时滞递归神经网络的周期动力学行为
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摘要
由于信号传递和开关的闭合需要一定的时间,时滞在神经网络中普遍存在,所以学者们在研究神经网络时,建立了时滞神经网络模型.时滞递归神经网络是一类重要的时滞神经网络,它是时滞动力系统的一个重要组成部分,具有十分丰富的动力学属性.由于它在人工智能、信号处理、动态图像处理以及全局优化等问题中的重要应用,近年来时滞递归神经网络的动力学问题引起了学术界的广泛关注和深入研究,出现了一系列深刻的结果.本文主要对递归神经网络的周期动力行为及其稳定性进行了一系列的研究,取得了一些较深刻的结果.这些结论将为设计具有某些特殊功能的神经网络提供理论依据.
全文共分为六个部分：
第一章概述了神经网络的结构特征、功能特点和发展历程,综述了神经网络的分类和动力学特征.同时描述了本文研究的递归神经网络模型并指出了本文的主要工作和创新之所在.
在第二章中,为了便于文章的理解,我们概述了竞争合作系统和混合单调系统的一些基本定义和文章中需要用到的结果.
在第三章中,我们研究了一类具分布时滞的2-神经元自治Hopfield神经网络的周期解,首先,我们通过一个变换将系统化为三维常自治系统,然后建立适当的条件将系统化为竞争系统,利用竞争系统的基本结果,我们证明了系统周期解的存在性和稳定性.
第四章研究了一类一般意义上的变连接权变时滞递归神经网络的周期解和反周期解的存在性和指数稳定性,通过耦合,我们建立了一个新的混合单调系统,利用混合单调算子的基本理论,我们证明了混合单调系统的解的单调有界性和收敛性,并由此得到了原系统的周期解的存在性.在此基础上,我们还得到了一个反周期解存在性和指数稳定性的判据.
在第五章中,我们研究了一类变时滞递归神经网络模型的多周期解的存在性和局部指数稳定性.通过耦合,我们建立了一个新的混合单调系统,这个系统在2n个独立区域的解具有单调有界性和收敛性,从而得出原系统存在2n个局部指数稳定的周期解.
第六章对递归神经网络最新的研究进展进行了综述,同时对未来研究的发展方向进行了展望.
Due to the signal transmission, the opening and closing of switch need some time, the time-delay exists universally in the neural network, so when scholars study the neural network, they have established the model of the time-delay neural network.. Of which the delay recurrent neural network is a kind of important one, and it is also an important part of the delay dynamic system and possesses very abundant dynamics properties. Because of its important applications in artificial intelligence, signal processing, dynamic image processing and the optimization problems in overall situation,in recent years, the dynamics problems of the delay recurrent neural network have aroused wide attention and deep research in academic circles, emerging a series of profound results. This paper mainly focuses on a series of study on its stability and periodic solutions of the recurrent neural network and have achieved some profound results. These conclusions will provide the theory bases for the neural network which is designed with periodic solutions of the exponential stability in overall situation.
The whole thesis is divided into six parts:
In the first chapter, we firstly summarize the structure, function, characteristics and the development process of the neural network, and analyze the structure and dynamics features of the neural network, and then the paper has also made a general description of the recurrent neural network model which is studied, meanwhile, the main work and innovation of this thesis have been pointed out as well.
In chapter 2, in order to facilitate our understanding, we summarized the competition and cooperation system, some basic definitions of the mixed monotone system and the results which are used in this article.
In chapter 3, we study a kind of periodic solutions with autonomous 2-neurons Hopfield neural networks with distribute delays, first, we transform the system into three-dimensional constant autonomous system by means of a change, and then set up appropriate conditions to transform the system into the competition system, by using the results of competition system, we have proved the existence and stability of periodic solutions of the system.
In the fourth chapter, we have studied a kind of the existence and exponential stability of periodic solution and anti-periodic solutions which transform connection weight into delay recurrent neural network, in general sense, by coupling, we established a new mixed monotone system, by using the basic theories of mixed monotone operator, we proved the boundness and convergence of the mixed monotone system solution, because of this, we can prove the existence of periodic solutions of the original system ,based on this, we have also given the existence of the anti-periodic solutions,exponential stability and a condition.
In chapter 5, we have studied a kind of generally variable delay recurrent neural network model which has the existence and local exponential stability of multi-periodic solutions, by coupling, we established a new mixed monotone system, which possesses monotonous boundness and convergence in 2" separate areas, thus we get to the conclusion that 2" periodic solutions with local exponential stability exists in the original system.
Chapter 6 gives a review of recent results on recurrent neural networks and the prospect of further research.
