全局耦合网络的特性及其混沌控制研究
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摘要
全局耦合映射(Globally Coupled Map, GCM)网络是一种特殊的混沌神经网络,网络的每个神经元的运动由混沌映射决定。网络有一类特殊的吸引子被称为“聚类冻结吸引子”(Cluster Frozen Attractors, CFA),可有效的用于信息处理和联想记忆等。本文通过对不同的GCM网络的研究,揭示了这一类型网络的整体性质；通过对网络的控制使得GCM网络可以用于动态联想记忆的应用。论文的主要工作有
1.研究了几个混沌映射的性质。首先提出了新的混沌映射——三次Logistic映射,并对三次Logistic映射、正(余)弦映射和一种特殊非线性映射所构成的离散混沌迭代系统进行了研究,通过Lyapunov指数和分岔图刻画了系统的混沌行为,给出了混沌系统的参数域。其次从映射密度的角度揭示了它们的性质,指出不同的分岔参数导致分布密度的遍历性产生变化,并且影响密度函数的形状。最后从相关性分析、功率谱和均匀分布的卡方检验等方而研究了混沌映射的性质,说明了不同的混沌映射具有不同的混沌特性,为后文分析出它们构成的GCM网络打下基础。
2.分析了网络的宏观和微观特性。首先针对不同GCM网络分析了网络的特性,指出聚类冻结吸引子是GCM网络的共性特征,不会随信息传递函数而改变。其次提出了两种延时耦合方法,分析了延时耦合方式下不同GCM网络的宏观特性,指出延时的引入不改变网络的聚类冻结吸引子,并且能够使得网络具有更多的动力学性质。最后研究了网络微观特性,从神经元的迭代方程出发,分析了不同耦合项对神经元运动产生的影响,结论指出GCM网络的混沌特性主要由每个神经元产生,而聚类特性主要受耦合项的影响。
3.分析了网络的动力学特性。首先从数学角度说明了GCM网络平衡点的存在性,这一结论与网络的信息传递函数无关,适用于所有的GCM网络。其次给出了S-GCM网络和CL-GCM网络零平衡点渐近稳定的一个充分条件；并针对一维和二维情形分析了两个网络平衡点的稳定性和分岔行为,仿真结果说明了理论分析的正确性。
4.研究了GCM网络的两种控制方法和网络的应用。提出的第一种方法为反馈控制方法,这种方法不需利用外部信息,而是将系统两次运行的状态之差反馈回系统,通过调节动态参数的值实现了网络的不动点和周期控制,说明了这种方法适用于绝大部分的GCM模型。第二种控制方法是对Ishii方法的一个改进,通过调整参数阈值实现GCM网络的控制,但这种方法不适用于所有的GCM网络。在两种控制模式中都说明了延时耦合GCM网络具有比常规耦合GCM网络更好的性质。最后利用改进的参数调制控制方法研究了网络的联想记忆,指出了CL-GCM网络和SI-GCM实现了混沌神经网络的动态联想记忆,网络既可以输出固定模式,也可以输出包含正确模式的周期模式。
A neural network called globally coupled map (GCM) model is a special kind of chaotic neural network composed of chaotic neurons, whose dynamical behavior depend on the chaotic map. It has cluster frozen attractors (CFA) which can be taken to represent information, therefore it can be used to information processing and associative memory and so on. Motivated by creating and developing GCM model to be effectively applied to information processing, we investigate the dynamic behaviors of several different GCM models and apply the networks under certain control methods to implement dynamic associative memory.
The contributions can be concluded as four main points listed as follows:
1. The characteristics of several chaotic map models are investigated. First, discrete chaotic systems iterated by a newly presented cubic Logistic map, sinusoidal (cosine) map and a special nonlinear map are considered. Their chaotic dynamic behaviors are demonstrated by calculating Lyapunov exponents and showing bifurcation diagrams, and their parameter regions are also given. Second, their features are illustrated from the point view of distribution of probability density. It is observed that both ergodicity of distribution and the shape of probability density function vary according to different bifurcation parameters. At last, different characteristics of different chaotic maps are developed from the view of relativity, power spectrum and chi-square test of mean distribution and so on. All the analyses lay the foundations for the improved GCM models iterated by the chaotic maps above.
