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Exponential periodicity and stability of neural networks with reaction-diffusion terms and both variable and unbounded delays. (English) Zbl 1104.35065
Summary: The exponential periodicity and stability of neural networks with Lipschitz continuous activation functions are investigated, without assuming the boundedness of the activation functions and the differentiability of time-varying delays, as needed in most other papers. The neural networks contain reaction-diffusion terms and both variable and unbounded delays. Some sufficient conditions ensuring the existence and uniqueness of periodic solution and stability of neural networks with reaction-diffusion terms and both variable and unbounded delays are obtained by analytic methods and inequality technique. Furthermore, the exponential converging index is also estimated. The method, which does not make use of Lyapunov functional, is simple and valid for the periodicity and stability analysis of neural networks with variable and/or unbounded delays.

MSC:
35R10Partial functional-differential equations
92B20General theory of neural networks (mathematical biology)
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References:
[1] Chua, L. O.; Yang, L.: Cellular neural networks: theory. IEEE trans. Circuits syst. 35, No. 10, 1257-1272 (1988) · Zbl 0663.94022
[2] Chua, L. O.; Yang, L.: Cellular neural networks: applications. IEEE trans. Circuits syst. 35, No. 10, 1273-1290 (1988)
[3] Roska, T.; Chua, L. O.: Cellular neural networks with non-linear and delay-type template elements nonuniform grids. Int. J. Circuit theory appl. 20, No. 5, 469-481 (1992) · Zbl 0775.92011
[4] Roska, T.; Wu, C. W.; Balsi, M.; Chua, L. O.: Stability and dynamics of delay-type general and cellular neural networks. IEEE trans. Circuits syst. I 39, No. 6, 487-490 (1992) · Zbl 0775.92010
[5] Civalleri, P. P.; Gill, M.; Pandolfi, L.: On stability of cellular neural networks with delay. IEEE trans. Circuits syst. I 40, No. 3, 157-165 (1993) · Zbl 0792.68115
[6] Zheng, Y. X.; Chen, T. P.: Global exponential stability of delayed periodic dynamical systems. Phys. lett. A 322, No. 5--6, 344-355 (2004) · Zbl 1118.81479
[7] Arik, S.; Tavsanoglu, V.: Equilibrium analysis of delayed cnns. IEEE trans. Circuits syst. I 45, No. 2, 168-171 (1998) · Zbl 0917.68223
[8] Cao, J.: Global stability analysis in delayed cellular neural networks. Phys. rev. E 59, No. 5, 5940-5944 (1999)
[9] Cao, J.; Wang, J.: Absolute exponential stability of recurrent neural networks with Lipschitz-continuous activation functions and time delays. Neural networks 17, No. 3, 379-390 (2004) · Zbl 1074.68049
[10] Zhang, J.: Absolutely exponential stability in delayed cellular neural networks. International journal of circuit theory and applications 30, No. 4, 395-409 (2002) · Zbl 1021.68077
[11] Zhang, J.: Global stability analysis in delayed cellular neural networks. Computers math. Applic. 30, No. 10, 1707-1720 (2003) · Zbl 1045.37057
[12] Zhang, J.; Jin, X.: Global stability analysis in delayed Hopfield neural network models. Neural networks 13, No. 7, 745-753 (2000)
[13] Gopalsamy, K.; He, X.: Stability in asymmetric Hopfield nets with transmission delays. Physica D 76, No. 4, 344-358 (1994) · Zbl 0815.92001
[14] Chen, Y.: Global stability of neural networks with distributed delays. Neural networks 15, No. 7, 867-871 (2002)
[15] Rao, V. S. H.; Phaneendra, Bh.R.M.: Global dynamics of bidirectional associative memory neural networks involving transmission delays and dead zones. Neural networks 12, No. 3, 455-465 (1999)
