 · doi:10.1016/j.physd.2004.10.010
[11] J. J. Wei, C. R. Zhang, and X. L. Li, “Bifurcation in a two-dimensional neural network model with delay,” Applied Mathematics and Mechanics, vol. 26, no. 2, pp. 193-200, 2005 (Chinese). · doi:10.1007/BF02438244
[12] J. Wei, M. G. Velarde, and V. A. Makarov, “Oscillatory phenomena and stability of periodic solutions in a simple neural network with delay,” Nonlinear Phenomena in Complex Systems, vol. 5, no. 4, pp. 407-417, 2002.
[13] H. Zhao and L. Wang, “Stability and bifurcation for discrete-time Cohen-Grossberg neural network,” Applied Mathematics and Computation, vol. 179, no. 2, pp. 787-798, 2006. · Zbl 1147.39303 · doi:10.1016/j.amc.2005.11.148
[14] H. Zhao and L. Wang, “Hopf bifurcation in Cohen-Grossberg neural network with distributed delays,” Nonlinear Analysis. Real World Applications, vol. 8, no. 1, pp. 73-89, 2007. · Zbl 1119.34052 · doi:10.1016/j.nonrwa.2005.06.002
[15] Q. Liu and R. Xu, “Stability and bifurcation of a Cohen-Grossberg neural network with discrete delays,” Applied Mathematics and Computation, vol. 218, no. 6, pp. 2850-2862, 2011. · Zbl 1283.34065 · doi:10.1016/j.amc.2011.08.029
[16] Q. Liu, R. Xu, and Z. Wang, “Stability and bifurcation of a class of discrete-time Cohen-Grossberg neural networks with delays,” Discrete Dynamics in Nature and Society, Article ID 403873, 14 pages, 2011. · Zbl 1215.39019 · doi:10.1155/2011/403873 · eudml:226661
[17] J. K. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 1977. · Zbl 0352.34001
[18] B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41, Cambridge University Press, Cambridge, UK, 1981. · Zbl 0474.34002
[19] J. Wu, “Symmetric functional-differential equations and neural networks with memory,” Transactions of the American Mathematical Society, vol. 350, no. 12, pp. 4799-4838, 1998. · Zbl 0905.34034 · doi:10.1090/S0002-9947-98-02083-2