Reconstruction of shared nonlinear dynamics in a network

We study a general physical network consisting of a collection of response systems with complex nonlinear dynamics, influenced by a common driver. The goal is to reconstruct dynamics, regular or chaotic, that are common to all of the response systems, working from simultaneous time series measured at the responses systems only. A fundamental theorem is stated concerning the reconstruction of the common driver. An algorithm is developed, based on the theorem, to carry out the reconstruction, and is demonstrated with several examples.     

Extracting messages masked by chaotic signals of time-delay systems

We show how to extract messages masked by a chaotic signal of a time-delay system with very high dimensions and many positive Lyapunov exponents. Using a special embedding coordinate, the infinite-dimensional phase space of the time-delay system is projected onto a special three-dimensional space, which enables us to identify the time delay of the system from the transmitted signal and reconstruct the chaotic dynamics to unmask the hidden message successfully. The message extraction procedure is illustrated by simulations with the Mackey-Glass time-delay system for two types of masking schemes and different kinds of messages.     

基于符号分析的极大联合熵延迟时间求取方法

本文通过符号分析法求取联合熵的极大值点, 进而得到相空间重构的最佳延迟时间, 通过对几个典型的混沌系统进行数值仿真试验, 结果表明, 该方法简化了计算, 提高了效率, 能够准确快速地获得最佳延迟时间, 从而有效地重构原系统的相空间, 为混沌信号识别提供一种快速有效的途径. 关键词： 相空间重构 延迟时间 联合熵 符号分析     

A Sound Field Separation and Reconstruction Technique Based on Reciprocity Theorem and Fourier Transform

We show a method to separate the sound field radiated by a signal source from the sound field radiated by noise sources and to reconstruct the sound field radiated by the signal source. The proposed method is based on reciprocity theorem and the Fourier transform. Both the sound field and its gradient on a measurement surface are needed in the method. Evanescent waves are considered in the method, which ensures a high resolution reconstruction in the near field region of the signal source when evanescent waves can be measured. A simulation is given to verify the method and the influence of measurement noise on the method is discussed.     

Model-based detector and extraction of weak signal frequencies from chaotic data

Detecting a weak signal from chaotic time series is of general interest in science and engineering. In this work we introduce and investigate a signal detection algorithm for which chaos theory, nonlinear dynamical reconstruction techniques, neural networks, and time-frequency analysis are put together in a synergistic manner. By applying the scheme to numerical simulation and different experimental measurement data sets (Henon map, chaotic circuit, and NH(3) laser data sets), we demonstrate that weak signals hidden beneath the noise floor can be detected by using a model-based detector. Particularly, the signal frequencies can be extracted accurately in the time-frequency space. By comparing the model-based method with the standard denoising wavelet technique as well as supervised principal components analysis detector, we further show that the nonlinear dynamics and neural network-based approach performs better in extracting frequencies of weak signals hidden in chaotic time series.     

相位光栅型曲率传感器的波前校正算法

 针对相位光栅曲率传感器能够测量波前曲率在光瞳面上分布的特点,提出了一种采用整个空间分布的曲率信号来实现波前校正的算法。用曲率型变形镜影响函数的曲率信号在光瞳面上的分布来拟合待校正波前的曲率信号,采用最小二乘方法得到变形镜的控制电压,实现波前校正过程。数值模拟了一种41单元曲率型自适应光学系统采用该算法的波前校正过程。结果表明,对4至28阶Zernike像差体现了校正效果,对曲率为0的Zernike像差校正效果略好于曲率不为0的Zernike像差。与传统分区法的校正效果相比,整体法对Zernike像差的校正效果基本相当。整体法无需对光瞳面上的曲率信号进行与电极分布相同的分区,降低了对系统校准的要求。     

一种无线传感器网络中的混沌信号重构算法

将无线传感器网络节点观测区域中的一个混沌信号发送到融合中心,进行信号重构.由于节点的通信带宽受限,信号传输之前需要进行量化,给信号带来量化噪声,使得信号重构工作变得更为棘手.本文提出用平方根容积卡尔曼滤波器对融合中心收集的信号进行重构.首先估计观测信号的概率密度函数,使用最优量化器量化观测信号,在有限的量化比特数下,取得最优的信号量化性能.平方根容积卡尔曼滤波器相对无先导卡尔曼算法具有较少的求容积分点,因此具有计算量小的优点,同时迭代过程采用传递误差矩阵的平方根矩阵,保证迭代过程的稳定性和提高数据估计精度.仿真结果表明,该算法能够有效和快速地重构观测信号,并且比基于无先导卡尔曼滤波的算法更快.     

