外部激励和惯性项对时滞神经网络动力学行为的影响研究
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摘要
众所周知,在网络系统中时滞的引入往往会带来系统动力学性质上很大的变化,而由于信号传输速度的有限性,时滞对系统的影响是不得不考虑的一个因素。由于时滞神经网络是时滞大系统的一个重要组成部分,所以同样也具有非常丰富动力学性质,尤其是鉴于它在信号处理、动态图象处理以及全局优化等问题中的重要应用。时滞神经网络的动力学问题一直都是学术界广泛关注的话题,特别是对时滞神经网络平衡点的全局稳定性、局部稳定性(包括渐近稳定性、指数稳定性、绝对稳定性)以及时滞神经网络的分岔和混沌等动力学现象,都得到了广大科学工作者的深入研究并取得了一系列深刻而有实际意义的理论成果。本论文主要对两种类型的时滞神经网络(即Hopfield神经网络和惯性神经网络),从两个方面即自治系统平衡点的全局稳定性以及自治系统周期解的局部稳定性、Hopf分岔、共振余维二分岔和混沌,以及非自治系统周期解的局部稳定性和Hopf分岔进行了一系列的研究,获取了一些有意义的成果。
本论文的主要内容和创新之处可概述如下:
①获得了分布时滞和有限时滞神经网络与时滞相关的全局稳定性和局部稳定性判据
由于神经元之间有限的信息传输速度以及电路系统中放大器的有限开关速度,同时因为存在大量的并行旁路以及各种不同长度和大小的轴突神经网络在空间上的扩展,信号传输不可能在一定时间内完成,因此建模上仅仅依靠有限时滞(常数时滞或者时变时滞)是不完整的,较精确的模型应该还应该包括无限时滞(分布时滞)。本论文中采用通常的Lyapunov函数构造法分别对带分布时滞或有限时滞的神经网络模型进行探讨,获得了与时滞相关的全局稳定性和局部稳定性判定准则。本研究的意义有两个方面:一、从两种时滞的角度分别给出稳定性的判定准则,这样对于更加精确的建模并预测模型从稳定至不稳定的演变过程是十分有益的;二、给出了与时滞相关的稳定性判据,这种准则对于小时滞神经网络不会过于苛刻,对系统参数也没有过多的限制,有利于预测系统模型动力学行为的鲁棒性和稳定性。
②对惯性时滞神经网络的局部稳定性和Hopf分岔进行了研究
惯性时滞神经网络具有较强的生物学背景,因此在满足一定条件的情况下,神经元的电路实现可以通过加入一个电感完成类似于带通滤波器、电调谐或者时空过滤的作用。由此我们研究了一个带有惯性项的时滞神经网络模型的局部稳定性和Hopf分岔,并用中心流形定理和正规型理论确定了分岔周期解的稳定性和分岔方向。其意义在于:一、由于Hopf分岔与振荡现象密切相关,对这种小规模网络Hopf分岔的研究可以使我们更好地解释现实世界中的许多大规模网络,如Internet、电网、生物神经网络中发生的对参数敏感的现象。二、若能深入地了解小规模网络中的分岔现象和规律,则通过利用比较成熟的分岔控制理论和方法,我们就可以将现实世界中大规模网络控制到所期望的有利的状态中去。
③对惯性神经网络的共振余维二分岔的研究
如何在人工神经网络系统中去预测并避免共振是十分重要的一个课题。在本论文中对惯性时滞神经网络模型进行了分析,在以时滞参数作为分岔参数的条件下,将研究重点放在对模型特征方程的根的讨论上,从而获取系统能够出现两对纯虚根的条件,当频率比ω1 :ω2为有理数时系统将会产生共振现象,所以本现象的研究结果以及分析方法可以为系统振幅的耦合和频率同步以及减少系统出现的共振提供理论依据。
④对惯性时滞神经网络在外部周期激励影响下的局部稳定性和Hopf分岔的研究
根据生物学实验,神经元在被施予外部周期激励时会产生同步振荡现象。因此作者研究了当惯性时滞神经网络引入了外部周期激励后,该非自治系统的局部稳定性以及周期解的存在和方向性。通过使用中心流形定理以及非线性振动中平均法技术的结合,我们首先获得了模型的中心流形,进而获取其平均方程,利用对方程的雅可比矩阵的分析得到分岔方程,由该分岔方程分析出周期解的方向和分岔点。由于目前主要的分岔理论都是针对自治系统的分岔问题,对非自治系统的分岔问题讨论的很少,本论文对这种类型系统的动力学性质做了一个初探,有利于为实际应用提供帮助。
⑤对Hopfield网络模型在时滞以及外部周期激励的共同影响下的局部稳定性和Hopf分岔的研究
对于Hopfield神经网络模型,它的应用已经渗透到生物学、物理学、地质学等诸多领域,并在智能控制、模式识别、非线性优化等方面获得了广泛的应用,针对这种实际应用非常广泛的网络模型讨论它的动力学行为是非常有意义的。本论文对该模型除了引入时滞外还增加了外部的周期激励,同时采用中心流形定理结合平均法获得系统的分岔周期解。虽然本论文该内容的研究方法与上一章的内容类似,但是因为该网络模型在实际应用中十分广泛,更具有重要意义。
As we all know, the introduce of time delay often brings great changes on the dynamical behavior in the fields of network systems. But the time delay has to be considered into the factors which influence systems due to the limitation of the signal transformation velocity. The delayed neural network is an important part of the delayed large-scale system, which exhibits the rich and colorful dynamical behavior. Due to the important applications in signal processing, moving image processing as well as optimizing problems, the dynamical behavior of the delayed nueral network has been studied by many researchers all the time, especially the dynamical behaviors of the global stability, the local stability (including asymptotic stability, exponential stability and absolute stability) as well as bifurcation and chaos, and many intensive and valuable results have been derived. This thesis mainly focuced on two types of neural network(Hopfield neural network and Inertial neural network). On the one hand, we also studied the global stability, the local stability, the Hopf bifurcation, the resonant codimension two bifurcation and chaos for autonomous system; On the other hand, we studied the local stability and the Hopf bifurcation for the nonautonomous system.
The main contents and originalities in this paper can be summarized as follows:①Delay-dependent asymptotic stability for neural networks with distributed delays and discrete delays
