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Hopf bifurcation and stability analysis on discrete-time Hopfield neural network with delay. (English) Zbl 1136.93039

Summary: A discrete-time Hopfield neural network with delay is considered. We give some sufficient conditions ensuring the local stability of the equilibrium point for this model. By choosing the delay as a bifurcation parameter, we demonstrated that Neimark-Sacker bifurcation (or Hopf bifurcation for map) would occur when the delay exceeds a critical value. A formula for determining the direction bifurcation and stability of bifurcation periodic solutions is given by applying the normal form theory and the center manifold theorem. Some numerical simulations for justifying the theoretical results are also provided.

MSC:

93D99 Stability of control systems
92B20 Neural networks for/in biological studies, artificial life and related topics
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[1] Cao, J., Global exponential stability of Hopfield neural networks, Int. J. Systems Sci., 32, 2, 233-236 (2001) · Zbl 1011.93091
[2] Cao, J., An estimation of the domain of attraction and convergence rate for Hopfield continuous feedback neural networks, Phys. Lett. A, 325, 5-6, 370-374 (2004) · Zbl 1161.82333
[3] Cao, J.; Song, Q., Stability in Cohen-Grossberg-type bidirectional associative memory neural networks with time-varying delays, Nonlinearity, 19, 7, 1601-1617 (2006) · Zbl 1118.37038
[4] Cao, J.; Yuan, K.; Ho, D. W.C.; Lam, J., Global point dissipativity of neural networks with mixed time-varying delays, Chaos, 16, 01310 (2006)
[5] Gopalsamy, K.; He, X., Stability in asymmetric Hopfield neural networks with transmission delay. Physica D, 76, 344-358 (1994) · Zbl 0815.92001
[6] K. Gopalsamy, P. Liu, Dynamics of a hysteretic neuron model, Nonlinear Anal. Real World Appl. 8 (2007) 375-398.; K. Gopalsamy, P. Liu, Dynamics of a hysteretic neuron model, Nonlinear Anal. Real World Appl. 8 (2007) 375-398. · Zbl 1115.34070
[7] Hopfield, J., Neural networks and physical systems with emergent collective computations abilities, Proc. Natl. Acad. USA, 79, 2554-2558 (1982) · Zbl 1369.92007
[8] Kuznetsov, Y. A., Elements of Applied Bifurcation Theory (1998), Springer: Springer New York · Zbl 0914.58025
[9] Mohamad, S.; Naim, A., Discrete-time analogues of integrodifferential equations modelling bidirectional neural networks, J. Comput. Appl. Math., 138, 1-20 (2002) · Zbl 1030.65137
[10] Ren, F.; Cao, J., LMI-based criteria for stability of high-order neural networks with time-varying delay, Nonlinear Anal. Real World Appl., 7, 967-979 (2006) · Zbl 1121.34078
[11] Stuart, A.; Humphries, A., Dynamical Systems and Numerical Analysis (1996), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0869.65043
[12] Xu, D.; Zhao, H., Invariant and attracting sets of Hopfield neural networks with delay, Int. J. Syst. Sci, 32, 863-866 (2001) · Zbl 1003.92002
[13] Xu, D.; Zhao, H., Global dynamics of Hopfield neural networks involving variable delays, Comput. Math. Appl., 42, 39-45 (2001) · Zbl 0990.34036
[14] Yu, W.; Cao, J., Stability and Hopf bifurcation analysis on a four-neuron BAM neural network with time delays, Phys. Lett. A, 351, 1-2, 64-78 (2006) · Zbl 1234.34047
[15] Yuan, Z.; Hu, D.; Huang, L., Stability and bifurcation analysis on a discrete-time system of two neurons, Appl. Math. Lett., 17, 1239-1245 (2004) · Zbl 1058.92012
[16] Yuan, Z.; Hu, D.; Huang, L., Stability and bifurcation analysis on a discrete-time neural network, J. Comput. Appl. Math., 177, 89-100 (2005) · Zbl 1063.93030
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