时标上Cohen-Grossberg神经网络的定性分析
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摘要
本文利用时标理论、压缩映射原理、重合度的全连续定理、M-矩阵、Poincare映射、拓扑度、Lyapunov泛函方法和不等式技巧,在时标上讨论了Cohen-Grossberg神经网络的全局指数稳定性,获得了一些新的结果,统一了在相同条件下连续系统和离散系统的结果,扩展了神经网络设计的范围,在理论和应用中都有重要的意义。
全文共分四部分。
在第1章中,我们简单介绍问题研究的背景,以及近年来许多学者研究的Cohen-Grossberg神经网络模型。然后,给出了本文的主要工作和本文中需要用到的一些基本定义和基本引理。
在第2章中,我们主要利用压缩映射原理,不等式技巧,通过构造适当的Lyapunov函数,在时标上讨论了具有分布时滞和有界时滞的Cohen-Grossberg神经网络的平衡点的存在性和全局指数稳定性,得到了新的充分条件,统一了连续和离散情形下两种系统。
在第3章中,我们主要利用M-矩阵,重合度的全连续定理,构造适当的Lyapunov函数,在时标上得到具有分布时滞和有界时滞的Cohen-Grossberg-BAM型神经网络的周期解的存在性和全局指数稳定性的充分条件。
在第4章中,我们主要利用拓扑度理论,不等式技术和Lyapunov泛函方法,在时标上研究了具有中立型有界时滞和分布时滞的Cohen-Grossberg神经网络,得到了该模型平衡点的存在性和全局指数稳定性的新条件,推广了以前的结论。
In this thesis, using the theory of calculus on time scales, contraction mapping principle, the continuation theorem of coincidence degree theory, M-matrix, Poincarémapping, Lyapunov functional method and inequality techniques, we study the global exponential stability for Cohen-Grossberg neural networks on time scales, and we obtain some new results. Those results unify the continuous and discrete system in the same framework, and to enlarge the area of designing neural networks. Our works has important significance in theory and application. This thesis is divided into 4 chapters.
In chapter 1, we introduce the applied background of research and Cohen-Grossberg neural networks which many researchers studied in recent years. Then, the main works and some useful notations and definitions are given in this article.
In chapter 2, we discuss the the global exponential stability of the Cohen-Grossberg neural network with distributed delays and bounded delays on time scales. By applying contraction mapping principle, inequality techniques and construct the suitable Lyapunov function, some sufficient conditions of existence and the global exponential stability are obtained. These results unify the continuous and discrete system.
In chapter 3, using M-matrix and the continuation theorem of coincidence degree theory, and construct the suitable Lyapunov function, some new sufficient conditions are obtain to ensure the existence and global exponential stability of the periodic solution to Cohen-Grossberg-BAM neural networks with distributed delays and bounded delays on time scales.
In chapter 4, by using topological degree, Lyapunov functional method and inequality techniques, we study the global exponential stability of the Cohen-Grossberg-BAM type neural networks with neutral bounded delays and distributed delays. Some sufficient conditions are obtained to ensure the existence, uniqueness and global exponential stability, which extend the formerly results.
全文共分四部分。
在第1章中,我们简单介绍问题研究的背景,以及近年来许多学者研究的Cohen-Grossberg神经网络模型。然后,给出了本文的主要工作和本文中需要用到的一些基本定义和基本引理。
在第2章中,我们主要利用压缩映射原理,不等式技巧,通过构造适当的Lyapunov函数,在时标上讨论了具有分布时滞和有界时滞的Cohen-Grossberg神经网络的平衡点的存在性和全局指数稳定性,得到了新的充分条件,统一了连续和离散情形下两种系统。
在第3章中,我们主要利用M-矩阵,重合度的全连续定理,构造适当的Lyapunov函数,在时标上得到具有分布时滞和有界时滞的Cohen-Grossberg-BAM型神经网络的周期解的存在性和全局指数稳定性的充分条件。
在第4章中,我们主要利用拓扑度理论,不等式技术和Lyapunov泛函方法,在时标上研究了具有中立型有界时滞和分布时滞的Cohen-Grossberg神经网络,得到了该模型平衡点的存在性和全局指数稳定性的新条件,推广了以前的结论。
In this thesis, using the theory of calculus on time scales, contraction mapping principle, the continuation theorem of coincidence degree theory, M-matrix, Poincarémapping, Lyapunov functional method and inequality techniques, we study the global exponential stability for Cohen-Grossberg neural networks on time scales, and we obtain some new results. Those results unify the continuous and discrete system in the same framework, and to enlarge the area of designing neural networks. Our works has important significance in theory and application. This thesis is divided into 4 chapters.
In chapter 1, we introduce the applied background of research and Cohen-Grossberg neural networks which many researchers studied in recent years. Then, the main works and some useful notations and definitions are given in this article.
In chapter 2, we discuss the the global exponential stability of the Cohen-Grossberg neural network with distributed delays and bounded delays on time scales. By applying contraction mapping principle, inequality techniques and construct the suitable Lyapunov function, some sufficient conditions of existence and the global exponential stability are obtained. These results unify the continuous and discrete system.
