几类时滞神经网络的稳定性
摘要
随着现代社会的发展,神经网络广泛的运用在工程,航空,经济,金融等方面.大多数情况下,时滞神经网络比神经网络能更好的解决问题.学者们已经取得了很大的成就,尤其是时滞神经网络平衡点和周期解的稳定性得到了深入的研究,也出现了一系列较好的结果.
本文对几类具有时滞的神经网络模型进行了定性研究,讨论了这些神经网络平衡点的存在性,唯一性和全局指数稳定性及周期解的存在性和全局指数稳定性.全文的内容共分为四章:在第一章中,本文在[5]的基础上放宽了对激活函数有界的要求,研究了模型运用了压缩影像原理证平衡点的存在唯一性,并且构造李雅普诺夫函数法证的平衡点指数稳定的两个充分条件.进一步构造庞加莱映射研究了模型存在周期解且该周期解全局指数稳定.在第二章中,研究了一类具有脉冲的时滞泛函微分方程(可以推广到脉冲细胞神经网络和脉冲BAM自动联想神经网络中去)的周期解的全局指数稳定性.模型如下:利用线性矩阵不等式,采用矩阵谱半径性质和M矩阵法证的其周期解全局指数稳定性.并且进一步运用偏微分知识研究带扩散项的模型构造李雅普诺夫函数证明平衡点存在唯一且指数稳定性.并进一步研究模型在第三章中,本文在[19]的基础上,通过使用Halanay不等式,数学归纳法和不动点定理,研究了有脉冲的高阶细胞神经网络的周期解的指数稳定性.模型如下:
在第四章中,本文在[31]的基础上,改变模型为:研究其平衡点的指数稳定性.通过用压缩影像原理证平衡点存在唯一,建立新的李雅普诺夫函数证的其平衡点的全局指数稳定性,其中所含时滞既不只是纯连续的,也不只是纯离散的.在原有文献的基础上改进时滞,推广了相关文献的结果.
With the development of modern society, neural networks is widely used in engineering,astronavigation,economy, financial etc.In most cases,many problems are solvedmore practically by delayed neural networks than neural networks. Great achievementshave been given by the respective researchers, especially about stability of fixed pointsand periodic solutions of neural networks with delayed, many specialists and scholarsapply themselves to the research of theory and achieve many perfect productions. Inthis paper, we perform re-searches of the stability of several of networks with delayedThe main contents of this paper include:
In the first chapter ,the model aswas studied. Based on Lyapunov functional method, we study the global exponential stability for delayed impulsive cellular neural networks (DCNN)and give a newsufficient condition.exponential stability and periodicity of the model are investigated
In the second chapter,firstly dynamics of a class of retarded impulsive differentialequations (IDE)are studied, which generalizes the delayed cellular neural networks (DCNN)and delayedbidirectional associative memory (BAM) neural networks. The approaches are basedon Banach's fixed point theorem, matrix theory and its spectral theory.
Secondly the global exponential stability stability and periodicity for a class ofreaction-diffusion differential equations with Dirichlet boundary conditions as andwhich are address by constructing suitable Lyapunor functional and utilizing someinequality techniques.
In the third chapter, a class of high-order neural networks with time delays andimpulsive effects asthe global exponential stability and periodicity are investigated for it. Some sufficientconditions are derived for checking the global exponential stability and the existenceof periodic solution for this system based on Halanay Inequality, induction and fixedpoint theorem.
In the fourth chapter, a class of general Cohen -Grossberg neural networks withimpulsive is studied,based on the method of Lyapunov function, which is model with discrete and distributedelay.sufficient conditions on global exponential stabitity are given.Moreover,comparisionsandmade with earier findings.
本文对几类具有时滞的神经网络模型进行了定性研究,讨论了这些神经网络平衡点的存在性,唯一性和全局指数稳定性及周期解的存在性和全局指数稳定性.全文的内容共分为四章:在第一章中,本文在[5]的基础上放宽了对激活函数有界的要求,研究了模型运用了压缩影像原理证平衡点的存在唯一性,并且构造李雅普诺夫函数法证的平衡点指数稳定的两个充分条件.进一步构造庞加莱映射研究了模型存在周期解且该周期解全局指数稳定.在第二章中,研究了一类具有脉冲的时滞泛函微分方程(可以推广到脉冲细胞神经网络和脉冲BAM自动联想神经网络中去)的周期解的全局指数稳定性.模型如下:利用线性矩阵不等式,采用矩阵谱半径性质和M矩阵法证的其周期解全局指数稳定性.并且进一步运用偏微分知识研究带扩散项的模型构造李雅普诺夫函数证明平衡点存在唯一且指数稳定性.并进一步研究模型在第三章中,本文在[19]的基础上,通过使用Halanay不等式,数学归纳法和不动点定理,研究了有脉冲的高阶细胞神经网络的周期解的指数稳定性.模型如下:
在第四章中,本文在[31]的基础上,改变模型为:研究其平衡点的指数稳定性.通过用压缩影像原理证平衡点存在唯一,建立新的李雅普诺夫函数证的其平衡点的全局指数稳定性,其中所含时滞既不只是纯连续的,也不只是纯离散的.在原有文献的基础上改进时滞,推广了相关文献的结果.
