采用LS-SVM计算时间序列的Lyapunov指数谱

为了计算未知系统的Lyapunov指数谱,首先,对一维观测数据序列进行相空间重构,然后,利用最小二乘支持向量机（LS-SVM）逼近重构系统的动力学方程,再通过雅克比矩阵计算Lyapunov指数谱。采用提出的方法计算Henon映射的Lyapunov指数谱,可以得到精确的计算结果且需要的序列步长小于1 000。计算了实测不同状态的交通流时间序列的Lyapunov指数谱。结果表明：在拥挤状态下,有多个Lyapunov指数大于零,说明系统是超混沌的;在同步状态下,有一个或多个Lyapunov指数大于零,说明系统是混沌的或超混沌的;在堵塞状态下,Lyapunov指数全小于零,说明系统不是混沌的。     

含噪声混沌信号最大Lyapunov指数计算

提出一种计算含噪声混沌序列最大Lyapunov指数的改进Rosentein算法,通过将原始含噪声的相空间与局部投影去噪后的相空间合并,使邻近点的选择更为准确,减小噪声的影响提高计算精度.分别对加入白噪声的Henon混沌序列和逐月太阳黑子混沌序列进行仿真,证明此方法有效性.     

基于最大Lyapunov指数的交通流仿真数据混沌状态识别

利用交通流模型产生交通流时间序列：再利用Lyapunov指数的矩形阵算法，计算出交通流时间序列的最大Lyapunov指数。由于Lyapunov指数是定量描述混沌吸引子的重要指标，可根据Lyapunov指数对混沌序列的辨别原理，进而识别该由基于模型的动力系统是否处于混沌状态。     

低分辨率小规模网络流量数据的混沌特性鉴别

使用非线性时间序列分析方法,对低分辨率采样的小规模网络流量数据进行混沌特性分析。给出了网络流量数据的平滑策略,计算了网络流量数据的最大Lyapunov指数,并对流量数据与噪声序列加以区分,从不同角度验证了小规模网络流量数据具有低维混沌特性。为采用混沌方法研究网络流量行为特性奠定了基础。     

Chua电路的数值研究

混沌运动有许多特征量,其中Lyapunov指数描述了混沌系统对初始状态的敏感性.本文研究了蔡式混沌非线性电路系统的Lyapunov指数以及如何利用最大Lvapunov指数判定Chua系统是否处于混沌运动状态.此外,在Matlab平台上设计了一种基于时间序列计算Chua电路系统最大Lyapunov指数的模型,该模型具有易于实现、计算量小、统计性能好等特点.     

非线性时间序列混沌特性分析及短期预测

为提高非线性时间序列预测的准确性和可靠性,采用基于混沌理论的方法对时间序列进行分析和预测.在研究关联维数和最大Lyapunov指数算法基础上,利用关联维数和最大Lyapunov指数判定时间序列的混沌特性,根据混沌特性参数建立预测模型,并对非线性时间序列进行预测.以上海证券交易所股票价格指数时间序列为实例验证预测模型,研究结果表明,基于混沌特性参数建立的预测模型具有较好的预测能力和预测精度,证明用该方法预测非线性时间序列具有可行性.     

混沌时间序列预测模型的比较研究

针对目前混沌时间序列预测模型预测结果差异较大的问题,归纳了4种混沌时间序列预测模型：BRF神经网络模型、最大Lyapunov指数模型、局域线性模型和Volterra滤波器自适应预测模型,并对这4种预测模型进行了比较研究。应用4种预测模型对几个典型的非线性系统进行预测仿真。结果表明,这4种预测模型对典型混沌时间序列预测都具有很好的预测效果;在预测精度上BRF模型和Volterra模型明显优于最大Lyapunov指数模型和局域线性模型。     

一种新的混沌序列的产生及其性能分析

通过对混沌理论深入的研究分析,提出了一种新的混沌映射,并且给出了序列二进制化的方法.在此基础上,针对能体现混沌序列的几个重要特性对改进的混沌映射进行分析、实验,得出了此混沌映射相比Logistic映射具有较好的自/互相关特性、初值敏感性、平衡性、Lyapunov指数和线性复杂度的特性.这种映射产生简单、对硬件的要求不高,对混沌序列在加密通信中的应用具有很好的参考价值.     

