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Periodic solutions for a class of higher-order Cohen-Grossberg type neural networks with delays. (English) Zbl 1143.34046
This paper deals with the existence and global attractivity of periodic solutions to a class of higher-order Cohen-Grossberg type neural networks with delays. Sufficient conditions are obtained to ascertain existence and global attractivity of a periodic solution without the constraints of symmetry of the connection matrix, monotonicity, and smoothness of the activation function. The proposed model is a generalization of the classical Cohen-Grossberg model, as well as Hopfield neural networks. The proofs are based on Gains and Mawhin’s continuation theorem of coincidence degree, a Lyapunov functional and a nonsingular \(M\)-matrix. An example is also given to illustrate the effectiveness of the proposed criteria.

MSC:
34K13 Periodic solutions to functional-differential equations
34K25 Asymptotic theory of functional-differential equations
37N25 Dynamical systems in biology
92B20 Neural networks for/in biological studies, artificial life and related topics
34K20 Stability theory of functional-differential equations
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[1] Cohen, M.A.; Grossberg, S., Absolute stability and global pattern formation and parallel memory storage by competitive neural networks, IEEE trans. syst. man cybern., 13, 815-821, (1983) · Zbl 0553.92009
[2] Hopfield, J.J., Neurons with graded response have collective computational properites like those of two-stage neurons, Proc. nat. acad. sci. USA (biophysics), 81, 3088-3092, (1984) · Zbl 1371.92015
[3] Chua, L.O.; Yang, L., Cellular neural networks: theory, IEEE trans. circuits syst., 35, 1257-1272, (1988) · Zbl 0663.94022
[4] Wang, L.; Zou, X., Exponential stability of cohen – grossberg neural networks, Neural netw., 15, 415-422, (2002)
[5] Ye, H.; Michel, A.N.; Wang, K., Qualitative analysis of cohen – grossberg neural networks with multiple delays, Phys. rev. E, 51, 2611-2618, (1995)
[6] Chen, T.; Rong, L., Delay-independent stability analysis of cohen – grossberg neural networks, Phys. lett. A, 317, 436-449, (2003) · Zbl 1030.92002
[7] Chen, T.; Rong, L., Robust global exponential stability of cohen – grossberg neural networks with time delays, IEEE trans. neural netw., 15, 203-206, (2004)
[8] Cao, J.; Liang, J., Boundedness and stability for cohen – grossberg neural network with time-varying delays, J. math. anal. appl., 296, 665-685, (2004) · Zbl 1044.92001
[9] Wang, L.; Zou, X., Harmless delays in cohen – grossberg neural networks, Physica D, 170, 162-173, (2002) · Zbl 1025.92002
[10] Wang, Z.; Liu, Y.; Li, M.; Liu, X., Stability analysis for stochastic cohen – grossberg neural networks with mixed time delays, IEEE trans. neural netw., 17, 3, 814-820, (2006)
[11] Abu-Mostafa, Y.; Jacques, J., Information capacity of the Hopfield model, IEEE trans. inform. theory, 31, 461-464, (1985) · Zbl 0571.94030
[12] McEliece, R.J.; Posner, E.C.; Rodemich, E.R.; Venkatesh, S.S., The capacity of the Hopfield associative memory, IEEE trans. inform. theory, 33, 461-482, (1983) · Zbl 0631.94016
[13] Baldi, P., Neural networks, orientations of the hypercube and algebraic threshold functions, IEEE trans. inform. theory, 34, 523-530, (1998) · Zbl 0653.94027
[14] Dembo, A.; Farotimi, O.; Kailath, T., High-order absolutely stable neural networks, IEEE trans. circuits syst., 38, 57-65, (1991) · Zbl 0712.92002
[15] Kosmatopoulos, E.B.; Christodoulou, M.A., Structural properties of gradient recurrent higher-order neural networks, IEEE trans. circuits syst. II, 42, 592-603, (1995) · Zbl 0943.68510
[16] Xu, B.; Liu, X.; Liao, X., Global asymptotic stability of high-order Hopfield type neural networks with time delays, Comput. math. appl., 45, 1729-1737, (2003) · Zbl 1045.37056
[17] Cao, J.; Liang, J.; Lam, J., Exponential stability of high-order bidirectional associative memory neural networks with time delays, Physica D, 199, 425-436, (2004) · Zbl 1071.93048
[18] Ren, F.; Cao, J., LMI-based criteria for stability of high-order neural networks with time-varying delay, Nonlinear anal. RWA, 7, 967-979, (2006) · Zbl 1121.34078
[19] Ren, F.; Cao, J., Periodic oscillation of higher-order BAM neural networks with periodic coefficients and delays, Nonlinearity, 20, 605-629, (2007) · Zbl 1136.34055
[20] Cao, J., Periodic oscillation and exponential stability of delayed cnns, Phys. lett. A, 270, 157-163, (2000)
[21] Zhou, J.; Liu, Z.; Chen, G., Dynamics of periodic delayed neural networks, Neural netw., 17, 87-101, (2004) · Zbl 1082.68101
[22] Cao, J.; Wang, L., Periodic oscillation solution of bidirectional associative memory networks with delays, Phys. rev. E, 61, 1825-1828, (2000)
[23] Fu, C.; He, H.; Liao, X., Globally stable periodic state of delayed cohen – grossberg neural networks, Lecture notes in comput. sci., 3496, 276, (2005) · Zbl 1082.68617
[24] Cao, J.; Wang, J., Global exponential stability and periodicity of recurrent neural networks with time delays, IEEE trans. circuits syst. I, 152, 920-931, (2005) · Zbl 1374.34279
[25] Cao, J.; Song, Q., Stability in cohen – grossberg type BAM neural networks with time-varying delays, Nonlinearity, 19, 1601-1617, (2006) · Zbl 1118.37038
[26] Berman, A.; Plemmons, R.J., Nonnegative matrices in the mathematical science, (1979), Academic Press New York
[27] Chen, J.; Chen, X., Special matrices, (2001), Tsinghua Univ. Press Beijing, China, pp. 277-299
[28] Gaines, R.E.; Mawhin, J.L., Coincidence degree and nonlinear differential equations, (1977), Springer-Verlag Berlin · Zbl 0326.34021
[29] Gopalsamy, K., Stability and oscillation in delay equation of population dynamics, (1992), Kluwer Acadenic Publishers Dordreeht · Zbl 0752.34039
[30] Wang, Z.; Liu, Y.; Fraser, K.; Liu, X., Stochastic stability of uncertain Hopfield neural networks with discrete and distributed delays, Phys. lett. A, 354, 4, 288-297, (2006) · Zbl 1181.93068
[31] Wang, Z.; Shu, H.; Liu, Y.; Ho, D.W.C.; Liu, X., Robust stability analysis of generalized neural networks with discrete and distributed time delays, Chaos solitons fractals, 30, 4, 886-896, (2006) · Zbl 1142.93401
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