Signal transformation can not be finished in time due to the finite signal propogation time in biological networks, or to the finite switching speed of amplifiers in electronic neural networks, so the models depending the discrete time delays are not complete, the more exact models should include the distributed time delays. Moreover, we study the delay-dependent asymptotic stability for neural networks with distributed delays and discrete delays via constructing suitable Lyapunov-Krasovskii functions. Our studies are very important, on the one hand, the stability criteria for two kinds of delays are derived, and it is significant to give the more exact models and to predict the dynamical behavior of the models; on the other hand, comparing with the delay-independent stability criteria, the delay-dependent stability criteria are easier to obtain for the small delay neural network and have less limitations for the parameters of system.
②Local stability and Hopf bifurcation in an inertial two-neuron system with time delay
Under certain conditions, neurons exhibite a quasi-active membrane, which can be modeled by a phenonmenokogical inductance that allows the membrane to behave like a bandpass filter, enabling electrical tuning, or spatio-temporal filtering. So there are some strong biological backgrounds for the inertial delayed neural networks. In this thesis, we have studied the local stability and the existence of Hopf bifurcation at first, then we derived the direction of Hopf bifurcation and stability criterias of bifurcating periodic solutions by applying the center manifold theorem and the normal form theory. These studies are very important. On the one hand, bifurcations, which involve emergence of oscillatory behaviors, may provide an explanation for the parameter sensitivity observed in practice in many realistic small-scale networks such as the Internet, the electrical power grids, and the biological neural networks; and on the other hand, if we understand more about the bifurcation behaviors of small-scale networks, we can apply the existing effective bifurcation control method to achieve some desirable system behaviors that benefit the networks.
③Local stability and resonant codimension-two bifurcation in an inertial two-neuron system with time delay
It is a very important topic to predict and avoid resonant in delayed neural network. In this dissertation, we focuse on the discussion for the roots of the system’s characteristic equation based on the studies of the above problem, then using the delay as bifurcation parameter, we derive the conditions for the occurrence of two pairs of pure imaginary roots(i.e.±iω1 ,±iω2) are derived. When the frequency ratioω1 :ω2 is rational number, the phenomena of resonant will occure. This result and analytical method can provide the theoretical basis for the couple of system amplitudes, the synchronization of frequency, and can also reduced the occurrence of system resonant.
④Local stability and Hopf bifurcation for delayed inertial neural network under periodic excitation
There is evidence from the experimental studies that assemblies of cells in the visual cortex oscillate synchronously in response to external stimuli. So we have studied the local stability, the existence and the direction for bifurcating periodic solutions of delayed inertial neural model with periodic stimuli (non-autonomous system). By using the center manifold theorem and the averaged method in the theory of nonlinear oscillation, first we derived the center manifold of the system is derived. Then the averaged equations of the system are obtained. Finally the bifurcating equations via analyzing the Jacobian matrix for the averaged equations are obtained, furthermore we got the direction of the periodic solution and the bifurcating points from the bifurcation equations. Since most of the existing literature on theoretical studies of bifurcation problem is predominantly concerned with autonomous systems, Literature dealing with bifurcation for non-autonomous systems appears to be scarce. The analyzing method for the system’s dynamical behavior in this dissertation may be helpful to solve these problems.