2. The macro and micro characteristics of the networks are analyzed. First, it indicates that cluster frozen attractor is a common feature of different GCM and it doesn't change according to the variance of information transfer function. Second, the macro features of different GCM with two newly presented ways of time-delay coupling are illustrated. It can be seen that the cluster frozen attractor doesn't change with the introduction of time delay and the networks exhibit richer running behaviors. Finally, the micro features of the networks and the influence on the neurons' behaviors by coupling are investigated from the view of iteration of neurons. It can be concluded that the chaotic dynamics of GCM networks originate from every neuron's chaotic running behavior and the ways of coupling mainly influence clusters.
3. The dynamics of GCM networks are demonstrated. First, the existence of equilibrium points is proved, which has nothing to do with the information transfer functions and is fit for all GCM models. Second, sufficient conditions on the asymptotical stability of zero equilibrium point for S-GCM and CL-GCM are given respectively. Furthermore, the stability of equilibrium points and bifurcation for one and two dimensional systems are analyzed. Numerical simulation results demonstrate the correctness and effectiveness of the analysis above.
4. Two control methods and the applications of the GCM model are given. On the one hand, GCM models are controlled by a kind of feedback control method which sends deviations of the states back instead of introducing outer information to realize the control of fixed points and periods by adjusting parameters. It is indicated that the control method is fit for most of GCM models. On the other hand, a control method presented as an improvement of Ishii's parameter modulated control method is given to control GCM models by adjusting parameter threshold. Unfortunately, it isn't suitable for all GCM models. However, it is delighted to find that more excellent characteristics are exhibited by time-delay GCM model than by common GCM model. Finally, the systems' associative memory is illustrated by the improved parameter modulated control method. It suggests that CL-GCM and SI-GCM can not only output fixed patterns but also output periodical patterns which contain correct pattern. So the associative memory is successful.
1.研究了几个混沌映射的性质。首先提出了新的混沌映射——三次Logistic映射,并对三次Logistic映射、正(余)弦映射和一种特殊非线性映射所构成的离散混沌迭代系统进行了研究,通过Lyapunov指数和分岔图刻画了系统的混沌行为,给出了混沌系统的参数域。其次从映射密度的角度揭示了它们的性质,指出不同的分岔参数导致分布密度的遍历性产生变化,并且影响密度函数的形状。最后从相关性分析、功率谱和均匀分布的卡方检验等方而研究了混沌映射的性质,说明了不同的混沌映射具有不同的混沌特性,为后文分析出它们构成的GCM网络打下基础。
2.分析了网络的宏观和微观特性。首先针对不同GCM网络分析了网络的特性,指出聚类冻结吸引子是GCM网络的共性特征,不会随信息传递函数而改变。其次提出了两种延时耦合方法,分析了延时耦合方式下不同GCM网络的宏观特性,指出延时的引入不改变网络的聚类冻结吸引子,并且能够使得网络具有更多的动力学性质。最后研究了网络微观特性,从神经元的迭代方程出发,分析了不同耦合项对神经元运动产生的影响,结论指出GCM网络的混沌特性主要由每个神经元产生,而聚类特性主要受耦合项的影响。
3.分析了网络的动力学特性。首先从数学角度说明了GCM网络平衡点的存在性,这一结论与网络的信息传递函数无关,适用于所有的GCM网络。其次给出了S-GCM网络和CL-GCM网络零平衡点渐近稳定的一个充分条件；并针对一维和二维情形分析了两个网络平衡点的稳定性和分岔行为,仿真结果说明了理论分析的正确性。
4.研究了GCM网络的两种控制方法和网络的应用。提出的第一种方法为反馈控制方法,这种方法不需利用外部信息,而是将系统两次运行的状态之差反馈回系统,通过调节动态参数的值实现了网络的不动点和周期控制,说明了这种方法适用于绝大部分的GCM模型。第二种控制方法是对Ishii方法的一个改进,通过调整参数阈值实现GCM网络的控制,但这种方法不适用于所有的GCM网络。在两种控制模式中都说明了延时耦合GCM网络具有比常规耦合GCM网络更好的性质。最后利用改进的参数调制控制方法研究了网络的联想记忆,指出了CL-GCM网络和SI-GCM实现了混沌神经网络的动态联想记忆,网络既可以输出固定模式,也可以输出包含正确模式的周期模式。
A neural network called globally coupled map (GCM) model is a special kind of chaotic neural network composed of chaotic neurons, whose dynamical behavior depend on the chaotic map. It has cluster frozen attractors (CFA) which can be taken to represent information, therefore it can be used to information processing and associative memory and so on. Motivated by creating and developing GCM model to be effectively applied to information processing, we investigate the dynamic behaviors of several different GCM models and apply the networks under certain control methods to implement dynamic associative memory.
