Existence and Global Convergence of Periodic Solutions in Recurrent Neural Network Models with a General Piecewise Alternately Advanced and Retarded Argument
Tóm tắt
This paper is concerned with existence, uniqueness and global exponential stability of a periodic solution for recurrent neural network described by a system of differential equations with piecewise constant argument of generalized type (in short DEPCAG). The model involves both advanced and delayed arguments. Employing Banach fixed point theorem combined with Green’s function and DEPCAG integral inequality of Gronwall type, we obtain some novel sufficient conditions ensuring the existence as well as the global convergence of the periodic solution. Our results are new, extend and improve earlier publications. Several numerical examples and simulations are also given to show the feasibility of our results.
Tài liệu tham khảo
Akhmet, M.U., Yımaz, E.: Hopfield-type neural networks systems with piecewise constant argument. Int. J. Qual. Theory Differ. Equ. Appl. 3(1–2), 8–14 (2009)
Akhmet, M.U., Arugaslan, D., Yımaz, E.: Stability analysis of recurrent neural networks with piecewise constant argument of generalized type. Neural Netw. 23, 805–811 (2010)
Belair, J., Campbell, S.A., Van den Driessche, P.: Frustration, stability, and delay-induced oscillations in a neural network model. SIAM J. Appl. Math. 56, 245–255 (1996)
Busenberg, S., Cooke, K.: Models of vertically transmitted diseases with sequential-continuous dynamics. In: Lakshmikantham, V. (ed.) Nonlinear Phenomena in Mathematical Sciences, pp. 179–187. Academic Press, New York (1982)
Campbell, S.A., Edwards, R., Van den Driessche, P.: Delayed coupling between two neural network loops. SIAM J. Appl. Math. 65, 316–335 (2004)
Cao, J.: Global asymptotic stability of neural networks with transmission delays. Int. J. Syst. Sci. 31, 1313–1316 (2000)
Cao, J., Liang, J.: Boundedness and stability for Cohen–Grossberg neural network with time-varying delays. J. Math. Anal. Appl. 296, 665–685 (2004)
Ch, E.G.: Dynamical behaviour of neural networks. SIAM J. Algebr. Discrete Methods 6, 749–754 (1985)
Cheng, C.-Y., Lin, K.-H., Shih, C.-W.: Multistability in recurrent neural networks. SIAM J. Appl. Math. 66, 1301–1320 (2006)
Chiu, K.-S.: Stability of oscillatory solutions of differential equations with a general piecewise constant argument. Electron. J. Qual. Theory Differ. Equ. 88, 1–15 (2011)
Chiu, K.-S.: Periodic solutions for nonlinear integro-differential systems with piecewise constant argument. Sci. World J. (2013, to appear)
Chiu, K.-S.: Existence and global exponential stability of equilibrium for impulsive cellular neural network models with piecewise alternately advanced and retarded argument. Abstr. Appl. Anal. (2013, to appear)
Chiu, K.-S., Pinto, M.: Stability of periodic solutions for neural networks with a general piecewise constant argument. In: First Joint International Meeting AMS-SOMACHI, Pucón, Chile, December 15–18 (2010)
Chiu, K.-S., Pinto, M.: Periodic solutions of differential equations with a general piecewise constant argument and applications. Electron. J. Qual. Theory Differ. Equ. 46, 1–19 (2010)
Chiu, K.-S., Pinto, M.: Variation of parameters formula and Gronwall inequality for differential equations with a general piecewise constant argument. Acta Math. Appl. Sin. 27(4), 561–568 (2011)
Chiu, K.-S., Pinto, M.: Oscillatory and periodic solutions in alternately advanced and delayed differential equations. Carpath. J. Math. 29(2), 149–158 (2013)
Chua, L.O., Yang, L.: Cellular neural networks: theory. IEEE Trans. Circuits Syst. 35, 1257–1272 (1988)
Chua, L.O., Yang, L.: Cellular neural networks: applications. IEEE Trans. Circuits Syst. 35, 1273–1290 (1988)
Civalleri, P.P., Gilli, M., Pandolfi, L.: On stability of cellular neural networks with delay. IEEE Trans. Circuits Syst. I 40, 157–164 (1993)
Cooke, K.L., Wiener, J.: Retarded differential equations with piecewise constant delays. J. Math. Anal. Appl. 99, 265–297 (1984)
Cooke, K.L., Wiener, J.: An equation alternately of retarded and advanced type. Proc. Am. Math. Soc. 99, 726–732 (1987)
Cooke, K.L., Wiener, J.: A survey of differential equations with piecewise continuous argument. In: Lecture Notes in Math., vol. 1475, pp. 1–15. Springer, Berlin (1991)