With the development of modern society, neural networks is widely used in engineering,astronavigation,economy, financial etc.In most cases,many problems are solvedmore practically by delayed neural networks than neural networks. Great achievementshave been given by the respective researchers, especially about stability of fixed pointsand periodic solutions of neural networks with delayed, many specialists and scholarsapply themselves to the research of theory and achieve many perfect productions. Inthis paper, we perform re-searches of the stability of several of networks with delayedThe main contents of this paper include:
In the first chapter ,the model aswas studied. Based on Lyapunov functional method, we study the global exponential stability for delayed impulsive cellular neural networks (DCNN)and give a newsufficient condition.exponential stability and periodicity of the model are investigated
In the second chapter,firstly dynamics of a class of retarded impulsive differentialequations (IDE)are studied, which generalizes the delayed cellular neural networks (DCNN)and delayedbidirectional associative memory (BAM) neural networks. The approaches are basedon Banach's fixed point theorem, matrix theory and its spectral theory.
Secondly the global exponential stability stability and periodicity for a class ofreaction-diffusion differential equations with Dirichlet boundary conditions as andwhich are address by constructing suitable Lyapunor functional and utilizing someinequality techniques.
In the third chapter, a class of high-order neural networks with time delays andimpulsive effects asthe global exponential stability and periodicity are investigated for it. Some sufficientconditions are derived for checking the global exponential stability and the existenceof periodic solution for this system based on Halanay Inequality, induction and fixedpoint theorem.
In the fourth chapter, a class of general Cohen -Grossberg neural networks withimpulsive is studied,based on the method of Lyapunov function, which is model with discrete and distributedelay.sufficient conditions on global exponential stabitity are given.Moreover,comparisionsandmade with earier findings.
引文
[1]Cao JD.Zhou D. Stability analysis of delayed cellular neural networks. Neural Networks 1998;11:1601-5.
[2]Chua LQ,YangL. Cellular neural networks: theory IEEE Trans Circuits Syst 1988;35:1257-72.
[3]Fang YG.Kincaid TG. Stability analysis of dynamical neural networks .IEEE Trans Neural Networks 1996;7:996-10006
[4]HopfieldJJ. Neural networks and physical system with emergent collective computational abilities. Proc Am Math Socl982;49:2554-8.
[5]Zai-Tang Huang,Qi-Gui Yang ,Xiao-shu Luo.Exponential stability of impulsive neural networks with time-varying delays.Chaos,Solitions and Fractals 35(2008)770-780.
[6]Arik S, Tavanoglu V. Equilibrium analysis Of delayed cellular neural networks.IEEE Tram CASI , 1998; 45(2):168-171
[4]Cao JD. Stability of a of a class of neural networks models with delays.Appl Math Mech 1999;8:851-5.
[7]Cao JD. Stability in Cohen-Grossberg type BAM neural networks withtimevarying delays.Nonlinearity 2006:19:1601-17.
[8]Chuangxia Huang, Lihong Huang. Dynamics of a class of Cohen-Grossberg neural networks with time-varying delays.NonlinearAnalysis:RealWorldApplications8(2007)40-52
[9]Huang C,Huang L,Yuan Z. global stability analysis of a class of delayed cellular neural networks.Math Comput Simul2005;70(3):133-48.
[10]HongyongZhao.Global exponential stabilityand periodicity of cellular neural.Physics LettersA 336 (2005) 331-341
[11]Mohamad S,Gopalsamy K. Exponential stability of continuous-time and discretetime cellular neural networks with delays..IEEE Trans Neural Networks2005;16:1694-6.
[12]ShoumingZhong ,XinzhiLiu.Enxponential stability and periodicity of cellular neural networks With time delay .Mathematical and Computer Modelling 45(2007)1231- 1240
[13]Xia YH,Cao JD,Cheng SS. Global exponential stability analysis of delayed cellular neural networks with impulse.Neurocomputing.doi:10.1016/j.neucom.2006.8.005.
