具有时滞的非自治模糊细胞神经网络动力学行为研究
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摘要
由于神经网络在应用方面的巨大潜力,很多学者都致力于神经网络的理论研究,并取得了许多很好的成果。本文主要涉及模糊细胞神经网络的稳定性和周期解的存在性研究.其中包括:具有变时滞非自治周期模糊细胞神经网络的稳定性以及周期解的存在性的研究和具有扩散项和变时滞的非自治周期模糊细胞神经网络的有界性和全局稳定性的研究.
本文的主要内容可以概述如下:
首先在第一节,第一部分,介绍了神经网络的产生及意义.在随后的第二部分,介绍了细胞神经网络的数学模型.在第三部分中,介绍了模糊细胞神经网络的研究成果.在第四部分中,给出了本文的组织结构.
在第二节中,讨论了具有离散时滞的非自治周期细胞神经网络,利用Young不等式,和构造Lyapunov泛函的方法,得到了该系统的一系列关于周期解存在性和指数稳定性的判定准则.并用数值模拟验证了结果的有效性.在这一部分中,我们没有要求激活函数必须是可微,单调和有界的,只需它满足Lipschitz连续条件即可.
在第三节中,讨论了具有分布时滞的非自治周期细胞神经网络,在这一部分,我们给出了周期解的弱全局指数稳定性定义,这是前面工作中没有的.通过利用矩阵理论和不等式分析技巧,得到了一系列确保周期解的存在性,全局指数稳定性和弱全局指数稳定性的充分准则.特别地,在讨论周期解的弱全局指数稳定性时,我们没要求系统中的变系数ci(t)(i = 1,2,···,n)是恒正的.
在第四节,我们研究了国内外文献较少涉及的具有扩散项和离散变时滞的非自治模糊细胞神经神经网络模型,通过利用不等式分析技巧,巧妙引入多个实参数,构造适当的辅助函数,得到了一系列关于解的有界性和全局指数稳定性的充分准则.最后给出了两个实例说明了这一理论的有效性.
Since neural networks have enormous potential in wide varieties of appli-cations, many specialists and scholars apply themselves to the research of thetheory and achieve many perfect productions. In this paper, we perform re-searches of the stability and existence of periodic solution of fuzzy cellular neuralnetworks (FCNNs). The main contents of this paper include: the existence ofperiodic solution and the stability analysis for a class of FCNNs with variable co-e?cients and time-varying delays,and boundedness and exponential stability fornonautonomous FCNNs with reaction-di?usion terms and time-varying delaysrespectively.
The main contents in this paper can be summarized as follows:
1. Firstly, in the first section of the first chapter, we introduce the develop-mental process and significance of neural networks. In the following section 2 weintroduce the models of cellular neural networks. In section 3, we introduce theresearch results for fuzzy cellular neural networks. In section 4, the organizationof this paper is given.
2. In the second chapter, we investigate the fuzzy cellular neural networks(FCNNs) with variable coe?cients and time-varying delays. A series of simplesu?cient conditions is obtained for checking the boundedness, global exponentialstability and the existence and global exponential stability of periodic solutionsof FCNNs with variable coe?cients and time-varying delays by using Younginequality and general Lyapunov functional.
3. In the third chapter, a class of fuzzy cellular neural networks (FCNNs)with distributed delays and variable coe?cients is discussed. By applying the matrix theory and the inequality analysis technique, some su?cient conditions onthe existence, global exponential stability and weak global exponential stabilityof periodic solutions are established. Particularly, di?erently from some previousworks, in this paper, we do not require the coe?cients ci(t)(i = 1,2,···,n) inthe system are positive.
4. In the forth chapter, nonautonomous fuzzy cellular neural networks withreaction-di?usion terms and time-varying delays are studied. By applying theinequality analysis technique, introducing ingeniously many real parameters andconstructing new auxiliary functions, some new and useful criteria on the bound-edness and global exponential stability are established. Finally, we give twoexamples to illustrate the e?ciency of our results.
本文的主要内容可以概述如下:
首先在第一节,第一部分,介绍了神经网络的产生及意义.在随后的第二部分,介绍了细胞神经网络的数学模型.在第三部分中,介绍了模糊细胞神经网络的研究成果.在第四部分中,给出了本文的组织结构.
