# zbMATH — the first resource for mathematics

Periodic solution to Cohen–Grossberg BAM neural networks with delays on time scales. (English) Zbl 1254.93111
Summary: In this paper, we investigate first the existence and uniqueness of periodic solution in a general Cohen-Grossberg BAM neural networks with delays on time scales by means of a contraction mapping principle. Then, by using the existence result of periodic solution and constructing a Lyapunov functional, we discuss the global exponential stability of periodic solution for above neural networks. In the last section, we also give examples to demonstrate the validity of our global exponential stability result of the periodic solution for above neural networks.

##### MSC:
 93C70 Time-scale analysis and singular perturbations in control/observation systems 92B20 Neural networks for/in biological studies, artificial life and related topics 34C25 Periodic solutions to ordinary differential equations 93D20 Asymptotic stability in control theory
Full Text:
##### References:
 [1] Cao, J.; Liang, J., Boundedness and stability of cohen – grossberg neural networks with time-varying delays, Journal of mathematical analysis and applications, 296, 665-685, (2004) · Zbl 1044.92001 [2] Zhang, S.; Li, C.; Liao, X., Global stability of discrete-time cohen – grossberg neural networks with impulses, Neurocomputing, 73, 3132-3138, (2010) [3] Cohen, M.; Grossberg, S., Absolute stability and global pattern formation and parallel memory storage by competitive neural networks, IEEE transactions on systems man and cybernetics, 13, 815-826, (1983) · Zbl 0553.92009 [4] Li, Y., Existence and stability of periodic solutions for cohen – grossberg neural networks with multiple delays, Chaos solitons fractals, 20, 459-466, (2004) · Zbl 1048.34118 [5] Xiong, W.; Cao, J., Global exponential stability of discrete-time cohen – grossberg neural networks, Neurocomputing, 64, 433-446, (2005) [6] Song, Q.K.; Cao, J.D., Robust stability in cohen – grossberg neural networks with both time-varying and distributed delays, Neural process letter, 27, 179-196, (2008) · Zbl 1396.34036 [7] Yu, W.W.; Cao, J.D.; Wang, J., An LMI approach to global asymptotic stability of the delayed cohen – grossberg neural network via nonsmooth analysis, Neural networks, 20, 810-818, (2007) · Zbl 1124.68100 [8] S. Mohamad, H. Akca, V. Covachev, Discrete-time Cohen-Grossberg neural networks with transmission delays and impulses, Tatra Mountains Mathematical Publications 43(2009) 145-161. · Zbl 1212.93243 [9] Liao, X.; Li, C.; Wong, K., Criteria for exponential stability of cohen – grossberg neural networks, Neural networks, 17, 1401-1414, (2004) · Zbl 1073.68073 [10] Rong, L.B.; Chen, T.P., New results on the robust stability of Cohen Grossberg neural networks with delays, Neural process letter, 24, 3, 193-202, (2006) · Zbl 1131.93365 [11] Wang, L.; Zou, X., Harmless delays in cohen – grossberg neural networks, Physica D, 170, 162-173, (2002) · Zbl 1025.92002 [12] Wang, L.; Zou, X., Exponential stability of cohen – grossberg neural networks, Neural networks, 15, 415-422, (2002) [13] Xiong, W.; Cao, J., Global exponential stability of discrete-time cohen – grossberg neural networks, Neurocomputing, 64, 433-446, (2005) [14] Zhang, J.; Suda, Y.; Komine, H., Global exponential stability of cohen – grossberg neural networks with variable delays, Physics letters A, 338, 44-50, (2005) · Zbl 1136.34347 [15] Lu, W.L.; Chen, T.P., $$R_+^n \operatorname{-} \operatorname{global}$$ stability of a cohen – grossberg neural network system with nonnegative equilibria, Neural networks, 20, 714-722, (2007) · Zbl 1129.68065 [16] Z.G. Zeng, J. Wang, Global exponential stability of recurrent networks with time-varying delays in the presence of strong external stimuli, Neural Networks 19 (2006) 1528-1537. · Zbl 1178.68479 [17] Wang, Z.D.; Liu, Y.R.; Li, M., Stability analysis for stochastic cohen – grossberg neural networks with mixed time delays, IEEE transactions on neural networks, 17, 3, 814-820, (2006) [18] Wang, L., Stability of cohen – grossberg neural networks with distributed delays, Applied mathematics and computation, 160, 93-110, (2005) · Zbl 1069.34113 [19] Wang, Z.D.; Liu, Y.R.; Liu, X.H., On global asymptotic stability of neural networks with discrete and distributed delays, Physics letters A, 345, 4-6, 299-308, (2005) · Zbl 1345.92017 [20] Zhao, H.Y.; Wang, L., Hopf bifurcation in cohen – grossberg neural network with distributed delays, Nonlinear analysis: real world applications, 8, 73-89, (2007) · Zbl 1119.34052 [21] Arik, S.; Orman, Z., Global stability analysis of cohen – grossberg neural networks