[14]Li Wan, QinghuaZhou.Global exponential stabilityof BAM neural networks withtime- varying delays and diffusion terms .Physics LettersA371 (2007) 83-89
[15] Yonghui Xia,Patricia J.Y.Wong. Global exponential stability of a class of retarded impulsive differential equatoins with applications. Chao,Solitons and Fractals xxx(2007)xxx-xxx
[16] Chen T ,Amari S.Stability of asymmetric Hopfield networks .Neural Networks 2001;12:159-63
[17] Jun Guo Lu . Global exponential stability and periodicity of reaction-diffusion delayed recurrent neural networks with Dirichlet boundary conditions .Chao,Solitons and Fractals.35 (2008) 116-125
[18] Haibo Gu,Haijun Jiang,Zhidong Teng .Stability and periodicity in high-order neural networks with impulsive effects.Nonlinear Analysis 68 (2008)3186-3200.
[19]Yongqing Yang,Jinde Cao. .Stability and periodicity in delayed cellular neural networks with impulsive effects. Nonlinear Analysis :Real World Applications 8 (2008)362-374.
[20] Chun-Hsien Li, Suh-Yuh Yang.A further analysis on harmless delays in Cohen- Grossberg neural networks. Chaos, Solitons and Fractals 34 (2007) 646-653
[21] Lu HT. Global exponential stability analysis of Cohen-Grcossberg neural networks .IEEE Trans Circuits Syst.II2005;52:476-9.
[22] Lu HT.Shen RM,Chung FL. Global exponential convergence of Cohen-Grcossberg neural networks with delays. IEEE Trans Neural Networks2005; 16:1694-6.
[23] Lin Wang , Xingfu Zou.Harmless delaysin Cohen-Grossberg neural networks PhysicaD170(2002) 162-173
[24] Qinghua Zhou , Li Wan , Jianhua Sun .Exponential stability of reaction-difusion generalized Cohen-Grossberg neural networks with time-varying delays .Chaos, Solitons and Fractals 32 (2007) 1713-1719
[25]Qinghua Zhou , Li Wan , Jianhua Sun Exponential stability of reaction-difusion generalized Cohen-Grossberg neural networks with time-varying delays. Chaos,Solitons and Fractals 32 (2007) 1713-1719
[26]Sun J , Wan L . Globally exponential stability and periodic solutions Cohen-Grossberg neural networks with continuously distributed delays [J]. Physics D: Nonlinear Phenomena, 2005. 208:1-20.
[27]Tingwen Huang ,AndrewChan , Yu Huang , Jinde Cao.Stability of Cohen-Grossberg neural networks with time-varying delays.Neural Networks 20(2007)868-873
[28]Tianping Chen , Libin Rong Delay-independent stability analysis of Cohen-Grossberg neural networks Physics Letters A 317 (2003) 436-449
[29]季策,张化光.一类具有时滞的广义Hopfield神经网络的动态分析[J].东北大学学报:自然科学版.2004,25(3):205-208.
[30]Zhang H G , Wang G . Global exponential stability of a class of generalized neural networks with variable coefficients and distributed delays [C] Advances in Intelligent Computing, ICIC 2005. Part Ⅰ: LNCS 3644. Berlin: Springer,2005:807-817.
[31]Zhang Chen,Jiong Ruan Cao, (2007). Global dynamic analysis of general Cohen-Grossberg neural networks with impluse .Chao,Solitons and Fractals 32 1830-1837.
[32]Changyin Sun , Chun-Bo Feng . Exponential periodicityand stabilityof delayed neural networks. Mathematics and Computers inSimulation 66 (2004) 469-478
[33]Chua L O . CNN :a vsions of complexity Bifurcation and Chaos in Appi Sci Eng , 1997;7(10) , 2219 2425 .
[34]Forti M .On asymptotic Stability of a of a class of nonlinear systems arusing in neural networks theory J Diff Eqns 1994;113:246-64.
[35]Forti M .Marini M.A conditions for global convergence of a class of symmetric neural networks. IEEE Trans Circuits Syst.l992;39:480-3.
[36]Forti M,Testi A.New conditions for global stability of neural networks with applications to linear and quadratic programming problems. IEEE Trans Circuits Syst..l995;42:354-66.
[37]Gopaisamy K, He XZ. Stability in asymmetric Hopfield nets with trammission delays.Physics D 1994;76:344-58.
[38] Gopaisamy K, He XZ.Delay-independent stabiliey in bi- directional associative memory networks. IEEE Trans Neural Networksl994;5998-1002.