In chapter 3, using M-matrix and the continuation theorem of coincidence degree theory, and construct the suitable Lyapunov function, some new sufficient conditions are obtain to ensure the existence and global exponential stability of the periodic solution to Cohen-Grossberg-BAM neural networks with distributed delays and bounded delays on time scales.
In chapter 4, by using topological degree, Lyapunov functional method and inequality techniques, we study the global exponential stability of the Cohen-Grossberg-BAM type neural networks with neutral bounded delays and distributed delays. Some sufficient conditions are obtained to ensure the existence, uniqueness and global exponential stability, which extend the formerly results.
引文
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[47]X.Liao,C.Li,Global attractivity of Cohen-Grossberg model with finite and infinite delays[J],Journal of Mathematical Analysis and Applications,2006,315(1):244-262.
[48]Y.K.Li,X.R.Chen,L.Zhao,Stability and existence of periodic solutions to delayed Cohen-Grossberg BAM neural networks with impulses on time scales[J],Neurocomputing,2009,72:1621-1630.
[49]J.Liang,J.Cao,Exponential stability of continuous-time and discrete-time bidirectional associative memory neural networks with delay[J],Chaos Solitons &Fractal,2004,22(4):773-785.
[50]X.Liu,M.Tang,R.Martin,X.Liu,Discrete-time BAM neural networks with variable delays[J],Physics Letters A,2007,367(4-5):322-330.
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[2]M.Bohner,A.Peterson,Advance in Dynamic Equations on Time Scales,Birkh(a|¨)user[M],Boston,2003.
[3]R.Agarwal,M.Bohner,D.O'Regan,A.Peterson,Dynamic equations on time scales:A survey(special issue on dynamic equations on time scales)[J],Journal of Computational and Applied Mathematics,2002,141(1-2):1-26.
[4]V.Lakshmikantham,A.S.Vatsala,Hybrid systems on time scales[J],Journal of Computational and Applied Mathematics,2002,141:227-235.
[5]M.Cohen,S.Grossberg,Absolute stability and global pattern formation and parallel memory storage by competitive neural networks[J],IEEE Trans.Syst.Man Cybernet.SMC,1983,13:815-826.
[6]J.Cao,X.Li,Stability in delayed Cohen-Grossberg neural networks:LMI optimization approach[J],Physica D:Nonlinear Phenomena,2005,212(1-2):54-65.
[7]J.Cao,J.Liang,Boundeeness and stability for Cohen-Grossberg neural networks with time-varying delays[J],Journal of Mathematical Analysis and Applications,2004,296:665-685.
[8]L.Wang,X.Zou Harmless delays in Cohen-Grossberg neural networks[J],Physiea D,2002,170:162-173.
[9]A.Chen,L.Huang,J.Cao,Exponential stability of delayed bidirectional associative memory neural networks with reaction diffusion terms[J],International Journal of Systems Science,2007,38(5):421-432.
[10]A.Chen,J.Cao,L.Huang,Exponential stability of BAM neural networks with transmission delays[J],Neurocomputing,2004,57:435-454.
[11]A.Chen,J.Cao,Periodic bi-directional Cohen-Grossberg neural networks with distributed delays[J],Nonlinear Analysis,2007,66(12):2947-2961.
[12]A.Chen,L.Huang,J.Cao,Existence and stability of almost periodic solution for BAM neural networks with delays[J],Applied Mathematics and Computation,2002,137(1):177-193.
[13]C.Cheng,T.Liao,J.Yan,C.Hwang,exponential synchronization of delayed chaotic networks with time-varying delays[J],IEEE Transactions on Systems,Man and Cybernet,2006,36:209-215.
[14]J.Cao,Daniel W.C.Ho,X.Huang,LMI-based criteria for global robust stability of bidirectional associative memory networks with time delay[J],Nonlinear Analysis,2007,66(7):1558-1572.
[15]J.Cao,L.Wang,Exponential stability and periodic oscillatory solution in BAM networks with delays[J],IEEE Transactions on Neural Networks,2002,13(2):457-463.
[16]W.Zhao Global exponential stability analysis of Cohen-Grossberg neural network with delays[J],Communications in Nonlinear Science and Numerical Simulation,2008,13:847-856.
[17]W.Wu,B.Cui,X.Lo,Global exponential stability of Cohen-Grossberg neural networks with distributed delays[J],Mathematical and Computer Modeling,2008,47:868-873.
[18]B.Wang,J.Jian,C.Guo,Global exponential stability of a class of BAM networks with time-varying delays and continuously distributed delays[J],Neurocomputing,2008,71:495-501.
[19]W.Xiong,J.Cao,Global exponential stability of discrete-time Cohen-Grossberg neural networks[J],Neurocomputing,2005,64:433-446.
[20]W.Zhao,Y.Tan,Harmless delays for global exponential stability of Cohen-Grossberg neural networks[J],Mathematics and Computers in Simulation,2007,74:47-57.