在第二节中,讨论了具有离散时滞的非自治周期细胞神经网络,利用Young不等式,和构造Lyapunov泛函的方法,得到了该系统的一系列关于周期解存在性和指数稳定性的判定准则.并用数值模拟验证了结果的有效性.在这一部分中,我们没有要求激活函数必须是可微,单调和有界的,只需它满足Lipschitz连续条件即可.
在第三节中,讨论了具有分布时滞的非自治周期细胞神经网络,在这一部分,我们给出了周期解的弱全局指数稳定性定义,这是前面工作中没有的.通过利用矩阵理论和不等式分析技巧,得到了一系列确保周期解的存在性,全局指数稳定性和弱全局指数稳定性的充分准则.特别地,在讨论周期解的弱全局指数稳定性时,我们没要求系统中的变系数ci(t)(i = 1,2,···,n)是恒正的.
在第四节,我们研究了国内外文献较少涉及的具有扩散项和离散变时滞的非自治模糊细胞神经神经网络模型,通过利用不等式分析技巧,巧妙引入多个实参数,构造适当的辅助函数,得到了一系列关于解的有界性和全局指数稳定性的充分准则.最后给出了两个实例说明了这一理论的有效性.
Since neural networks have enormous potential in wide varieties of appli-cations, many specialists and scholars apply themselves to the research of thetheory and achieve many perfect productions. In this paper, we perform re-searches of the stability and existence of periodic solution of fuzzy cellular neuralnetworks (FCNNs). The main contents of this paper include: the existence ofperiodic solution and the stability analysis for a class of FCNNs with variable co-e?cients and time-varying delays,and boundedness and exponential stability fornonautonomous FCNNs with reaction-di?usion terms and time-varying delaysrespectively.
The main contents in this paper can be summarized as follows:
1. Firstly, in the first section of the first chapter, we introduce the develop-mental process and significance of neural networks. In the following section 2 weintroduce the models of cellular neural networks. In section 3, we introduce theresearch results for fuzzy cellular neural networks. In section 4, the organizationof this paper is given.
2. In the second chapter, we investigate the fuzzy cellular neural networks(FCNNs) with variable coe?cients and time-varying delays. A series of simplesu?cient conditions is obtained for checking the boundedness, global exponentialstability and the existence and global exponential stability of periodic solutionsof FCNNs with variable coe?cients and time-varying delays by using Younginequality and general Lyapunov functional.
3. In the third chapter, a class of fuzzy cellular neural networks (FCNNs)with distributed delays and variable coe?cients is discussed. By applying the matrix theory and the inequality analysis technique, some su?cient conditions onthe existence, global exponential stability and weak global exponential stabilityof periodic solutions are established. Particularly, di?erently from some previousworks, in this paper, we do not require the coe?cients ci(t)(i = 1,2,···,n) inthe system are positive.
4. In the forth chapter, nonautonomous fuzzy cellular neural networks withreaction-di?usion terms and time-varying delays are studied. By applying theinequality analysis technique, introducing ingeniously many real parameters andconstructing new auxiliary functions, some new and useful criteria on the bound-edness and global exponential stability are established. Finally, we give twoexamples to illustrate the e?ciency of our results.
引文
[1]胡守仁,余少波,戴葵,神经网络导论, 1993,国防科技大学出版社.
[2]黄立宏,李雪梅,细胞神经网络动力学, 2007,科学出版社.
[3] A. Berman, R. J. Plemmons, Nonnegative Matrices in the MathematicalScience, Academic Press, New york, 1979.
[4] J. Cao, Global stability conditions for delayed CNNs,IEEE Trans. CircuitsSystems I 48 (11) (2001) 1330-1333.
[5] J.Cao and L.Wang, Exponential stability and periodic oscillatory solutionin BAM networks with delays, IEEE Trans.Neural Networks 13(2002)457-463.
[6] J.Cao, Global asymptotic stability of delayed bi-directional associa-tive memory neural networks. Applied Mathematics and Computa-tion,142(2003) 333-339.