全文共分为六个部分：
第一章概述了神经网络的结构特征、功能特点和发展历程,综述了神经网络的分类和动力学特征.同时描述了本文研究的递归神经网络模型并指出了本文的主要工作和创新之所在.
在第二章中,为了便于文章的理解,我们概述了竞争合作系统和混合单调系统的一些基本定义和文章中需要用到的结果.
在第三章中,我们研究了一类具分布时滞的2-神经元自治Hopfield神经网络的周期解,首先,我们通过一个变换将系统化为三维常自治系统,然后建立适当的条件将系统化为竞争系统,利用竞争系统的基本结果,我们证明了系统周期解的存在性和稳定性.
第四章研究了一类一般意义上的变连接权变时滞递归神经网络的周期解和反周期解的存在性和指数稳定性,通过耦合,我们建立了一个新的混合单调系统,利用混合单调算子的基本理论,我们证明了混合单调系统的解的单调有界性和收敛性,并由此得到了原系统的周期解的存在性.在此基础上,我们还得到了一个反周期解存在性和指数稳定性的判据.
在第五章中,我们研究了一类变时滞递归神经网络模型的多周期解的存在性和局部指数稳定性.通过耦合,我们建立了一个新的混合单调系统,这个系统在2n个独立区域的解具有单调有界性和收敛性,从而得出原系统存在2n个局部指数稳定的周期解.
第六章对递归神经网络最新的研究进展进行了综述,同时对未来研究的发展方向进行了展望.
Due to the signal transmission, the opening and closing of switch need some time, the time-delay exists universally in the neural network, so when scholars study the neural network, they have established the model of the time-delay neural network.. Of which the delay recurrent neural network is a kind of important one, and it is also an important part of the delay dynamic system and possesses very abundant dynamics properties. Because of its important applications in artificial intelligence, signal processing, dynamic image processing and the optimization problems in overall situation,in recent years, the dynamics problems of the delay recurrent neural network have aroused wide attention and deep research in academic circles, emerging a series of profound results. This paper mainly focuses on a series of study on its stability and periodic solutions of the recurrent neural network and have achieved some profound results. These conclusions will provide the theory bases for the neural network which is designed with periodic solutions of the exponential stability in overall situation.
The whole thesis is divided into six parts:
In the first chapter, we firstly summarize the structure, function, characteristics and the development process of the neural network, and analyze the structure and dynamics features of the neural network, and then the paper has also made a general description of the recurrent neural network model which is studied, meanwhile, the main work and innovation of this thesis have been pointed out as well.
In chapter 2, in order to facilitate our understanding, we summarized the competition and cooperation system, some basic definitions of the mixed monotone system and the results which are used in this article.
In chapter 3, we study a kind of periodic solutions with autonomous 2-neurons Hopfield neural networks with distribute delays, first, we transform the system into three-dimensional constant autonomous system by means of a change, and then set up appropriate conditions to transform the system into the competition system, by using the results of competition system, we have proved the existence and stability of periodic solutions of the system.
In the fourth chapter, we have studied a kind of the existence and exponential stability of periodic solution and anti-periodic solutions which transform connection weight into delay recurrent neural network, in general sense, by coupling, we established a new mixed monotone system, by using the basic theories of mixed monotone operator, we proved the boundness and convergence of the mixed monotone system solution, because of this, we can prove the existence of periodic solutions of the original system ,based on this, we have also given the existence of the anti-periodic solutions,exponential stability and a condition.
In chapter 5, we have studied a kind of generally variable delay recurrent neural network model which has the existence and local exponential stability of multi-periodic solutions, by coupling, we established a new mixed monotone system, which possesses monotonous boundness and convergence in 2" separate areas, thus we get to the conclusion that 2" periodic solutions with local exponential stability exists in the original system.
Chapter 6 gives a review of recent results on recurrent neural networks and the prospect of further research.
引文
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[2]Hebb D O. The organization of Behavior:a neuropsychological theory. Hebb Wiley, New York,1949.
[3]Minsky L. Papert S. Pereeptrons. Cambridge, MA:MIT Press,1969.
[4]Hopfield J. Network and physical systems with emergment collective computational abilities. Proc.Nath.Acad.Sci,1982,79:2554-2558.
[5]Chua L O, Yang L. Cellular neural networks:Theory. IEEE Trans.Circuits Syst,1988, 35(10):1257-1272.
[6]Cohen M, Grossberg S. Absolute stability and global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans.Syst.Man Cybernet, 1983,13:815-826.
[7]Kosko B. Bi-directional associative memories. IEEE Trans.Syst.Man Cybernet,1998, 18:49-60.
