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Dynamical behaviors of fuzzy reaction-diffusion periodic cellular neural networks with variable coefficients and delays. (English) Zbl 1185.35129
Summary: When modeling neural networks in a real world, not only diffusion effect and fuzziness cannot be avoided, but also self-inhibitions, interconnection weights, and inputs should vary as time varies. In this paper, we discuss the dynamical behaviors of delayed reaction-diffusion fuzzy cellular neural networks with varying periodic self-inhibitions, interconnection weights as well as inputs. By using Halanay’s delay differential inequality, $M$-matrix theory and analytic methods, some new sufficient conditions are obtained to ensure the existence, uniqueness, and global exponential stability of the periodic solution, and the exponentially convergent rate index is also estimated. In particular, the traditional assumption on the differentiability of the time-varying delays is no longer needed. The methodology developed in this paper is shown to be simple and effective for the exponential periodicity and stability analysis of neural networks with time-varying delays. Two examples are given to show the usefulness of the obtained results that are less restrictive than recently known criteria.
##### MSC:
 35K57 Reaction-diffusion equations 37N25 Dynamical systems in biology 92B20 General theory of neural networks (mathematical biology)
##### References:
 [1] Chua, L. O.; Yang, L.: Cellular neural networks: theory, IEEE trans. Circ. syst. I 35, No. 10, 1257-1272 (1988) · Zbl 0663.94022 · doi:10.1109/31.7600 [2] Yi, Z.; Heng, P. A.; Leung, K. S.: Convergence analysis of cellular neural networks with unbounded delay, IEEE trans. Circ. syst. I 48, No. 6, 680-687 (2001) · Zbl 0994.82068 · doi:10.1109/81.928151 [3] Yu, G. J.; Lu, C. Y.; Tsai, J. S.; Su, T. J.; Liu, B. D.: Stability of cellular neural networks with time-varying delay, IEEE trans. Circ. syst. I 50, No. 5, 677-678 (2003) [4] Singh, V.: A generalized LMI-based approach to the global asymptotic stability of delayed cellular neural networks, IEEE trans. Neural networks 15, No. 1, 223-225 (2004) [5] Jiang, H. J.; Zhang, L.; Teng, Z. D.: Existence and global exponential stability of almost periodic solution for cellular neural networks with variable coefficients and time-varying delays, IEEE trans. Neural networks 16, No. 6, 1340-1351 (2005) [6] Senan, S.; Arik, S.: New results for exponential stability of delayed cellular neural networks, IEEE trans. Circ. syst. II 52, No. 3, 154-158 (2005) [7] Liao, X. X.; Wang, J.; Zeng, Z. G.: Global asymptotic stability and global exponential stability of delayed cellular neural networks, IEEE trans. Circ. syst. II 52, No. 7, 403-409 (2005) [8] Liang, J.; Wang, Z.; Liu, Y.; Liu, X.: Global synchronization control of general delayed discrete-time networks with stochastic coupling and disturbances, IEEE trans. Syst., man, and cyber. – part B 38, No. 4, 1073-1083 (2008) [9] Huang, L. H.; Huang, C. X.; Liu, B. W.: Dynamics of a class of cellular neural networks with time-varying delays, Phys. lett. A 345, No. 4 – 6, 330-344 (2005) [10] Zhao, H.: Global exponential stability and periodicity of cellular neural networks with variable delays, Phys. lett. A 336, No. 4-5, 331-341 (2005) · Zbl 1136.34348 · doi:10.1016/j.physleta.2004.12.001 [11] Zhang, Q.; Wei, X. P.; Xu, J.: Global exponential stability for nonautonomous cellular neural networks with delays, Phys. lett. A 