[7] J. Cao, New results concerning exponential stability and periodic solutionsof delayed cellular neural networks, Phys. Lett. A 307(2003) (2-3)
[8] J.Cao, K.Yuan, H.Li, Global Asymptotical stability of generalized recurrentneural networks with multiple discrete delays and distributed delays,IEEETrans, Neural Networks,17:6(2006) 1646-1651.
[9] J.Cao, Q.Song, Stability in Cohen-Grossbery type BAM neyral networkswith time-varying delays,Nonliearity,19:7(2006)1601-1617.
[10] J.Cao, Daniel W.C.Ho, X.Huang, LMI-based criteria for global robust sta-bility of bidirectional associative memory networks with time delay, Non-liear Analysis, series.A66:7(2007) 1558-1572.
[11] A. Chen, J. Cao and L. Huang, Global robust stability of interval cellularneural networks with time-varying delays, Chaos, Solitons & Fractals,23(2005), 787-799.
[12] Hyun J. Cho and Ju H. Park, Novel delay-dependent robust stability cri-terion of delayed cellular neural networks, Chaos, Solitons & Fractals,32(2007), 1194-1200.
[13] L. O. Chua and L. Yang, Cellular neural networks: theory, IEEE Trans.Circ. Syst., 35(1988), 1257-1272.
[14] L. O. Chua and L. Yang, Cellular neural networks: applications, IEEETrans. Circ. Syst., 35(1988), 1273-1290.
[15] R.S. Gau, C.H. Lien and J.G. Hsieh, Global exponential stability for un-certain cellular neural networks with multiple time-varying delays via LMIapproach, Chaos, Solitons & Fractals, 32(2007), 1258-1267.
[16] Z. Gui, X. Yang and W. Ge, Existence and global exponential stability ofperiodic solutions of recurrent cellular neural networks with impulses anddelays, Math. Comput. Simul., 2007, In Press.
[17] J. Hu, S. Zhong and L. Liang, Exponential stability analysis of stochasticdelayed cellular neural networks, Chaos, Solitons & Fractals, 27(2006),1006-1010.
[18] T. Huang, Exponential stability of fuzzy cellular neural networks with dis-tributed delay, Phys. Letters A, 351(2006), 48-52.
[19] T. Huang, Exponential stability of delayed fuzzy cellular neural networkswith di?usion, Chaos, Solitons & Fractals, 31(2007), 658-664.
[20] H. Jiang, Z. Li and Z. Teng, Boundedness and stability for nonautonomouscellular neural networks with delay, Phys. Letters A, 306(2003), 313-325.
[21] H. Jiang and Z. Teng, Global exponential stability of cellular neuralnetworks with time-varying coe?cients and delays, Neural Networks,17(2004), 1415-1425.
[22] H. Jiang and Z.Teng, Some new results for recurrent neural networks withvarying-time coe?cients and delays, Phys. Letters A, 308(2005), 446-460.
[23] H. Jiang and Z. Teng, A new criterion on the global exponential stability forcellular neural networks with multiple time-varying delays, Phys. LettersA, 338(2005), 461-471.
[24] H. Jiang, L. Zhang and Z. Teng, Existence and global exponential stabilityof almost periodic solution for cellular neural networks with variable coe?-cients and time-varying delays, IEEE Trans. Neural Networks, 16(2005),1340-1351.
[25] H. Jiang, J. Cao and Z. Teng, Dynamics of Cohen-Grossberg neural net-works with time-varying delays, Phys. Letters A, 354(2006), 414-422.
[26] J.P. Lasalle, The Stability of Dynamical System, Society of Industrial andApplied Mathematics, Philadephia, 1976.
[27] J. Liang and J. Cao, Global exponential stability of reaction-di?usionrecurrent neural networks with time-varying delays, Phys. Letters A,314(2003), 434-442.
[28] Y. Li and Z. Xing, Existence and global exponential stability of periodicsolution of CNNs with impulses, Chaos, Solitons & Fractals, 33(2007),1686-1693.
[29] B. Liu and L. Huang, Existence and exponential stability of almost periodicsolutions for cellular neural networks with mixed delays, Chaos, Solitons& Fractals, 32(2007), 95-103.
