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Periodic solutions and its exponential stability of reaction-diffusion recurrent neural networks with continuously distributed delays. (English) Zbl 1094.35128
Summary: Both exponential stability and periodic oscillatory solutions are considered for reaction-diffusion recurrent neural networks with continuously distributed delays. By constructing suitable Lyapunov functional, using $M$-matrix theory and some analysis techniques, some simple sufficient conditions are given ensuring the global exponential stability and the existence of periodic oscillatory solutions for reaction-diffusion recurrent neural networks with continuously distributed delays. Moreover, the exponential convergence rate is estimated. These results have leading significance in the design and applications of globally exponentially stable and periodic oscillatory neural circuits for reaction-diffusion recurrent neural networks with continuously distributed delays. Two examples are given to illustrate the correctness of the obtained results.

##### MSC:
 35Q80 Applications of PDE in areas other than physics (MSC2000) 92B20 General theory of neural networks (mathematical biology) 35K57 Reaction-diffusion equations 35B35 Stability of solutions of PDE
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##### References:
 [1] Arik, S.; Tavsanoglu, V.: Equilibrium analysis of delayed CNNS. IEEE trans. Circuits and systems I 45, 168-171 (1998) [2] Baldi, P.; Atiya, A. F.: How delays affect neural dynamics and learning. IEEE trans. Neural networks 5, 312-621 (1994) [3] Cao, J. D.: Global stability analysis in delayed cellular neural networks. Phys. rev. E 59, 5940-5944 (1999) [4] Cao, J. D.: New results concerning exponential stability and periodic solutions of delayed cellular neural networks. Phys. lett. A 307, 136-147 (2003) · Zbl 1006.68107 [5] Cao, J. D.; Wang, J.: Global asymptotic stability of a general class of recurrent neural networks with time-varying delays. IEEE tran. Cricuit and systems I 50, 34-44 (2003) [6] Cao, J. D.; Wang, J.: Absolute exponential stability of recurrent neural networks with Lipschitz-continuous activation functions and time delays. Neural netwoks 17, 379-390 (2004) · Zbl 1074.68049 [7] Cao, J. D.; Zhou, D. M.: Stability analysis of delayed cellular neural networks. Neural networks 11, 1601-1605 (1998) [8] Carpenter, G.: A geometric approach to singular perturbation problems with application to nerve implus equations. J. differential equations 23, 355-367 (1977) · Zbl 0341.35007 [9] Chua, L. O.; Yang, L.: Cellular neural networkstheory. IEEE trans. Circuits and systems 35, 1257-1272 (1988) · Zbl 0663.94022 [10] Cohen, M. A.; Grossberg, S.: Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. IEEE trans. Systems man. Cybernet 35, 815-826 (1983) · Zbl 0553.92009 [11] Evans, J. W.: Nerve axon equs iistability at rest. Indiana univ. Math. J. 22, 75-90 (1973) [12] Gopalsamy, K.; He, X. Z.: Stability in asymmetric Hopfield nets with transmission delays. Physica D 76, 344-358 (1994) · Zbl 0815.92001 [13] Hastings, A.: Global stability in Lotka -- Volterra systems with diffusion. J. math. Biol. 6, 163-168 (1978) · Zbl 0393.92013 [14] He, Q. M.; Kang, L. S.: Existence and stability of global solution for generalized Hopfield neural networks system. Neural parallel sci. Comput. 2, 165-176 (1994) · Zbl 0815.92002 [15] Hopfield, J. J.: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. nat. Acad. sci. USA 81, 3088-3092 (1984) [16] Hopfield, J. J.; Tank, D. W.: Computing with neural circuitsa model. Science 233, 625-633 (1986) [17] Liang, J. L.; Cao, J. D.: Global exponential stability of reaction -- diffusion recurrent neural networks with time-varying delays. Phys. lett. A 314, 434-442 (2003) · Zbl 1052.82023 [18] Marcuss, C. M.; Westervelt, R. M.: Stability of analog neural networks with delay. Phys. rev. A 39, 347-359 (1989) [19] Mohamad, S.; Gopalsamy, K.: Dynamics of a class of discrete-time neurals networks and their continuous-time counterparts. Math. comput. Simulat. 53, 1-39 (2000) [20] 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, 455-465 (1999) [21] Roska, T.; Wu, C. W.; Balsi, M.; Chua, L. O.: Stability and dynamics of delay-type general and cellular neural networks. IEEE trans. Circuits and systems I 39, 487-490 (1992) · Zbl 0775.92010 [22] Roska, T.; Wu, C. W.; Chua, L. O.: Stability of cellular neural networks with dominant nonlinear and delay-type templates. IEEE trans. Circuits and systems I 40, 270-272 (1993) · Zbl 0800.92044 [23] Rothe, F.: Convergence to the equilibrium state in the Volterra -- Lotka diffusion equations. J. math. Biol. 3, 319-324 (1976) · Zbl 0355.92013 [24] Wang, L. S.; Xu, D. Y.: Global exponential stability of Hopfield reaction -- diffusion neural networks with variable delays. Sci. China ser. F 46, 466-474 (2003) · Zbl 1186.82062 [25] Zhang, J. Y.: Global exponential stability of neural networks with variable delays. IEEE trans. Circuits and systems I 50, 288-290 (2003) [26] Zhang, J. Y.; Jin, X. S.: Global stability analysis in delayed Hopfield neural network models. Neural networks 13, 745-753 (2000) [27] Zhang, Q.; Wei, X. P.; Xu, J.: Global exponential stability of Hopfield neural networks with continuously distributed delays. Phys. lett. A 315, 431-436 (2003) · Zbl 1038.92002 [28] Zhao, H. Y.: Global stability of bidirectional associative memory neural networks with distributed delays. Phys. lett. A 297, 182-190 (2002) · Zbl 0995.92002 [29] Cao, J. D.; Liang, J. L.: Boundedness and stability for Cohen-Grossberg neural networks with time-varying delays. J. math. Anal. appl. 296, 665-685 (2004) · Zbl 1044.92001 [30] Cao, J. D.; Ho, D. W. C.: A general framework for global asympotic stability analysis of delayed neural networks based on LMI approach. Chaos, solitons and fractals 24, 1317-1329 (2005) · Zbl 1072.92004 [31] Cao, J. D.; Huang, D. -S.; Qu, Y. Z.: Global robust stability of delayed recurrent neural networks. Chaos, solitons and fractals 23, 221-229 (2005) · Zbl 1075.68070 [32] Cao, J. D.; Liang, J. L.; Lam, J.: Exponential stability of high-order bidirectional associative memory neural networks with time delays. Physica D 199, 425-436 (2004) · Zbl 1071.93048 [33] Cao, J. D.: Periodic solutions and exponential stability in delayed cellular neural networks. Physical review E 60, 3244-3248 (1999) [34] Cao, J. D.: Global stability conditions for delayed cnns. IEEE trans. Circuits and systems I 48, 1330-1333 (2001) · Zbl 1006.34070 [35] Cao, J. D.: A set of stability criteria for delayed cellular neural networks. IEEE trans. Circuits and systems I 48, 494-498 (2001) · Zbl 0994.82066 [36] Cao, J. D.: Exponential stability and periodic solution of delayed cellular neural networks, sci. China ser. E. 43, 328-336 (2000) · Zbl 1019.94041 [37] Cao, J. D.; Wang, J.; Liao, X. F.: Novel stability criteria of delayed cellular neural networks, int. J. neural systems. 13, 367-375 (2003)