时滞神经网络稳定性及复杂网络同步的研究
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摘要
为了易于分析和应用,许多系统建模时忽略了传输所带来的时间延迟。但是,时滞是客观存在的。同时,时滞的存在也给系统的稳定性带来影响,产生振荡行为或其它不稳定现象甚至出现混沌现象。近年来,时滞系统的研究吸引了大批的研究人员,对时滞系统的稳定性研究也取得了大量深刻的结果。神经网络是一种复杂的大规模动力学系统,其动力学属性十分广泛。由于其在人工智能、信号处理、图像处理和全局优化等问题中的重要应用,近年来时滞神经网络的动力学问题引起了学术界的广泛关注。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。复杂网络近年来成为科学和工程各个领域的研究热点。由于复杂网络在信号传输速度有限,节点间竞争和通道拥塞等情况下,必然存在时滞,这种现象在生物或物理网络中非常普遍。因此,带时滞的复杂网络系统吸引了越来越多研究者的关注。
本论文主要致力于对几类时滞系统的全局稳定性、指数稳定性和同步状态稳定性分析,取得了一些较深刻的结果。本文的主要内容和创新之处可概述如下:
①时滞Hopfield神经网络的全局渐近稳定性分析
在对时滞Hopfield神经网络模型进行“线性化”后,对模型进行变换,通过构造一个新颖的Lyapunov泛函,得到了时滞Hopfield神经网络平衡点全局渐近稳定的充分判定准则。我们的结果适用于典型的带时滞的Hopfield神经网络。
②时滞细胞神经网络的全局渐近稳定性分析
通过一个恒等变形将所研究的时滞细胞神经网络系统转换为线性模型,然后利用Lyapunov稳定性定理和线性矩阵不等式方法建立新颖的与时滞有关的渐近稳定性判定准则。给出了数值例子来验证我们结果的可行性。
③带常时滞的BAM神经网络的稳定性分析
首先对时滞BAM模型进行“线性化”,然后进行参数化模型变换,通过Lyapunov-Krasovskii泛函和线性矩阵不等式技术,建立了几个新颖的关于时滞BAM神经网络的与时滞无关和与时滞有关的渐近稳定性和指数稳定性判定准则。尤其是在激活函数的适当假设下将一般的时滞BAM神经网络转换为一类自治的线性系统。理论分析和数值模拟显示本文结果为时滞BAM神经网络提供了一些新的稳定性判定准则。
④带耦合时滞的复杂网络的同步分析
首先将非线性的复杂网络系统变换为线性系统,通过构造适当的Lyapunov-Krasovskii泛函,结合线性矩阵不等式(LMI)技术,研究了这种时滞复杂网络的同步问题,得到一个与时滞相关的同步条件。数值模拟说明我们的结果是有效的。
In order to analyze and apply easily, the transmission delays are ignored in modeling for most systems. But it is known that time delay is unavoidable. At the same time, time delays may affect the stability of the system, and even lead to instability, oscillation or chaos phenomena. Recently, research of delayed systems have attracted a large number of researchers, and many profound results have been established. The neural network, as a large-scale complex system, exhibits the rich and colorful dynamical behaviors. Due to its important and potential applications in artificial intelligence, signal processing, image processing as well as optimizing problems and so on, the dynamical issues of delayed neural networks have attracted worldwide attention in recent years, and many interesting stability criteria for the equilibriums and periodic solutions of delayed neural networks have been derived. Recently, complex networks attract more and more attentions from various fields of science and engineering. Due to the finite speeds of transmission and spreading as well as traffic congestions, a signal or influence traveling through a complex network often is associated with time delays, which is very common in biological and physical networks. So, complex dynamical networks with delays have become a focal research topic in recent years, and are attracting more and more attention from many fields of scientific research.
This dissertation focuses on the global exponential and synchronous stability for several delayed system. The main achievements and originality contained in this dissertation are as follows:
①Global asymptotic stability for Hopfield neural networks with delays
After the“linearization”of the delayed Hopfield neural network model, the considered neural network model is transformed. By employing suitable Lyapunov functional, delay-dependent criteria to ensure global asymptotic stability of the equilibrium of the Hopfield neural networks are established. Our results are suitable to the general delayed Hopfield neural networks.
②Global asymptotic stability for cellular neural networks with delays
The cellular neural networks are transformed into the linear models via some equivalent transformation. By constructing a novel Lyapunov functional, sufficient criteria for the existence of a unique equilibrium and global asymptotic stability of the cellular neural network are derived. A numerical simulation is also given to illustrate the validity of our result.
③Stability criteria for BAM neural networks with delays
First, the delayed BAM neural network model is linearized, and then, the model is parameterized transformed. Some novel delay-dependent and delay-independent asymptotical stability and exponential stability criteria for delayed BAM neural networks are derived and an estimation of the exponential convergence rate is then established by constructing an appropriate Lyapunov functional and using the linear matrix inequality (LMI) approach. Particularly, the general delayed BAM neural networks have been shifted into a class of non-autonomic linear systems under the appropriate assumption on the activation functions. The theoretical analysis and numerical simulations show that our results give some new criteria for the stability of delayed BAM neural networks.