基于多次重构映射的相干混沌数字通信系统*

利用相空间多次重构的混沌映射,为相干方式的混沌数字通信问题,提供一种解决方案。新映射产生的混沌码序列性能更加优异,自相关性明显改善,最大Lyapunov指数由0.697增加到5.545,复杂度也明显提高,仿真证实,利用新的混沌映射,系统抗多址性能提高;基于新映射混沌码良好的相关性能和抗多址性能,设计了一种滑膜相关峰检测同步法,完成离散混沌系统同步,实现相干方式混沌数字通信。     

基于无退化混沌系统的序列密码研究

针对连续时间混沌系统的退化问题,提出一种基于矩阵特征值配置的方法来构造具有多个正Lyapunov指数的连续时间混沌系统。提出一种基于特征值定义的特征值配置方法,通过设计一个线性反馈控制器,可以配置任何系统为以稳定焦点为原点的渐近稳定线性系统;通过设计一个非线性反馈控制器来配置多个正Lyapunov指数。相比于现有算法,对于任意受控系统,该方法都能系统地配置该受控系统的Lyapunov指数,使之成为无退化混沌系统。将该方法得到的无退化混沌系统转换为二进制序列,对得到的混沌序列进行分析后证明该序列具有良好的加密特性。     

Neural learning of chaotic dynamics

In recent years, considerable progress has been made in modeling chaotic time series with neural networks. Most of the work concentrates on the development of architectures and learning paradigms that minimize the prediction error. A more detailed analysis of modeling chaotic systems involves the calculation of the dynamical invariants which characterize a chaotic attractor. The features of the chaotic attractor are captured during learning only if the neural network learns the dynamical invariants. The two most important of these are the largest Lyapunov exponent which contains information on how far in the future predictions are possible, and the Correlation or Fractal Dimension which indicates how complex the dynamical system is. An additional useful quantity is the power spectrum of a time series which characterizes the dynamics of the system as well, and this in a more thorough form than the prediction error does. In this paper, we introduce recurrent networks that are able to learn chaotic maps, and investigate whether the neural models also capture the dynamical invariants of chaotic time series. We show that the dynamical invariants can be learned already by feedforward neural networks, but that recurrent learning improves the dynamical modeling of the time series. We discover a novel type of overtraining which corresponds to the forgetting of the largest Lyapunov exponent during learning and call this phenomenondynamical overtraining. Furthermore, we introduce a penalty term that involves a dynamical invariant of the network and avoids dynamical overtraining. As examples we use the Hénon map, the logistic map and a real world chaotic series that corresponds to the concentration of one of the chemicals as a function of time in experiments on the Belousov-Zhabotinskii reaction in a well-stirred flow reactor.     

Finding chaos in Finnish GDP

The goal of this paper is to analyze the Finnish gross domestic product(GDP) and to find chaos in the Finnish GDP. We chose Finland where data has been available since 1975, because we needed the longest time series possible. At first we estimated the time delay and the embedding dimension, which is needed for the Lyapunov exponent estimation and for the phase space reconstruction.Subsequently, we computed the largest Lyapunov exponent, which is one of the important indicators of chaos. Then we calculated the 0-1 test for chaos. Finally we computed the Hurst exponent by rescaled range analysis and by dispersional analysis. The Hurst exponent is a numerical estimate of the predictability of a time series. In the end, we executed a recurrent analysis and displayed recurrence plots of detrended GDP time series. The results indicated that chaotic behaviors obviously exist in GDP.     

Chaotic time series method combined with particle swarm optimization and trend adjustment for electricity demand forecasting

Electricity demand forecasting plays an important role in electric power systems planning. In this paper, nonlinear time series modeling technique is applied to analyze electricity demand. Firstly, the phase space, which describes the evolution of the behavior of a nonlinear system, is reconstructed using the delay embedding theorem. Secondly, the largest Lyapunov exponent forecasting method (LLEF) is employed to make a prediction of the chaotic time series. In order to overcome the limitation of LLEF, a weighted largest Lyapunov exponent forecasting method (WLLEF) is proposed to improve the prediction accuracy. The particle swarm optimization algorithm (PSO) is used to determine the optimal weight parameters of WLLEF. The trend adjustment technique is used to take into account the seasonal effects in the data set for improving the forecasting precision of WLLEF. A simulation is performed using a data set that was collected from the grid of New South Wales, Australia during May 14–18, 2007. The results show that chaotic characteristics obviously exist in electricity demand series and the proposed prediction model can effectively predict the electricity demand. The mean absolute relative error of the new prediction model is 2.48%, which is lower than the forecasting errors of existing methods.     