[30] B. Liu and L. Huang, Existence of periodic solutions for cellular neural net-works with complex deviating arguments, Appl. Math. Letters, 20(2007),103-109.
[31] Y. Liu and W. Tang, Exponential stability of fuzzy neural networks withconstant and time-varying delays, Phys. Letters A, 323(2004), 224-233.
[32] X. Lou and B. Cui, Boundedness and exponential stability for nonau-tonomous cellular neural networks with reaction-di?usion terms, Chaos,Solitons & Fractals, 33(2007), 653-662.
[33] X. Lou and B. Cui, Boundedness and exponential stability for nonau-tonomous RCNNs with distributed delays, Comput. Math. Applic., 2007,In press.
[34] X. Lou and B. Cui, Boundedness and exponential stability for nonau-tonomous RCNNs with distributed delays, Comput. Math. Applic.,54(2007), 589-598.
[35] S. Mohamad, Global exponential stability in DCNNs with distributed de-lays and unbounded activations, J. Comput. Appl. Math., 205(2007), 161-173.
[36] Ju H. Park, An analysis of global robust stability of uncertain cellularneural networks with discrete and distributed delays, Chaos, Solitons &Fractals, 32(2007), 800-807.
[37] Ju H. Park and Hyun J. Cho, A delay-dependent asymptotic stability crite-rion of cellular neural networks with time-varying discrete and distributeddelays, Chaos, Solitons & Fractals, 33(2007), 436-442.
[38] Vimal Singh, New global robust stability results for delayed cellular neuralnetworks based on normbounded uncertainties, Chaos, Solitons & Frac-tals, 30(2006), 1165-1171.
[39] Q. Song and J. Cao, Global exponential stability and existence of peri-odic solutions in BAM networks with delays and reaction-di?usion terms,Chaos, Solitons & Fractals, 23(2005), 421-430.
[40] Q. Song and J. Cao, Dynamical behaviors of discrete-time fuzzy cellularneural networks with variable delays and impulses, J. Frank. Inst., 2007,In press.
[41] Q. Song and J. Cao, Impulsive e?ects on stability of fuzzy Cohen-Grossbergneural networks with time-varying delays, IEEE Trans. Syst. Man Cy-bern., B: Cybernetics, 37(2007), 733-741.
[42] Q. Song, J. Cao and Z. Zhao, Periodic solutions and its exponential sta-bility of reaction-di?usion recurrent neural networks with continuously dis-tributed delays, Nonl. Anal.: RWA, 7(2006), 65-80.
[43] Q. Song, Z. Zhao and Y. Li, Global exponential stability of BAM neu-ral networks with distributed delays and reaction-di?usion terms, Phys.Letters A, 335(2005), 213-225.
[44] J. Sun and L. Wan, Convergence dynamics of stochastic reaction-di?usionrecurrent neural networks with delays, Int. J. Bifurc. Chaos, 15(2005),2131-2144.
[45] J. Wang and J. Lu, Global exponential stability of fuzzy cellular neuralnetworks with delays and reaction-di?usion terms, Chaos, Solitons &Fractals, 2007, In press.
[46] L. Wang and D. Xu, Asymptotic behavior of a class of reaction-di?usionequations with delays, J. Math. Anal. Appl., 281(2003), 439-453.
[47] L. Wang and Y. Gao, Global exponential robust stability of reaction-di?usion interval neural networks with time-varying delays, Phys. LettersA, 350(2006), 342-348.– 56–
[48] S. Wang, D. Fu, M. Xu and D. Hu, Advanced fuzzy cellular neural network:Application to CT liver images, Artif. Intel. Medicine, 39(2007), 65-77.
[49] S. Wang, Korris F. L. Chung and D. Fu, Applying the improved fuzzycellular neural network IFCNN to white blood cell detection, Neurocom-puting, 2007, In press.
[50] S. Wang, D. Fu, M. Xu and D.Hu, Advanced fuzzy cellular neural network:Application to CT liver images, Artif. Intell. Medic., 39(2007), 65―77.
[51] J. Wang and J.G. Lu, Global exponential stability of fuzzy cellular neu-ral networks with delays and reaction-di?usion terms, Chaos, Solitons &Fractals, 2007, In press.
