Liu, Kaiyu; Zhang, Zhengqiu; Wang, Liping Existence and global exponential stability of periodic solution to Cohen-Grossberg BAM neural networks with time-varying delays. (English) Zbl 1242.93102 Abstr. Appl. Anal. 2012, Article ID 805846, 21 p. (2012). Summary: We investigate first the existence of periodic solution in general Cohen-Grossberg BAM neural networks with multiple time-varying delays by means of using degree theory. Then using the existence result of periodic solution and constructing a Lyapunov functional, we discuss global exponential stability of periodic solution for the above neural networks. Our result on global exponential stability of periodic solution is different from the existing results. In our result, both the hypothesis for monotonicity inequality conditions on the behaved functions and the assumption for boundedness from earlier works are removed. We just require that the behaved functions satisfy sign conditions and activation functions are globally Lipschitz continuous. Cited in 3 Documents MSC: 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory 93C95 Application models in control theory × Cite Format Result Cite Review PDF Full Text: DOI References: [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] Z. Liu, S. Lü, S. Zhong, and M. Ye, “Improved robust stability criteria of uncertain neutral systems with mixed delays,” Abstract and Applied Analysis, vol. 2009, Article ID 294845, 18 pages, 2009. · Zbl 1184.93096 · doi:10.1155/2009/294845 [3] M. De la Sen, “Sufficiency-type stability and stabilization criteria for linear time-invariant systems with constant point delays,” Acta Applicandae Mathematicae, vol. 83, no. 3, pp. 235-256, 2004. · Zbl 1067.34078 · doi:10.1023/B:ACAP.0000039018.13226.ed [4] C. Xu and X. He, “Stability and bifurcation analysis in a class of two-neuron networks with resonant bilinear terms,” Abstract and Applied Analysis, vol. 2011, Article ID 697630, 21 pages, 2011. · Zbl 1218.37122 · doi:10.1155/2011/697630 [5] J. Cao and J. Liang, “Boundedness and stability for Cohen-Grossberg neural network with time-varying delays,” Journal of Mathematical Analysis and Applications, vol. 296, no. 2, pp. 665-685, 2004. · Zbl 1044.92001 · doi:10.1016/j.jmaa.2004.04.039 [6] T. Chen and L. Rong, “Delay-independent stability analysis of Cohen-Grossberg neural networks,” Physics Letters. A, vol. 317, no. 5-6, pp. 436-449, 2003. · Zbl 1030.92002 · doi:10.1016/j.physleta.2003.08.066 [7] Y. Li, “Existence and stability of periodic solutions for Cohen-Grossberg neural networks with multiple delays,” Chaos, Solitons and Fractals, vol. 20, no. 3, pp. 459-466, 2004. · Zbl 1048.34118 · doi:10.1016/S0960-0779(03)00406-5 [8] K. Lu, D. Xu, and Z. Yang, “Global attraction and stability for Cohen-Grossberg neural networks with delays,” Neural Networks, vol. 19, no. 10, pp. 1538-1549, 2006. · Zbl 1178.68437 · doi:10.1016/j.neunet.2006.07.006 [9] Q. K. Song and J. D. Cao, “Robust stability in cohen-grossberg neural network with both time-varying and distributed delays,” Neural Processing Letters, vol. 27, no. 2, pp. 179-196, 2008. · Zbl 1396.34036 · doi:10.1007/s11063-007-9068-3 [10] W. Yu, J. Cao, and J. Wang, “An LMI approach to global asymptotic stability of the delayed Cohen-Grossberg neural network via nonsmooth analysis,” Neural Networks, vol. 20, no. 7, pp. 810-818, 2007. · Zbl 1124.68100 · doi:10.1016/j.neunet.2007.07.004 [11] J. Sun and L. Wan, “Global exponential stability and periodic solutions of Cohen-Grossberg neural networks with continuously distributed delays,” Physica D, vol. 208, no. 1-2, pp. 1-20, 2005. · Zbl 1086.34061 · doi:10.1016/j.physd.2005.05.009 [12] X. Liao, C. Li, and K. -W. Wong, “Criteria for exponential stability of Cohen-Grossberg neural networks,” Neural Networks, vol. 17, no. 10, pp. 1401-1414, 2004. · Zbl 1073.68073 · doi:10.1016/j.neunet.2004.08.007 [13] L. Rong and T. Chen, “New results on the robust stability of Cohen-Grossberg neural networks with delays,” Neural Processing Letters, vol. 24, no. 3, pp. 193-202, 2006. · Zbl 1131.93365 · doi:10.1007/s11063-006-9010-0 [14] L. Wang and X. Zou, “Harmless delays in Cohen-Grossberg neural networks,” Physica D, vol. 170, no. 2, pp. 162-173, 2002. · Zbl 1025.92002 · doi:10.1016/S0167-2789(02)00544-4 [15] B. Kosko, “Bidirectional associative memories,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 18, no. 1, pp. 49-60, 1988. · doi:10.1109/21.87054 [16] B. Kosko, “Adaptive bidirectional associative memories,” Applied Optics, vol. 26, no. 23, pp. 4947-4960, 1987. [17] Z. Q. Zhang, Y. Yang, and Y. S. Huang, “Global exponential stability of interval general BAM neural networks with reaction-diffusion terms and multiple time-varying delays,” Neural Networks, vol. 24, pp. 457-465, 2011. · Zbl 1221.35057 · doi:10.1016/j.neunet.2011.02.003 [18] J. Liu and G. Zong, “New delay-dependent asymptotic stability conditions concerning BAM neural networks of neutral type,” Neurocomputing, vol. 72, no. 10-12, pp. 2549-2555, 2009. · doi:10.1016/j.neucom.2008.11.006 [19] J. Cao, D. W. C. Ho, and X. Huang, “LMI-based criteria for global robust stability of bidirectional associative memory networks with time delay,” Nonlinear Analysis, vol. 66, no. 7, pp. 1558-1572, 2007. · Zbl 1120.34055 · doi:10.1016/j.na.2006.02.009 [20] Z. Zhang, K. Liu, and Y. Yang, “New LMI-based condition on global asymptotic stability concerning BAM neural networks of neutral type,” Neurocomputing, vol. 81, pp. 24-32, 2012. · doi:10.1016/j.neucom.2011.10.006 [21] Z. Zhang, W. Liu, and D. Zhou, “Global asymptotic stability to a generalized Cohen-Grossberg BAM neural networks of neutral type delays,” Neural Networks, vol. 25, pp. 94-105, 2012. · Zbl 1266.34124 · doi:10.1016/j.neunet.2011.07.006 [22] C. Feng and R. Plamondon, “Stability analysis of bidirectional associative memory networks with time delays,” IEEE Transactions on Neural Networks, vol. 5, pp. 998-1002, 1994. [23] H. Jiang and J. Cao, “BAM-type Cohen-Grossberg neural networks with time delays,” Mathematical and Computer Modelling, vol. 47, no. 1-2, pp. 92-103, 2008. · Zbl 1143.34048 · doi:10.1016/j.mcm.2007.02.020 [24] X. Nie and J. Cao, “Stability analysis for the generalized Cohen-Grossberg neural networks with inverse Lipschitz neuron activations,” Computers & Mathematics with Applications, vol. 57, no. 9, pp. 1522-1536, 2009. · Zbl 1186.34099 · doi:10.1016/j.camwa.2009.01.003 [25] Z. Zhang and D. Zhou, “Global robust exponential stability for second-order Cohen-Grossberg neural networks with multiple delays,” Neurocomputing, vol. 73, no. 1-3, pp. 213-218, 2009. · doi:10.1016/j.neucom.2009.09.003 [26] Y. Xia, “Impulsive effect on the delayed cohen-grossberg-type BAM neural networks,” Neurocomputing, vol. 73, no. 13-15, pp. 2754-2764, 2010. · doi:10.1016/j.neucom.2010.04.011 [27] A. Chen and J. Cao, “Periodic bi-directional Cohen-Grossberg neural networks with distributed delays,” Nonlinear Analysis A, vol. 66, no. 12, pp. 2947-2961, 2007. · Zbl 1122.34055 · doi:10.1016/j.na.2006.04.016 [28] H. Xiang and J. Cao, “Exponential stability of periodic solution to Cohen-Grossberg-type BAM networks with time-varying delays,” Neurocomputing, vol. 72, no. 7-9, pp. 1702-1711, 2009. · doi:10.1016/j.neucom.2008.07.006 [29] Z. Zhang, G. Peng, and D. Zhou, “Periodic solution to Cohen-Grossberg BAM neural networks with delays on time scales,” Journal of the Franklin Institute, vol. 348, pp. 2759-2781, 2011. · Zbl 1254.93111 [30] Y. Li, X. Chen, and L. Zhao, “Stability and existence of periodic solutions to delayed Cohen-Grossberg BAM neural networks with impulses on time scales,” Neurocomputing, vol. 72, no. 7-9, pp. 1621-1630, 2009. · doi:10.1016/j.neucom.2008.08.010 [31] C. Bai, “Periodic oscillation for Cohen-Grossberg-type bidirectional associative memory neural networks with neutral time-varying delays,” in Proceedings of the 5th International Conference on Natural Computation (ICNC ’09), pp. 18-23, August 2009. · doi:10.1109/ICNC.2009.630 [32] R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, Springer, Berlin, 1977. · Zbl 0339.47031 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.