with time varying delays, Physics letters A, 341, 410-421, (2005) · Zbl 1171.37337 [22] Wu, W.; Cui, B.T.; Lou, X.Y., Some criteria for asymptotic stability of cohen – grossberg neural networks with time varying delays, Neurocomputing, 70, 1085-1088, (2007) [23] Orman, Z.; Arik, S., New results for global stability of cohen – grossberg neural networks with multiple time delays, Neurocomputing, 71, 3053-3063, (2008) [24] Cui, B.T.; Wu, W., Global exponential stability of cohen – grossberg neural networks with distributed delays, Neurocomputing, 72, 386-391, (2008) [25] Feng, J.; Xu, S., New criteria on global robust stability of cohen – grossberg neural networks with time varying delays, Neurocomputing, 72, 445-457, (2008) [26] Xiang, H.J.; Cao, J.D., Exponential stability of periodic solution to cohen – grossberg type BAM neural networks with time-varying delay, Neurocomputing, 72, 1702-1711, (2009) [27] Chen, A.P.; Cao, J.D., Periodic bidirectional cohen – grossberg neural networks with distributed delays, Nonlinear analysis, 66, 2947-2961, (2007) · Zbl 1122.34055 [28] Hilger, S., Analysis on measure chains a unified approach to continuous and discrete calculus, Results in mathematics, 18, 18-56, (1990) · Zbl 0722.39001 [29] Hilger, S., Differential and difference calculus-unified, Nonlinear analysis, 30, 2683-2694, (1997) · Zbl 0927.39002 [30] Agarwal, R.P.; Bohner, M.; O’Regan, D.; Peterson, A., Dynamic equations on time scales: a survey, Journal of computational and applied mathematics, 141, 1-26, (2002) · Zbl 1020.39008 [31] Bohner, M.; Peterson, A., Dynamic equations on time scales: an introduction with applications, (2001), Birkhauser Boston · Zbl 0978.39001 [32] Bohner, M.; Peterson, A., Advances in dynamic equations on time scales, (2003), Birkhaser Boston · Zbl 1025.34001 [33] Li, Y.K.; Chen, X.R.; Zhao, L., Stability and existence of periodic solutions to delayed cohen – grossberg BAM neural networks with impulses on time scales, Neurocomputing, 72, 1621-1630, (2009) [34] Chen, A.P.; Chen, F.L., Periodic solution to BAM neural network with delays on time scales, Neurocomputing, 73, 274-282, (2009) [35] Cohen, M.A.; Grossberg, S., Absolute stability and global pattern formation and partial memory storage by competitive neural networks, IEEE transactions on systems man and cybernetics, 13, 815-826, (1983) · Zbl 0553.92009 [36] Kosko, B., Bi-directional associative memories, IEEE transactions on systems man and cybernetics, 18, 49-60, (1988) [37] Feng, C.; Plamondon, R., Stability analysis of bidirectional associative memory networks with time delays, IEEE transactions on neural networks, 14, 1560-1565, (2003) [38] Gopalsamy, K.; He, X., Delay-independent stability in bidirectional associative memory networks, IEEE transactions on neural networks, 5, 998-1002, (1994) [39] Zhang, Z.Q.; Zhou, D.M., Existence and global exponential stability of a periodic solution for a discrete time interval general BAM neural networks, Journal of the franklin institute, 347, 5, 763-780, (2010) · Zbl 1286.34065 [40] Yang, X.S.; Li, F.; Long, Y.; Cui, X.Z., Existence of periodic solution for discrete time cellular neural networks with complex deviating argument and impulses, Journal of the franklin institute, 347, 2, 559-566, (2010) · Zbl 1185.93077 [41] Wu, R.C., Exponential convergence of BAM neural networks with time-varying coefficients and distributed delays, Nonlinear analysis: real world applications, 11, 562-573, (2010) · Zbl 1186.34121 [42] Gu, H.B.; Jiang, H.J.; Teng, Z.D., Existence and global exponential stability of equilibrium of competitive neural networks with differential time scales and multiple delays, Journal of the franklin institute, 347, 5, 719-731, (2010) · Zbl 1286.93146 [43] Tiam, A.; Cai, M.; Shi, B.; Zhang, Q., Existence and exponential stability of periodic solution for a class of cohen – grossberg BAM neural networks, Neurocomputing, 73, 3147-3159, (2010) [44] Chen, Z.; Zhao, D.H.; Fu, X.L., Discrete analogue of high-order periodic cohen – grossberg neural networks with delays, Applied mathematics and computation, 214, 210-217, (2009) · Zbl 1172.39020 [45] Zhang, Z.; Wang, L., Existence and global exponential stability of a periodic solution to discrete-time cohen – grossberg BAM neural networks with delays, Journal of Korean mathematics society, 48, 4, 727-747, (2011) · Zbl 1236.39014 [46] Fu, X.; Chen, Z., New discrete and analogue of neural networks with nonlinear amplification functional and its periodic dynamical analysis, discrete and continuous dynamical system, Supplement, 391-398, (2007) · Zbl 1163.39300
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.