The contributions can be concluded as four main points listed as follows:
1. The characteristics of several chaotic map models are investigated. First, discrete chaotic systems iterated by a newly presented cubic Logistic map, sinusoidal (cosine) map and a special nonlinear map are considered. Their chaotic dynamic behaviors are demonstrated by calculating Lyapunov exponents and showing bifurcation diagrams, and their parameter regions are also given. Second, their features are illustrated from the point view of distribution of probability density. It is observed that both ergodicity of distribution and the shape of probability density function vary according to different bifurcation parameters. At last, different characteristics of different chaotic maps are developed from the view of relativity, power spectrum and chi-square test of mean distribution and so on. All the analyses lay the foundations for the improved GCM models iterated by the chaotic maps above.
2. The macro and micro characteristics of the networks are analyzed. First, it indicates that cluster frozen attractor is a common feature of different GCM and it doesn't change according to the variance of information transfer function. Second, the macro features of different GCM with two newly presented ways of time-delay coupling are illustrated. It can be seen that the cluster frozen attractor doesn't change with the introduction of time delay and the networks exhibit richer running behaviors. Finally, the micro features of the networks and the influence on the neurons' behaviors by coupling are investigated from the view of iteration of neurons. It can be concluded that the chaotic dynamics of GCM networks originate from every neuron's chaotic running behavior and the ways of coupling mainly influence clusters.
3. The dynamics of GCM networks are demonstrated. First, the existence of equilibrium points is proved, which has nothing to do with the information transfer functions and is fit for all GCM models. Second, sufficient conditions on the asymptotical stability of zero equilibrium point for S-GCM and CL-GCM are given respectively. Furthermore, the stability of equilibrium points and bifurcation for one and two dimensional systems are analyzed. Numerical simulation results demonstrate the correctness and effectiveness of the analysis above.
4. Two control methods and the applications of the GCM model are given. On the one hand, GCM models are controlled by a kind of feedback control method which sends deviations of the states back instead of introducing outer information to realize the control of fixed points and periods by adjusting parameters. It is indicated that the control method is fit for most of GCM models. On the other hand, a control method presented as an improvement of Ishii's parameter modulated control method is given to control GCM models by adjusting parameter threshold. Unfortunately, it isn't suitable for all GCM models. However, it is delighted to find that more excellent characteristics are exhibited by time-delay GCM model than by common GCM model. Finally, the systems' associative memory is illustrated by the improved parameter modulated control method. It suggests that CL-GCM and SI-GCM can not only output fixed patterns but also output periodical patterns which contain correct pattern. So the associative memory is successful.
引文
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