[21]L.Wang,X.Zou,Harmless delays in Cohen-Grossberg neural networks[J],Physica D,2002,170:162-173.
[22]F.Tu and X.Liao,Harmless delays for global asymptotic stability of Cohen-Grossberg neural networks[J],Chaos,Solitons & Fractals,2005,26:927-933.
[23]H.Ye,A.N.Michel,K.Wang,Qualitative analysis of Cohen-Grossberg neural networks with multiple delays[J],Phys Rev E,1995,51:2611-2618
[24]Z.Ping,J.Lu,Global exponential stability of impulsive Cohen-Grossberg neural networks with continuously distributed delays[J],Chaos,Solitons & Fractals,Available online 2007,31.
[25]T.Li,S.Fei,Stability analysis of Cohen-Grossberg neural networks with time-varying and distributed delays[J],Neurocomputing,2008,71:1069-1081.
[26]T.Huang,C.Li,G.Chen,Stability of Cohen-Grossberg neural networks with unbounded distributed delays[J],Chaos,Solitons & Fractals,2007,34(3):992-996.
[27]Q.Song,J.Cao,Stability analysis of Cohen-Grossberg neural network with both time-varying and continuously distributed delays[J],Journal of Computational and Applied Mathematics,2006,197:188-203.
[28]L.Wang,Stability of Cohen-Grossberg neural networks with distributed delays[J],Applied Mathematics and Computation,2005,160:93-110.
[29]C.Huang,L.Huang,Dynamics of a class of Cohen-Grossberg neural networks with time-varying delays[J],Nonlinear Analysis:Real World Applications,2007,8(1):40-52.
[30]H.Jiang,J.Cao,Z.Teng,Dynamics of Cohen-Grossberg neural networks with time-varying delays[J],Physics Letters A,2006,354(5-6):414-422.
[31]X.Liao,C.Li,Global attractivity of Cohen-Grossberg model with finite and infinite delays[J],Journal of Mathematical Analysis and Applications,2006,315(1):244-262.
[32]J.Liu,Global exponential stability of Cohen-Grossberg neural networks with time-varying delays[J],Chaos,Solitons & Fractals,2005,26(3):935-945.
[33]Z.Mao,H.Zhao,X.Wang,Dynamics of Cohen-Grossberg neural networks with variable and distributed delays[J],Physica D:Nonlinear Phenomena,2007,234(1):11-22.
[34]Q.Song,J.Zhang,Global exponential stability of impulsive Cohen-Grossberg neural network with time-varying delays[J],Nonlinear Analysis:Real World Applications 2008,9(2):500-510.
[35]H.Zhao,L.Wang,Hopf bifurcation in Cohen-Grossberg neural network with distributed delays[J],Nonlinear Analysis:Real World Applications,2007,8(1):73-89.
[36]T.Chen,L.Rong,Delay-independent stability analysis of Cohen-Grossberg neural networks[J],Physics Letters A,2003,317:436-449.
[37]T.Chen,L.Rong,Robust global exponential stability of Cohen-Grossberg neural networks with time delays[J],IEEE Transactions on Neural Networks,2004,15:203-206.
[38]B.Kosko,Neural Networks and Fuzzy Systems-A Dynamical System Approach to Machine Intelligence[J],Prentice-Hall,Englewood Cliffs,NJ,1992,38-108.
[39]B.Kosko,Bi-directional associative memories[J],IEEE Transactions on Systems,Man and Cybernet,1988,18(1):49-60.
[40]B.Kosko,Adaptive bi-directional associative memories[J],Applied Optimization,1987,26(23):4947-4960.
[41]A.Chen,D.Du,Global exponential stability of delayed BAM network on time scale[J],Neurocomputing,2008,71:3582-3588.
[42]S.Guo,L.Huang,L.Wang,Exponential stability of discrete-time Hopfield neural networks[J],Computers & Mathematics with Applications,2004,47(8-9):1249-1256.
[43]T.Huang,C.Li,G.Chen,Stability of Cohen-Grossberg neural networks with unbounded distributed delays[J],Chaos,Solitons & Fractals,2007,34(3):992-996.
[44]G.Guseinov,Integration on time scales[J],J.Math.Anal.Appl.,2003,285:107-127.
[45]E.R.Kaufmann,Y.N.Rraffoul,Periodic solutions for a neutral nonlinear dynamical equation on a time scales[J],J.Math.Anal.Appl.,2006,319:315-325.
[46]Y.Li,Existence and stability of periodic solutions for Cohen-Grossberg neural networks with multiple delays[J],Chaos,Solitons & Fractals,2004,20(3):459-466.
[47]X.Liao,C.Li,Global attractivity of Cohen-Grossberg model with finite and infinite delays[J],Journal of Mathematical Analysis and Applications,2006,315(1):244-262.
[48]Y.K.Li,X.R.Chen,L.Zhao,Stability and existence of periodic solutions to delayed Cohen-Grossberg BAM neural networks with impulses on time scales[J],Neurocomputing,2009,72:1621-1630.
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