⑤Local stability and Hopf bifurcation in Hopfield neural network with time delay and periodic excitation
The application of the Hopfield neural network has permeated many fields such as biology, physics, geology, etc. And it has also been widely applied to intelligent control, pattern recognition and nonlinear optimization. Moreover it is significant to study the dynamic behavior of this model which has wide use. In this dissertation, we study the dynamic behavior of the Hopfield model under the influence of time delay and periodic stimuli by using the center manifold theorem as well as the perturbation techniques.
本论文的主要内容和创新之处可概述如下:
①获得了分布时滞和有限时滞神经网络与时滞相关的全局稳定性和局部稳定性判据
由于神经元之间有限的信息传输速度以及电路系统中放大器的有限开关速度,同时因为存在大量的并行旁路以及各种不同长度和大小的轴突神经网络在空间上的扩展,信号传输不可能在一定时间内完成,因此建模上仅仅依靠有限时滞(常数时滞或者时变时滞)是不完整的,较精确的模型应该还应该包括无限时滞(分布时滞)。本论文中采用通常的Lyapunov函数构造法分别对带分布时滞或有限时滞的神经网络模型进行探讨,获得了与时滞相关的全局稳定性和局部稳定性判定准则。本研究的意义有两个方面:一、从两种时滞的角度分别给出稳定性的判定准则,这样对于更加精确的建模并预测模型从稳定至不稳定的演变过程是十分有益的;二、给出了与时滞相关的稳定性判据,这种准则对于小时滞神经网络不会过于苛刻,对系统参数也没有过多的限制,有利于预测系统模型动力学行为的鲁棒性和稳定性。
②对惯性时滞神经网络的局部稳定性和Hopf分岔进行了研究
惯性时滞神经网络具有较强的生物学背景,因此在满足一定条件的情况下,神经元的电路实现可以通过加入一个电感完成类似于带通滤波器、电调谐或者时空过滤的作用。由此我们研究了一个带有惯性项的时滞神经网络模型的局部稳定性和Hopf分岔,并用中心流形定理和正规型理论确定了分岔周期解的稳定性和分岔方向。其意义在于:一、由于Hopf分岔与振荡现象密切相关,对这种小规模网络Hopf分岔的研究可以使我们更好地解释现实世界中的许多大规模网络,如Internet、电网、生物神经网络中发生的对参数敏感的现象。二、若能深入地了解小规模网络中的分岔现象和规律,则通过利用比较成熟的分岔控制理论和方法,我们就可以将现实世界中大规模网络控制到所期望的有利的状态中去。
③对惯性神经网络的共振余维二分岔的研究
如何在人工神经网络系统中去预测并避免共振是十分重要的一个课题。在本论文中对惯性时滞神经网络模型进行了分析,在以时滞参数作为分岔参数的条件下,将研究重点放在对模型特征方程的根的讨论上,从而获取系统能够出现两对纯虚根的条件,当频率比ω1 :ω2为有理数时系统将会产生共振现象,所以本现象的研究结果以及分析方法可以为系统振幅的耦合和频率同步以及减少系统出现的共振提供理论依据。
④对惯性时滞神经网络在外部周期激励影响下的局部稳定性和Hopf分岔的研究
根据生物学实验,神经元在被施予外部周期激励时会产生同步振荡现象。因此作者研究了当惯性时滞神经网络引入了外部周期激励后,该非自治系统的局部稳定性以及周期解的存在和方向性。通过使用中心流形定理以及非线性振动中平均法技术的结合,我们首先获得了模型的中心流形,进而获取其平均方程,利用对方程的雅可比矩阵的分析得到分岔方程,由该分岔方程分析出周期解的方向和分岔点。由于目前主要的分岔理论都是针对自治系统的分岔问题,对非自治系统的分岔问题讨论的很少,本论文对这种类型系统的动力学性质做了一个初探,有利于为实际应用提供帮助。
⑤对Hopfield网络模型在时滞以及外部周期激励的共同影响下的局部稳定性和Hopf分岔的研究
对于Hopfield神经网络模型,它的应用已经渗透到生物学、物理学、地质学等诸多领域,并在智能控制、模式识别、非线性优化等方面获得了广泛的应用,针对这种实际应用非常广泛的网络模型讨论它的动力学行为是非常有意义的。本论文对该模型除了引入时滞外还增加了外部的周期激励,同时采用中心流形定理结合平均法获得系统的分岔周期解。虽然本论文该内容的研究方法与上一章的内容类似,但是因为该网络模型在实际应用中十分广泛,更具有重要意义。
As we all know, the introduce of time delay often brings great changes on the dynamical behavior in the fields of network systems. But the time delay has to be considered into the factors which influence systems due to the limitation of the signal transformation velocity. The delayed neural network is an important part of the delayed large-scale system, which exhibits the rich and colorful dynamical behavior. Due to the important applications in signal processing, moving image processing as well as optimizing problems, the dynamical behavior of the delayed nueral network has been studied by many researchers all the time, especially the dynamical behaviors of the global stability, the local stability (including asymptotic stability, exponential stability and absolute stability) as well as bifurcation and chaos, and many intensive and valuable results have been derived. This thesis mainly focuced on two types of neural network(Hopfield neural network and Inertial neural network). On the one hand, we also studied the global stability, the local stability, the Hopf bifurcation, the resonant codimension two bifurcation and chaos for autonomous system; On the other hand, we studied the local stability and the Hopf bifurcation for the nonautonomous system.