④Synchronization stability criterion for complex network with coupling delays
Nonlinear complex network system is transformed to linear system. A new criterion for global synchronization stability for the complex networks with coupling delays has been derived using an approach combining the Lyapunov-Krasovskii functional with LMI techniques. It has been illustrated that the proposed result generalizes and improves those reported in the literature. A numerical simulation is also given to illustrate the validity of our result.
本论文主要致力于对几类时滞系统的全局稳定性、指数稳定性和同步状态稳定性分析,取得了一些较深刻的结果。本文的主要内容和创新之处可概述如下:
①时滞Hopfield神经网络的全局渐近稳定性分析
在对时滞Hopfield神经网络模型进行“线性化”后,对模型进行变换,通过构造一个新颖的Lyapunov泛函,得到了时滞Hopfield神经网络平衡点全局渐近稳定的充分判定准则。我们的结果适用于典型的带时滞的Hopfield神经网络。
②时滞细胞神经网络的全局渐近稳定性分析
通过一个恒等变形将所研究的时滞细胞神经网络系统转换为线性模型,然后利用Lyapunov稳定性定理和线性矩阵不等式方法建立新颖的与时滞有关的渐近稳定性判定准则。给出了数值例子来验证我们结果的可行性。
③带常时滞的BAM神经网络的稳定性分析
首先对时滞BAM模型进行“线性化”,然后进行参数化模型变换,通过Lyapunov-Krasovskii泛函和线性矩阵不等式技术,建立了几个新颖的关于时滞BAM神经网络的与时滞无关和与时滞有关的渐近稳定性和指数稳定性判定准则。尤其是在激活函数的适当假设下将一般的时滞BAM神经网络转换为一类自治的线性系统。理论分析和数值模拟显示本文结果为时滞BAM神经网络提供了一些新的稳定性判定准则。
④带耦合时滞的复杂网络的同步分析
首先将非线性的复杂网络系统变换为线性系统,通过构造适当的Lyapunov-Krasovskii泛函,结合线性矩阵不等式(LMI)技术,研究了这种时滞复杂网络的同步问题,得到一个与时滞相关的同步条件。数值模拟说明我们的结果是有效的。
In order to analyze and apply easily, the transmission delays are ignored in modeling for most systems. But it is known that time delay is unavoidable. At the same time, time delays may affect the stability of the system, and even lead to instability, oscillation or chaos phenomena. Recently, research of delayed systems have attracted a large number of researchers, and many profound results have been established. The neural network, as a large-scale complex system, exhibits the rich and colorful dynamical behaviors. Due to its important and potential applications in artificial intelligence, signal processing, image processing as well as optimizing problems and so on, the dynamical issues of delayed neural networks have attracted worldwide attention in recent years, and many interesting stability criteria for the equilibriums and periodic solutions of delayed neural networks have been derived. Recently, complex networks attract more and more attentions from various fields of science and engineering. Due to the finite speeds of transmission and spreading as well as traffic congestions, a signal or influence traveling through a complex network often is associated with time delays, which is very common in biological and physical networks. So, complex dynamical networks with delays have become a focal research topic in recent years, and are attracting more and more attention from many fields of scientific research.
This dissertation focuses on the global exponential and synchronous stability for several delayed system. The main achievements and originality contained in this dissertation are as follows:
①Global asymptotic stability for Hopfield neural networks with delays
After the“linearization”of the delayed Hopfield neural network model, the considered neural network model is transformed. By employing suitable Lyapunov functional, delay-dependent criteria to ensure global asymptotic stability of the equilibrium of the Hopfield neural networks are established. Our results are suitable to the general delayed Hopfield neural networks.
②Global asymptotic stability for cellular neural networks with delays
The cellular neural networks are transformed into the linear models via some equivalent transformation. By constructing a novel Lyapunov functional, sufficient criteria for the existence of a unique equilibrium and global asymptotic stability of the cellular neural network are derived. A numerical simulation is also given to illustrate the validity of our result.
③Stability criteria for BAM neural networks with delays
First, the delayed BAM neural network model is linearized, and then, the model is parameterized transformed. Some novel delay-dependent and delay-independent asymptotical stability and exponential stability criteria for delayed BAM neural networks are derived and an estimation of the exponential convergence rate is then established by constructing an appropriate Lyapunov functional and using the linear matrix inequality (LMI) approach. Particularly, the general delayed BAM neural networks have been shifted into a class of non-autonomic linear systems under the appropriate assumption on the activation functions. The theoretical analysis and numerical simulations show that our results give some new criteria for the stability of delayed BAM neural networks.
④Synchronization stability criterion for complex network with coupling delays
Nonlinear complex network system is transformed to linear system. A new criterion for global synchronization stability for the complex networks with coupling delays has been derived using an approach combining the Lyapunov-Krasovskii functional with LMI techniques. It has been illustrated that the proposed result generalizes and improves those reported in the literature. A numerical simulation is also given to illustrate the validity of our result.
引文
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