[52] Y. Xia, J. Cao and Z. Huang, Existence and exponential stability of almostperiodic solution for shunting inhibitory cellular neural networks, Chaos,Solitons & Fractals, 34(2007), 1599-1607.
[53] T. Yang and L. B. Yang, The global stability of fuzzy cellular neural net-works, IEEE Trans. Circ. Syst.: I, 43(1996), 880-883.
[54] T. Yang, L. B. Yang, C. W. Wu and L. O. Chua, Fuzzy cellular neural net-works: theory, Proceedings of IEEE International Workshop on CellularNeural Networks and Applications, 1996, 181-186.
[55] K. Yuan, J. Cao and J. Deng, Exponential stability and periodic solutionsof fuzzy cellular neural networks with time-varying delays, Neurocom-puting, 69(2006), 1619-1627.
[56] Z. Zeng, D. Huang and Z. Wang, Pattern memory analysis based on stabil-ity theory of cellular neural networks, Appl. Math. Modelling, 32(2008),112-121.
[57] Q. Zhang, X. Wei and J. Xu, Delay-dependent exponential stability of cellu-lar neural networks with time-varying delays, Chaos, Solitons & Fractals,23(2005), 1363-1369.
[58] S. Zhong and X. Liu, Exponential stability and periodicity of cellular neuralnetworks with time delay, Mathl. Comput. Modelling, 45(2007), 1231-1240.
[59] Q. Zhou, J. Sun and G. Chen, Global exponential stability and periodicoscillations of reaction-di?usion BAM neural networks with periodic coef-ficients and general delays, J. Bifur. Chaos, 17(2007), 129-142.
[2]黄立宏,李雪梅,细胞神经网络动力学, 2007,科学出版社.
[3] A. Berman, R. J. Plemmons, Nonnegative Matrices in the MathematicalScience, Academic Press, New york, 1979.
[4] J. Cao, Global stability conditions for delayed CNNs,IEEE Trans. CircuitsSystems I 48 (11) (2001) 1330-1333.
[5] J.Cao and L.Wang, Exponential stability and periodic oscillatory solutionin BAM networks with delays, IEEE Trans.Neural Networks 13(2002)457-463.
[6] J.Cao, Global asymptotic stability of delayed bi-directional associa-tive memory neural networks. Applied Mathematics and Computa-tion,142(2003) 333-339.
[7] J. Cao, New results concerning exponential stability and periodic solutionsof delayed cellular neural networks, Phys. Lett. A 307(2003) (2-3)
[8] J.Cao, K.Yuan, H.Li, Global Asymptotical stability of generalized recurrentneural networks with multiple discrete delays and distributed delays,IEEETrans, Neural Networks,17:6(2006) 1646-1651.
[9] J.Cao, Q.Song, Stability in Cohen-Grossbery type BAM neyral networkswith time-varying delays,Nonliearity,19:7(2006)1601-1617.
[10] J.Cao, Daniel W.C.Ho, X.Huang, LMI-based criteria for global robust sta-bility of bidirectional associative memory networks with time delay, Non-liear Analysis, series.A66:7(2007) 1558-1572.
[11] A. Chen, J. Cao and L. Huang, Global robust stability of interval cellularneural networks with time-varying delays, Chaos, Solitons & Fractals,23(2005), 787-799.
[12] Hyun J. Cho and Ju H. Park, Novel delay-dependent robust stability cri-terion of delayed cellular neural networks, Chaos, Solitons & Fractals,32(2007), 1194-1200.
[13] L. O. Chua and L. Yang, Cellular neural networks: theory, IEEE Trans.Circ. Syst., 35(1988), 1257-1272.
[14] L. O. Chua and L. Yang, Cellular neural networks: applications, IEEETrans. Circ. Syst., 35(1988), 1273-1290.
[15] R.S. Gau, C.H. Lien and J.G. Hsieh, Global exponential stability for un-certain cellular neural networks with multiple time-varying delays via LMIapproach, Chaos, Solitons & Fractals, 32(2007), 1258-1267.