[39]Guan Z.Chen G,Qin Y.On equilibria stability and instability of Hopfield neural networks .IEEE Trans Neural Networks2000;2:534-40.
[40] Jin C. stability analysis of discrete-time Hopfield BAM neural networks.Acta Auto Sin 1999;25(5):606-12.
[41] Jinde Cao , Tianping Chen, Globally exponentially robust stability and periodicity of delayed neural networks .Chaos, Solitons and Fractals 22 (2004) 957-963
[42] Juang J. Stability Hopfield neural networks .IEEE Trans Neural Networks analysis of Hopfield neural networks .IEEE Trans Neural Networksl999; 10:1366-74.
[43] Jun Guo Lu , Lin Ji Lu.Global exponential stability and periodicity of reaction- difusion recurrent neural networks with distributed delays and Dirichlet boundary conditions. Chaos, Solitons and Fractals xxx (2007) xxx-xxx
[44] Xia YH,Cao JD,Huang Z.Existence and stability of neural networks with discrete and distributed delays.Phys Lett A2006;345(4-6):299-308.
[45] Xia YH,Huang Z,Han M. Existence and global exponential stability of equilibrium for BAM neural networks with impulses. Chao,Solitons and Fractals.doi:10.1016/j.chaos.2006.08.045.
[2]Chua LQ,YangL. Cellular neural networks: theory IEEE Trans Circuits Syst 1988;35:1257-72.
[3]Fang YG.Kincaid TG. Stability analysis of dynamical neural networks .IEEE Trans Neural Networks 1996;7:996-10006
[4]HopfieldJJ. Neural networks and physical system with emergent collective computational abilities. Proc Am Math Socl982;49:2554-8.
[5]Zai-Tang Huang,Qi-Gui Yang ,Xiao-shu Luo.Exponential stability of impulsive neural networks with time-varying delays.Chaos,Solitions and Fractals 35(2008)770-780.
[6]Arik S, Tavanoglu V. Equilibrium analysis Of delayed cellular neural networks.IEEE Tram CASI , 1998; 45(2):168-171
[4]Cao JD. Stability of a of a class of neural networks models with delays.Appl Math Mech 1999;8:851-5.
[7]Cao JD. Stability in Cohen-Grossberg type BAM neural networks withtimevarying delays.Nonlinearity 2006:19:1601-17.
[8]Chuangxia Huang, Lihong Huang. Dynamics of a class of Cohen-Grossberg neural networks with time-varying delays.NonlinearAnalysis:RealWorldApplications8(2007)40-52
[9]Huang C,Huang L,Yuan Z. global stability analysis of a class of delayed cellular neural networks.Math Comput Simul2005;70(3):133-48.
[10]HongyongZhao.Global exponential stabilityand periodicity of cellular neural.Physics LettersA 336 (2005) 331-341
[11]Mohamad S,Gopalsamy K. Exponential stability of continuous-time and discretetime cellular neural networks with delays..IEEE Trans Neural Networks2005;16:1694-6.
[12]ShoumingZhong ,XinzhiLiu.Enxponential stability and periodicity of cellular neural networks With time delay .Mathematical and Computer Modelling 45(2007)1231- 1240
[13]Xia YH,Cao JD,Cheng SS. Global exponential stability analysis of delayed cellular neural networks with impulse.Neurocomputing.doi:10.1016/j.neucom.2006.8.005.
[14]Li Wan, QinghuaZhou.Global exponential stabilityof BAM neural networks withtime- varying delays and diffusion terms .Physics LettersA371 (2007) 83-89
[15] Yonghui Xia,Patricia J.Y.Wong. Global exponential stability of a class of retarded impulsive differential equatoins with applications. Chao,Solitons and Fractals xxx(2007)xxx-xxx
[16] Chen T ,Amari S.Stability of asymmetric Hopfield networks .Neural Networks 2001;12:159-63
[17] Jun Guo Lu . Global exponential stability and periodicity of reaction-diffusion delayed recurrent neural networks with Dirichlet boundary conditions .Chao,Solitons and Fractals.35 (2008) 116-125
[18] Haibo Gu,Haijun Jiang,Zhidong Teng .Stability and periodicity in high-order neural networks with impulsive effects.Nonlinear Analysis 68 (2008)3186-3200.
[19]Yongqing Yang,Jinde Cao. .Stability and periodicity in delayed cellular neural networks with impulsive effects. Nonlinear Analysis :Real World Applications 8 (2008)362-374.