[8]Marcus C, Westervelt. Stability of analog neural networks with delay. Physical Review A,1989,39:347-359.
[9]Gopalsamy K, He X Z. Delay-independent stability in bidirectional associative memory networks. IEEE Trans. Neural Networks,1994,5:988-1002.
[10]Wang L, Zou X. Harmless delays in Cohen-Grossberg neural networks. Phys. D, 2002,170:162-173.
[11]Liao X F, Yu J B. Qualitative analysis of bidrectional associative Networks with time delays. Int. J. Circuit Theory Appl,1998,26:219-229.
[12]Song Q, Zhao Z, Li Y. Global exponential stability of BAM neural networks with distributed delays and reaction-diffusion terms. Phys.Lett.A,2005,335:213-225.
[13]Chen B, Wang J. Global exponential stability and global exponential stability of a class of recurrent neural networks with various activation functions and time-varying delays. Neu.Net,2007,20:1067-1080.
[14]Cao J, Wang L. Periodic oscillatory solution of bidirectional associative memory networks with delays. Phy. Rev. E,2000,61(2):1852-1828.
[15]Cao J. Global exponential stability and periodic solutions of CNNs with delays. Phys.Lett.A,2000,267:312-318.
[16]Guan Z H, Chen G R. On delayed impulsive Hopfield neural networks. Neural Networks,1999,12:273-280.
[17]Li Y K, Lu L. Global exponential stability and existence of periodic solution of Hopfield-type neural networks with impulses. Physics Letters A,2004,333:62-71.
[18]Akca H, Alassar R, Covacheveta V. Continuous-time additive Hopfield-type neural networks with impulses. J. Math. Anal. Appl.,2004,290:436-451.
[19]Yang T, Yang L B, Wu C W, et al. Fuzzy cellular neural networks:theory, In :Proceedings of IEEE International Workshop on Cellular Neural Networks and Applications,1996,181-186.
[20]Yang T,Yang L B,Wu C W, et al. Fuzzy cellular neural networks:theory, In :Proceedings of IEEE International Workshop on Cellular Neural Networks and Applications,1996,225-230.
[21]Yang T, Yang L B. The global stability of fuzzy cellular neural network. IEEE Trans, Circuits Systems 1,1996,43:880-883.
[22]Tan K, Tang H, Yi Z. Global exponential stability of discrete-time neural networks for constrained quadratic optimization. Neucomp,2004,56:399-406.
[23]Liu Y. Discrete-time recurrent neural networks with time-varying delays: exponential stability analysis. Phy. Let.A,2007,362:480-488.
[24]Liu X. Discrete-time BAM neural networks with variable delays. Physics Letters A, 3007,367:322-330.
[25]Gao H, Chen T. New results on stability of discrete-time systems with time-varying state delay. IEEE Transactions on Automatic Control,2007,52(2):328-334.
[26]Huang C, He Y, Chen P. Dynamic analysis of stochastic recurrent neural networks. Neural Processing Letters,2008,27(3):267-276.
[27]Huang C, He Y, Wang H. Mean square exponential stability of stochastic recurrent neural networks with time-varying delays. Computers & Mathematics with Applications,2008,56(7):1773-1778.
[28]Guo S, Huang L. Hopf bifurcating periodic orbits in a ring of neurons with delays. Physica D:Nonlinear Phenomena,2003,183(1-2):19-44.
[29]Zou S F, Huang L, Chen Y. Linear stability and Hopf bifurcation in a Three-unit neural network with two delays. Neurocomputing,2006,70:219-228.
[30]Cheng Y C, Lin K H, Shin C W. Multistability and convergence in delayed neural networks. Physica D,2007,225:61-74.
[31]Cheng Y C, Lin K H, Shin C W. Multistability in recurrent neural networks. SIAM.J.Appl.Math,2006,66(64):1301-1320.
[32]Zeng Z, Wang J. Multiperiodicity and exponential attractivity evoked by periodic external inputs in delayed cellular neural networks. Neural Comput,2006,18: 848-870.
[33]Cao J, Feng G, Wang YY. Multistability and multiperiodicity of delayed Cohen-Grossberg neural networks with a general class of activation function. Physica D: Nonlinear Phenomena,2008,237(13):1734-1749.
[34]Mao Y. Multistability of competitive neural networks with different time scales, in: Communications, Circuits and Systems. Proceedings.2005 International Conference on,2005,939-943.
[35]Hirsch M. Systems of differential equations which are competitive or cooperative, Ⅳ.SIAM.Math.Anal,1990,21:1225-1234.
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