[16] Z. Gui, X. Yang and W. Ge, Existence and global exponential stability ofperiodic solutions of recurrent cellular neural networks with impulses anddelays, Math. Comput. Simul., 2007, In Press.
[17] J. Hu, S. Zhong and L. Liang, Exponential stability analysis of stochasticdelayed cellular neural networks, Chaos, Solitons & Fractals, 27(2006),1006-1010.
[18] T. Huang, Exponential stability of fuzzy cellular neural networks with dis-tributed delay, Phys. Letters A, 351(2006), 48-52.
[19] T. Huang, Exponential stability of delayed fuzzy cellular neural networkswith di?usion, Chaos, Solitons & Fractals, 31(2007), 658-664.
[20] H. Jiang, Z. Li and Z. Teng, Boundedness and stability for nonautonomouscellular neural networks with delay, Phys. Letters A, 306(2003), 313-325.
[21] H. Jiang and Z. Teng, Global exponential stability of cellular neuralnetworks with time-varying coe?cients and delays, Neural Networks,17(2004), 1415-1425.
[22] H. Jiang and Z.Teng, Some new results for recurrent neural networks withvarying-time coe?cients and delays, Phys. Letters A, 308(2005), 446-460.
[23] H. Jiang and Z. Teng, A new criterion on the global exponential stability forcellular neural networks with multiple time-varying delays, Phys. LettersA, 338(2005), 461-471.
[24] H. Jiang, L. Zhang and Z. Teng, Existence and global exponential stabilityof almost periodic solution for cellular neural networks with variable coe?-cients and time-varying delays, IEEE Trans. Neural Networks, 16(2005),1340-1351.
[25] H. Jiang, J. Cao and Z. Teng, Dynamics of Cohen-Grossberg neural net-works with time-varying delays, Phys. Letters A, 354(2006), 414-422.
[26] J.P. Lasalle, The Stability of Dynamical System, Society of Industrial andApplied Mathematics, Philadephia, 1976.
[27] J. Liang and J. Cao, Global exponential stability of reaction-di?usionrecurrent neural networks with time-varying delays, Phys. Letters A,314(2003), 434-442.
[28] Y. Li and Z. Xing, Existence and global exponential stability of periodicsolution of CNNs with impulses, Chaos, Solitons & Fractals, 33(2007),1686-1693.
[29] B. Liu and L. Huang, Existence and exponential stability of almost periodicsolutions for cellular neural networks with mixed delays, Chaos, Solitons& Fractals, 32(2007), 95-103.
[30] B. Liu and L. Huang, Existence of periodic solutions for cellular neural net-works with complex deviating arguments, Appl. Math. Letters, 20(2007),103-109.
[31] Y. Liu and W. Tang, Exponential stability of fuzzy neural networks withconstant and time-varying delays, Phys. Letters A, 323(2004), 224-233.
[32] X. Lou and B. Cui, Boundedness and exponential stability for nonau-tonomous cellular neural networks with reaction-di?usion terms, Chaos,Solitons & Fractals, 33(2007), 653-662.
[33] X. Lou and B. Cui, Boundedness and exponential stability for nonau-tonomous RCNNs with distributed delays, Comput. Math. Applic., 2007,In press.
[34] X. Lou and B. Cui, Boundedness and exponential stability for nonau-tonomous RCNNs with distributed delays, Comput. Math. Applic.,54(2007), 589-598.
[35] S. Mohamad, Global exponential stability in DCNNs with distributed de-lays and unbounded activations, J. Comput. Appl. Math., 205(2007), 161-173.
[36] Ju H. Park, An analysis of global robust stability of uncertain cellularneural networks with discrete and distributed delays, Chaos, Solitons &Fractals, 32(2007), 800-807.
[37] Ju H. Park and Hyun J. Cho, A delay-dependent asymptotic stability crite-rion of cellular neural networks with time-varying discrete and distributeddelays, Chaos, Solitons & Fractals, 33(2007), 436-442.
[38] Vimal Singh, New global robust stability results for delayed cellular neuralnetworks based on normbounded uncertainties, Chaos, Solitons & Frac-tals, 30(2006), 1165-1171.