The main contents and originalities in this paper can be summarized as follows:①Delay-dependent asymptotic stability for neural networks with distributed delays and discrete delays
Signal transformation can not be finished in time due to the finite signal propogation time in biological networks, or to the finite switching speed of amplifiers in electronic neural networks, so the models depending the discrete time delays are not complete, the more exact models should include the distributed time delays. Moreover, we study the delay-dependent asymptotic stability for neural networks with distributed delays and discrete delays via constructing suitable Lyapunov-Krasovskii functions. Our studies are very important, on the one hand, the stability criteria for two kinds of delays are derived, and it is significant to give the more exact models and to predict the dynamical behavior of the models; on the other hand, comparing with the delay-independent stability criteria, the delay-dependent stability criteria are easier to obtain for the small delay neural network and have less limitations for the parameters of system.
②Local stability and Hopf bifurcation in an inertial two-neuron system with time delay
Under certain conditions, neurons exhibite a quasi-active membrane, which can be modeled by a phenonmenokogical inductance that allows the membrane to behave like a bandpass filter, enabling electrical tuning, or spatio-temporal filtering. So there are some strong biological backgrounds for the inertial delayed neural networks. In this thesis, we have studied the local stability and the existence of Hopf bifurcation at first, then we derived the direction of Hopf bifurcation and stability criterias of bifurcating periodic solutions by applying the center manifold theorem and the normal form theory. These studies are very important. On the one hand, bifurcations, which involve emergence of oscillatory behaviors, may provide an explanation for the parameter sensitivity observed in practice in many realistic small-scale networks such as the Internet, the electrical power grids, and the biological neural networks; and on the other hand, if we understand more about the bifurcation behaviors of small-scale networks, we can apply the existing effective bifurcation control method to achieve some desirable system behaviors that benefit the networks.
③Local stability and resonant codimension-two bifurcation in an inertial two-neuron system with time delay
It is a very important topic to predict and avoid resonant in delayed neural network. In this dissertation, we focuse on the discussion for the roots of the system’s characteristic equation based on the studies of the above problem, then using the delay as bifurcation parameter, we derive the conditions for the occurrence of two pairs of pure imaginary roots(i.e.±iω1 ,±iω2) are derived. When the frequency ratioω1 :ω2 is rational number, the phenomena of resonant will occure. This result and analytical method can provide the theoretical basis for the couple of system amplitudes, the synchronization of frequency, and can also reduced the occurrence of system resonant.
④Local stability and Hopf bifurcation for delayed inertial neural network under periodic excitation
There is evidence from the experimental studies that assemblies of cells in the visual cortex oscillate synchronously in response to external stimuli. So we have studied the local stability, the existence and the direction for bifurcating periodic solutions of delayed inertial neural model with periodic stimuli (non-autonomous system). By using the center manifold theorem and the averaged method in the theory of nonlinear oscillation, first we derived the center manifold of the system is derived. Then the averaged equations of the system are obtained. Finally the bifurcating equations via analyzing the Jacobian matrix for the averaged equations are obtained, furthermore we got the direction of the periodic solution and the bifurcating points from the bifurcation equations. Since most of the existing literature on theoretical studies of bifurcation problem is predominantly concerned with autonomous systems, Literature dealing with bifurcation for non-autonomous systems appears to be scarce. The analyzing method for the system’s dynamical behavior in this dissertation may be helpful to solve these problems.
⑤Local stability and Hopf bifurcation in Hopfield neural network with time delay and periodic excitation
The application of the Hopfield neural network has permeated many fields such as biology, physics, geology, etc. And it has also been widely applied to intelligent control, pattern recognition and nonlinear optimization. Moreover it is significant to study the dynamic behavior of this model which has wide use. In this dissertation, we study the dynamic behavior of the Hopfield model under the influence of time delay and periodic stimuli by using the center manifold theorem as well as the perturbation techniques.
引文
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