[16] Mohamad, S.; Gopalsamy, K.: Dynamics of a class of discrete-time neural networks and their continuoustime counterparts. Math. comput. Simulat. 53, No. 1, 1-39 (2001)
[17] Zhang, Q.; Wei, X.; Xu, J.: Global exponential stability of Hopfield neural networks with continuously distributed delays. Phys. lett. A 315, No. 6, 431-436 (2003) · Zbl 1038.92002
[18] Zhang, Q.; Ma, R.; Xu, J.: Global stability of bidirectional associative memory neural networks with continuously distributed delays. Science in China (Series F) 46, No. 5, 327-334 (2003) · Zbl 1185.92004
[19] Feng, C.; Plamondon, R.: On the stability analysis of delayed neural networks system. Neural networks 14, No. 9, 1181-1188 (2001)
[20] Liao, X. F.; Wu, F. H.; Yu, J. B.: Stability analyses of cellular neural networks with continuous time delay. J. comput. Appl. math. 143, No. 1, 29-47 (2002) · Zbl 1032.34072
[21] Liao, X. F.; Wong, K. W.; Yang, S. H.: Convergence dynamics of hybrid bidirectional associative memory neural networks with distributed delays. Physics letters A 316, No. 1--2, 55-64 (2003) · Zbl 1038.92001
[22] Zhao, H.: Global asymptotic stability of Hopfield neural network involving distributed delays. Neural networks 17, No. 1, 47-53 (2004) · Zbl 1082.68100
[23] Zhang, Y.; Heng, P.; Leung, K.: Convergence analysis of cellular neural networks with unbounded delay. IEEE trans. Circuit syst. I 48, No. 6, 680-687 (2001) · Zbl 0994.82068
[24] Zhang, J.: Absolute stability analysis in cellular neural networks with variable delays and unbounded delay. Computers math. Applic. 47, No. 2/3, 183-194 (2004) · Zbl 1052.45008
[25] Zhang, J.; Suds, Y.; Iwasa, T.: Absolutely exponential stability of a class of neural networks with unbounded delay. Neural networks 17, No. 3, 391-397 (2004) · Zbl 1074.68057
[26] He, Q.; Kang, L.: Existence and stability of global solution for generalized Hopfield neural network system. Neural parallel and scientific computations 2, No. 2, 165-176 (1994) · Zbl 0815.92002
[27] Carpenter, G.: A geometric approach to singular perturbation problems with application to nerve impulse equations. J. differential equations 23, No. 3, 355-367 (1997)
[28] Evans, J. W.: Nerve axon equations. II: stability at rest. Indiana univ. Math. J. 22, No. 1, 75-90 (1972) · Zbl 0236.92010
[29] Liang, J.; Cao, J.: Global exponential stability of reaction-diffusion recurrent neural networks with timevarying delays. Phys. lett. A 314, No. 5--6, 434-442 (2003) · Zbl 1052.82023
[30] Wang, L.; Xu, D.: Global exponential stability of reaction-diffusion Hopfield neural networks with variable delays. Science in China (Series F) 33, No. 6, 488-495 (2003)
[31] Song, Q.; Cao, J.: Global exponential stability and existence of periodic solutions in BAM neural networks with delays and reaction-diffusion terms. Chaos solitons and fractals 23, No. 2, 421-430 (2005) · Zbl 1068.94534
[32] Hastings, A.: Global stability in Lotka^Volterra systems with diffusion. J. math. Biol. 6, No. 2, 163-168 (1978) · Zbl 0393.92013
[33] Rothe, F.: Convergence to the equilibrium state in the Volterra-Lotka diffusion equations. J. math. Biol. 3, No. 3--4, 319-324 (1976) · Zbl 0355.92013
[34] Joy, M.: On the global convergence of a class of functional differential equations with applications in neural network theory. Journal of mathematical analysis and applications 232, No. 1, 61-81 (1999) · Zbl 0958.34057
[35] Joy, M.: Results concerning the absolute stability of delayed neural networks. Neural networks 13, No. 6, 613-616 (2000)