[20] Chun-Hsien Li, Suh-Yuh Yang.A further analysis on harmless delays in Cohen- Grossberg neural networks. Chaos, Solitons and Fractals 34 (2007) 646-653
[21] Lu HT. Global exponential stability analysis of Cohen-Grcossberg neural networks .IEEE Trans Circuits Syst.II2005;52:476-9.
[22] Lu HT.Shen RM,Chung FL. Global exponential convergence of Cohen-Grcossberg neural networks with delays. IEEE Trans Neural Networks2005; 16:1694-6.
[23] Lin Wang , Xingfu Zou.Harmless delaysin Cohen-Grossberg neural networks PhysicaD170(2002) 162-173
[24] Qinghua Zhou , Li Wan , Jianhua Sun .Exponential stability of reaction-difusion generalized Cohen-Grossberg neural networks with time-varying delays .Chaos, Solitons and Fractals 32 (2007) 1713-1719
[25]Qinghua Zhou , Li Wan , Jianhua Sun Exponential stability of reaction-difusion generalized Cohen-Grossberg neural networks with time-varying delays. Chaos,Solitons and Fractals 32 (2007) 1713-1719
[26]Sun J , Wan L . Globally exponential stability and periodic solutions Cohen-Grossberg neural networks with continuously distributed delays [J]. Physics D: Nonlinear Phenomena, 2005. 208:1-20.
[27]Tingwen Huang ,AndrewChan , Yu Huang , Jinde Cao.Stability of Cohen-Grossberg neural networks with time-varying delays.Neural Networks 20(2007)868-873
[28]Tianping Chen , Libin Rong Delay-independent stability analysis of Cohen-Grossberg neural networks Physics Letters A 317 (2003) 436-449
[29]季策,张化光.一类具有时滞的广义Hopfield神经网络的动态分析[J].东北大学学报:自然科学版.2004,25(3):205-208.
[30]Zhang H G , Wang G . Global exponential stability of a class of generalized neural networks with variable coefficients and distributed delays [C] Advances in Intelligent Computing, ICIC 2005. Part Ⅰ: LNCS 3644. Berlin: Springer,2005:807-817.
[31]Zhang Chen,Jiong Ruan Cao, (2007). Global dynamic analysis of general Cohen-Grossberg neural networks with impluse .Chao,Solitons and Fractals 32 1830-1837.
[32]Changyin Sun , Chun-Bo Feng . Exponential periodicityand stabilityof delayed neural networks. Mathematics and Computers inSimulation 66 (2004) 469-478
[33]Chua L O . CNN :a vsions of complexity Bifurcation and Chaos in Appi Sci Eng , 1997;7(10) , 2219 2425 .
[34]Forti M .On asymptotic Stability of a of a class of nonlinear systems arusing in neural networks theory J Diff Eqns 1994;113:246-64.
[35]Forti M .Marini M.A conditions for global convergence of a class of symmetric neural networks. IEEE Trans Circuits Syst.l992;39:480-3.
[36]Forti M,Testi A.New conditions for global stability of neural networks with applications to linear and quadratic programming problems. IEEE Trans Circuits Syst..l995;42:354-66.
[37]Gopaisamy K, He XZ. Stability in asymmetric Hopfield nets with trammission delays.Physics D 1994;76:344-58.
[38] Gopaisamy K, He XZ.Delay-independent stabiliey in bi- directional associative memory networks. IEEE Trans Neural Networksl994;5998-1002.
[39]Guan Z.Chen G,Qin Y.On equilibria stability and instability of Hopfield neural networks .IEEE Trans Neural Networks2000;2:534-40.
[40] Jin C. stability analysis of discrete-time Hopfield BAM neural networks.Acta Auto Sin 1999;25(5):606-12.
[41] Jinde Cao , Tianping Chen, Globally exponentially robust stability and periodicity of delayed neural networks .Chaos, Solitons and Fractals 22 (2004) 957-963
[42] Juang J. Stability Hopfield neural networks .IEEE Trans Neural Networks analysis of Hopfield neural networks .IEEE Trans Neural Networksl999; 10:1366-74.
[43] Jun Guo Lu , Lin Ji Lu.Global exponential stability and periodicity of reaction- difusion recurrent neural networks with distributed delays and Dirichlet boundary conditions. Chaos, Solitons and Fractals xxx (2007) xxx-xxx
[44] Xia YH,Cao JD,Huang Z.Existence and stability of neural networks with discrete and distributed delays.Phys Lett A2006;345(4-6):299-308.
[45] Xia YH,Huang Z,Han M. Existence and global exponential stability of equilibrium for BAM neural networks with impulses. Chao,Solitons and Fractals.doi:10.1016/j.chaos.2006.08.045.