Ermentrout, G.B.: Period doublings and possible chaos in neural models. SIAM J. Appl. Math. 44, 80–95 (1984)
Hopfield, J.J.: Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA 79(8), 2554–2558 (1982)
Huang, Z.K., Wang, X.H., Gao, F.: The existence and global attractivity of almost periodic sequence solution of discrete-time neural networks. Phys. Lett. A 350, 182–191 (2006)
Huang, T., Chan, A., Huang, Y., Cao, J.: Stability of Cohen–Grossberg neural networks with time-varying delays. Neural Netw. 20, 868–873 (2007)
Huang, Z.K., Xia, Y.H., Wang, X.H.: The existence and exponential attractivity of κ-almost periodic sequence solution of discrete time neural networks. Nonlinear Dyn. 50, 13–26 (2007)
Huang, T., Huang, Y., Li, C.: Stability of periodic solution in fuzzy BAM neural networks with finite distributed delays. Neurocomputing 71, 3064–3069 (2008)
Huang, Z.K., Mohamad, S., Feng, C.H.: New results on exponential attractivity of multiple almost periodic solutions of cellular neural networks with time–varying delays. Math. Comput. Model. 52, 1521–1531 (2010)
Li, Y., Huang, L.: Exponential convergence behavior of solutions to shunting inhibitory cellular neural networks with delays and time–varying coefficients. Math. Comput. Model. 48, 499–504 (2008)
Li, Y.-t., Yang, C.-b.: Global exponential stability analysis on impulsive BAM neural networks with distributed delays. J. Math. Anal. Appl. 324, 1125–1139 (2006)
Liu, Q., Cao, J.: Improved global exponential stability criteria of cellular neural networks with time–varying delays. Math. Comput. Model. 43, 423–432 (2006)
Liu, X., Dickson, R.: Stability analysis of Hopfield neural networks with uncertainty. Math. Comput. Model. 34, 353–363 (2001)
Liu, Z., Liao, L.: Existence and global exponential stability of periodic solutions of cellular neural networks with time–varying delays. J. Math. Anal. Appl. 290, 247–262 (2004)
Lu, C., Ding, X., Liu, M.: Numerical simulation of periodic solutions for a class of numerical discretization neural networks. Math. Comput. Model. 52, 386–396 (2010)
Oliveira, J.J.: Global asymptotic stability for neural network models with distributed delays. Math. Comput. Model. 50, 81–91 (2009)
Pinto, M.: Asymptotic equivalence of nonlinear and quasilinear differential equations with piecewise constant arguments. Math. Comput. Model. 49, 1750–1758 (2009)
Pinto, M.: Cauchy and Green matrices type and stability in alternately advanced and delayed differential systems. J. Differ. Equ. Appl. 17(2), 235–254 (2011)
Shah, S.M., Wiener, J.: Advanced differential equations with piecewise constant argument deviations. Int. J. Math. Math. Sci. 6, 671–703 (1983)
Sirovich, L.: Boundary effects in neural networks. SIAM J. Appl. Math. 39, 142–160 (1980)
Terman, D., Lee, E.: Partial synchronization in a network of neural oscillators. SIAM J. Appl. Math. 57, 252–293 (1997)
Van den Driessche, P., Zou, X.: Global attractivity in delayed Hopfield neural network models. SIAM J. Appl. Math. 58, 1878–1890 (1998)
Wu, W., Tong Cui, B., Yang Lou, X.: Global exponential stability of Cohen–Grossberg neural networks with distributed delays. Math. Comput. Model. 47, 868–873 (2008)
Wiener, J.: Differential equations with piecewise constant delays. In: Lakshmikantham, V. (ed.) Trends in the Theory and Practice of Nonlinear Differential Equations, pp. 547–580. Dekker, New York (1983)
Wiener, J.: Generalized Solutions of Functional Differential Equations. World Scientific, Singapore (1993)
Wu, B., Liu, Y., Lu, J.: New results on global exponential stability for impulsive cellular neural networks with any bounded time–varying delays. Math. Comput. Model. 55, 837–843 (2012)
Xu, S., Chu, Y., Lu, J.: Global exponential stability of delayed Hopfield neural networks. Neural Netw. 14, 977–980 (2001)
Xu, S., Chu, Y., Lu, J.: An analysis of global asymptotic stability of delayed cellular neural networks. IEEE Trans. Neural Netw. 13, 1239–1242 (2002)
Yu, Y., Cai, M.: Existence and exponential stability of almost–periodic solutions for high–order Hopfield neural networks. Math. Comput. Model. 47, 943–951 (2008)
Yuan, Z., Yuan, L.: Existence and global convergence of periodic solution of delayed neural networks. Math. Comput. Model. 48, 101–113 (2008)
Zhong, S., Liu, X.: Exponential stability and periodicity of cellular neural networks with time delay. Math. Comput. Model. 45, 1231–1240 (2007)