Effect of noise on chaotic behavior in Roessler-type nonlinear system

The effects of noise on chaotic behaviors of a nonlinear dynamic model were described from a point of view of the system analysis and the previous studies associated with chaos and noise were reviewed as well. The quasi-white noise was used as the observation noise as well as the system noise to clarify the deterioration of the chaotic patterns of the Roessler model. The effects of the noise intensity on the chaotic signal were observed through the deformation of the attractors, increase of the correlation dimension, and change of the maximum Lyapunov exponent. It has been found that the deterioration of the chaotic patterns is more pronounced in the case of the observation noise than the system noise for the Roessler model. As an example of noisy time series data, the laser speckles time series data was employed and discussed from the point of view of the necessity of noise reduction and possible chaos extraction. © 1997 John Wiley & Sons, Inc.     

基于神经网络的混沌时间序列建模及预测

该文从相空间重构理论出发,讨论了基于神经网络的混沌时间序列建模及预测方法,并以Logistic方程产生的混沌时间序列作为研究对象,采用BP和RBF两种神经网络分别对其进行了仿真分析,实验结果表明:最大Lyapunov指数越大,可预测步长越短;基于RBF网络的混沌时间序列建模及预测效果优于BP网络。     

基于离均差的时间序列相似性度量

 时间序列的相似性度量是数据挖掘领域研究的一个热点,高维多元时间序列数据一般含有大量的噪声不利于相似性的比较。针对现有的时间序列度量方法存在的问题,在改进的时间序列自底向上融合算法的基础上,提出一种新的基于离均差的时间序列相似性度量的夹角余弦算法(Angle Cosine Metric Similarity,ACMS)。ACMS算法将时间序列等价为一个多维度的向量,充分考虑2个向量的方向和大小特征,增强振幅变化的鲁棒性,减少人为干扰,对数据挖掘中的聚类和预测具有帮助作用。     

Asymptotic Lyapunov stability with probability one of quasi-integrable Hamiltonian systems with delayed feedback control

The asymptotic Lyapunov stability with probability one of multi-degree-of-freedom (MDOF) quasi-integrable and nonresonant Hamiltonian systems with time-delayed feedback control subject to multiplicative (parametric) excitation of Gaussian white noise is studied. First, the time-delayed feedback control forces are expressed approximately in terms of the system state variables without time delay and the system is converted into ordinary quasi-integrable and nonresonant Hamiltonian system. Then, the averaged Itô stochastic differential equations are derived by using the stochastic averaging method for quasi-integrable Hamiltonian systems and the expression for the largest Lyapunov exponent of the linearized averaged Itô equations is derived. Finally, the necessary and sufficient condition for the asymptotic Lyapunov stability with probability one of the trivial solution of the original system is obtained approximately by letting the largest Lyapunov exponent to be negative. An example is worked out in detail to illustrate the above mentioned procedure and its validity and to show the effect of the time delay in feedback control on the largest Lyapunov exponent and the stability of the system.     

一种基于始末距离的时间序列符号聚合近似表示方法

时间序列数据的特征表示方法是时间序列数据挖掘任务的关键技术,符号聚合近似表示(SAX)是特征表示方法中比较常用的一种。针对SAX算法在各序列段表示符号一致时无法区分时间序列间的相似性这一缺陷,提出了一种基于始末距离的时间序列符号聚合近似表示方法(SAX_SM)。由于时间序列有很强的形态趋势,因此文中提出的方法选用起点和终点来表示各个序列段的形态特征,并使用各序列段的形态特征和表示符号来近似表示时间序列数据,以将其从高维空间映射到低维空间;然后,针对起点和终点构建始末距离来计算两序列段间的形态距离;最后, 结合 始末距离和符号距离定义一种新的距离度量方式,以更客观地度量时间序列间的相似性。理论分析表明,该距离度量满足下界定理。在20组UCR时间序列数据集上的实验表明,所提SAX_SM方法在13个数据集中获得了最高的分类准确率(包含并列最大的),而SAX只在6个数据集中获得了最高的分类准确率(包含并列最大的),因此SAX_SM具有比SAX更优的分类效果。