[39] Q. Song and J. Cao, Global exponential stability and existence of peri-odic solutions in BAM networks with delays and reaction-di?usion terms,Chaos, Solitons & Fractals, 23(2005), 421-430.
[40] Q. Song and J. Cao, Dynamical behaviors of discrete-time fuzzy cellularneural networks with variable delays and impulses, J. Frank. Inst., 2007,In press.
[41] Q. Song and J. Cao, Impulsive e?ects on stability of fuzzy Cohen-Grossbergneural networks with time-varying delays, IEEE Trans. Syst. Man Cy-bern., B: Cybernetics, 37(2007), 733-741.
[42] Q. Song, J. Cao and Z. Zhao, Periodic solutions and its exponential sta-bility of reaction-di?usion recurrent neural networks with continuously dis-tributed delays, Nonl. Anal.: RWA, 7(2006), 65-80.
[43] Q. Song, Z. Zhao and Y. Li, Global exponential stability of BAM neu-ral networks with distributed delays and reaction-di?usion terms, Phys.Letters A, 335(2005), 213-225.
[44] J. Sun and L. Wan, Convergence dynamics of stochastic reaction-di?usionrecurrent neural networks with delays, Int. J. Bifurc. Chaos, 15(2005),2131-2144.
[45] J. Wang and J. Lu, Global exponential stability of fuzzy cellular neuralnetworks with delays and reaction-di?usion terms, Chaos, Solitons &Fractals, 2007, In press.
[46] L. Wang and D. Xu, Asymptotic behavior of a class of reaction-di?usionequations with delays, J. Math. Anal. Appl., 281(2003), 439-453.
[47] L. Wang and Y. Gao, Global exponential robust stability of reaction-di?usion interval neural networks with time-varying delays, Phys. LettersA, 350(2006), 342-348.– 56–
[48] S. Wang, D. Fu, M. Xu and D. Hu, Advanced fuzzy cellular neural network:Application to CT liver images, Artif. Intel. Medicine, 39(2007), 65-77.
[49] S. Wang, Korris F. L. Chung and D. Fu, Applying the improved fuzzycellular neural network IFCNN to white blood cell detection, Neurocom-puting, 2007, In press.
[50] S. Wang, D. Fu, M. Xu and D.Hu, Advanced fuzzy cellular neural network:Application to CT liver images, Artif. Intell. Medic., 39(2007), 65―77.
[51] J. Wang and J.G. Lu, Global exponential stability of fuzzy cellular neu-ral networks with delays and reaction-di?usion terms, Chaos, Solitons &Fractals, 2007, In press.
[52] Y. Xia, J. Cao and Z. Huang, Existence and exponential stability of almostperiodic solution for shunting inhibitory cellular neural networks, Chaos,Solitons & Fractals, 34(2007), 1599-1607.
[53] T. Yang and L. B. Yang, The global stability of fuzzy cellular neural net-works, IEEE Trans. Circ. Syst.: I, 43(1996), 880-883.
[54] T. Yang, L. B. Yang, C. W. Wu and L. O. Chua, Fuzzy cellular neural net-works: theory, Proceedings of IEEE International Workshop on CellularNeural Networks and Applications, 1996, 181-186.
[55] K. Yuan, J. Cao and J. Deng, Exponential stability and periodic solutionsof fuzzy cellular neural networks with time-varying delays, Neurocom-puting, 69(2006), 1619-1627.
[56] Z. Zeng, D. Huang and Z. Wang, Pattern memory analysis based on stabil-ity theory of cellular neural networks, Appl. Math. Modelling, 32(2008),112-121.
[57] Q. Zhang, X. Wei and J. Xu, Delay-dependent exponential stability of cellu-lar neural networks with time-varying delays, Chaos, Solitons & Fractals,23(2005), 1363-1369.
[58] S. Zhong and X. Liu, Exponential stability and periodicity of cellular neuralnetworks with time delay, Mathl. Comput. Modelling, 45(2007), 1231-1240.
[59] Q. Zhou, J. Sun and G. Chen, Global exponential stability and periodicoscillations of reaction-di?usion BAM neural networks with periodic coef-ficients and general delays, J. Bifur. Chaos, 17(2007), 129-142.