351, No. 3, 152-160 (2006) [12] Li, X. M.; Ma, C. Q.; Huang, L. H.: Invariance principle and complete stability for cellular neural networks, IEEE trans. Circ. syst. II 53, No. 3, 202-206 (2006) [13] Li, Y. K.; Zhu, L. F.; Liu, P.: Existence and stability of periodic solutions of delayed cellular neural networks, Nonlinear anal.: real world appl. 7, No. 2, 225-234 (2006) · Zbl 1086.92002 · doi:10.1016/j.nonrwa.2005.02.004 [14] T. Yang, L.B. Yang, C.W. Wu, L.O. Chua, Fuzzy celluar neural networks: theory, in: Proceedings of IEEE International Workshop on Cellular Neural Networks and Applications, 1996, pp. 181 – 186. [15] Yang, T.; Yang, L. B.: The global stability of fuzzy cellular neural network, IEEE trans. Circ. syst. I 43, No. 10, 880-883 (1996) [16] Liu, Y. Q.; Tang, W. S.: Exponential stability of fuzzy cellular neural networks with constant and time-varying delays, Phys. lett. A 323, No. 3 – 4, 224-233 (2004) · Zbl 1118.81400 · doi:10.1016/j.physleta.2004.01.064 [17] Yuan, K.; Cao, J. D.; Deng, J. M.: Exponential stability and periodic solutions of fuzzy cellular neural networks with time-varying delays, Neurocomputing 69, No. 13 – 15, 1619-1627 (2006) [18] Huang, T. W.: Exponential stability of fuzzy cellular neural networks with distributed delay, Phys. lett. A 351, No. 1 – 2, 48-52 (2006) [19] Huang, T. W.: Exponential stability of delayed fuzzy cellular neural networks with diffusion, Chaos, solitons fract. 31, No. 3, 658-664 (2007) · Zbl 1138.35414 · doi:10.1016/j.chaos.2005.10.015 [20] Serrano-Gotarredona, T.; Linares-Barranco, B.: Log-domain implementation of complex dynamics reaction – diffusion neural networks, IEEE trans. Neural networks 14, No. 5, 1337-1355 (2003) [21] Wang, L. S.; Xu, D. Y.: Global exponential stability of Hopfield reaction – diffusion neural networks with time-varying delays, Sci. China (Ser. F) 46, No. 6, 466-474 (2005) · Zbl 1186.82062 · doi:10.1360/02yf0146 [22] Chen, T. P.; Lu, W. L.; Chen, G. R.: Dynamical behaviors of a large class of general delayed neural networks, Neural comput. 17, 949-968 (2005) · Zbl 1080.68615 · doi:10.1162/0899766053429417 [23] Gopaldamy, K.; Sariyasa, S.: Time delays and stimulus-dependent pattern formation in periodic environments in isolated neurons, IEEE trans. Neural networks 13, No. 3, 551-563 (2002) [24] Zhou, J.; Liu, Z.; Chen, G.: Dynamics of periodic delayed neural networks, Neural networks 17, 87-101 (2004) · Zbl 1082.68101 · doi:10.1016/S0893-6080(03)00208-9 [25] Huang, T. W.; Cao, J. D.; Li, C. D.: Necessary and sufficient condition for the absolute exponential stability of a class of neural networks with finite delay, Phys. lett. A 352, No. 1 – 2, 94-98 (2006) · Zbl 1187.34100 · doi:10.1016/j.physleta.2005.11.038 [26] Gao, H.; Lam, J.; Wang, C.: Robust energy-to-peak filter design for stochastic time-delay systems, Syst. control lett. 55, No. 2, 101-111 (2006) · Zbl 1129.93538 · doi:10.1016/j.sysconle.2005.05.005 [27] Gao, H.; Lam, J.; Chen, G.: New criteria for synchronization stability of general complex dynamical networks with coupling delays, Phys. lett. A 360, No. 2, 263-273 (2006) [28] Gao, H.; Chen, T.; Lam, J.: A new delay system approach to network-based control, Automatica 44, No. 1, 39-52 (2008) · Zbl 1138.93375 · doi:10.1016/j.automatica.2007.04.020 [29] Roska, T.; Chua, L. O.: Cellular neural networks with nonlinear and delay-type template elements nonumiform grids, Int. J. Circ. theory appl. 20, 469-481 (1992) · Zbl 0775.92011 · doi:10.1002/cta.4490200504