Identification of deterministic chaos by an information-theoretic measure of the sensitive dependence on the initial conditions

One of the most difficult problems in the field of non-linear time series analysis is the unequivocal characterization of a measured signal. We present a practicable procedure which allows to decide if a given time series is pure noise, chaotic but distorted by noise, purely chaotic, or a Markov process. Furthermore, the method gives an estimate of the Kolmogorov-Sinai (KS) entropy and the noise level. The procedure is based on a measure of the sensitive dependence on the initial conditions which is called ε-information flow. This measure generalizes the concept of KS entropy and characterizes the underlying dynamics. The ε-information flow is approximated by the calculation of various correlation integrals.     

Image Encryption Scheme with Compressed Sensing Based on a New Six-Dimensional Non-Degenerate Discrete Hyperchaotic System and Plaintext-Related Scrambling

Digital images can be large in size and contain sensitive information that needs protection. Compression using compressed sensing performs well, but the measurement matrix directly affects the signal compression and reconstruction performance. The good cryptographic characteristics of chaotic systems mean that using one to construct the measurement matrix has obvious advantages. However, existing low-dimensional chaotic systems have low complexity and generate sequences with poor randomness. Hence, a new six-dimensional non-degenerate discrete hyperchaotic system with six positive Lyapunov exponents is proposed in this paper. Using this chaotic system to design the measurement matrix can improve the performance of image compression and reconstruction. Because image encryption using compressed sensing cannot resist known- and chosen-plaintext attacks, the chaotic system proposed in this paper is introduced into the compressed sensing encryption framework. A scrambling algorithm and two-way diffusion algorithm for the plaintext are used to encrypt the measured value matrix. The security of the encryption system is further improved by generating the SHA-256 value of the original image to calculate the initial conditions of the chaotic map. A simulation and performance analysis shows that the proposed image compression-encryption scheme has high compression and reconstruction performance and the ability to resist known- and chosen-plaintext attacks.     

Synchronizing chaotic systems with positive conditional Lyapunov exponents by using convex combinations of the drive and response systems

We introduce a new method for synchronizing chaotic systems with positive conditional Lyapunov exponents, i.e., systems that do not synchronize in the Pecora-Carroll sense. This method works by considering a convex combination of the drive and response systems as a new driving signal. In this combination, the compoent associated with the response system acts as a chaos suppression method stabilizing the dynamics of the response system. This allows the chaotic component from the drive signal to synchronize both systems. The method is applied to synchronize some connections of the Rössler, Lorenz and van der Pol-Duffing systems that do not synchronize using the Pecora-Carroll scheme.     

基于小波变换的混沌信号相空间重构研究

应用小波变换和非线性动力学方法研究了混沌信号在相空间中的行为，指出混沌时间序 列的小波变换实质上是在重构的相空间中，混沌吸引子向小波滤波器向量所张的空间中的投 影，与Packard等人提出的相空间重构方法本质上是一致的.实验结果表明，混沌信号经过 小波变换后，吸引子轨迹与原有轨迹具有相似的结构，同时，系统的关联维数、Kolmogorov 熵等非线性不变量仍然得到保留.这些结果表明，利用小波变换研究混沌信号是有效的. 关键词： 小波变换 相空间重构 混沌信号 脑电信号     

二进制信号的混沌压缩测量与重构

二进制信号的压缩感知问题对应超奈奎斯特信号系统中未编码的二进制符号的检测问题, 具有重要的研究意义. 已有的二进制信号压缩测量采用高斯随机矩阵, 信号重构采用经典的l₁最小化方法. 本文利用混沌映射构造基于Cat序列的循环测量矩阵, 并提出一种针对二进制信号的全新的重构算法——平滑函数逼近法. 文章构造的混沌循环测量矩阵兼具确定性和随机性的优点, 能够抵御低信令效率和低信噪比的影响, 取得更好的压缩测量效果. 文章提出的平滑函数逼近法利用非凸函数代替原问题不连续的目标函数, 将组合优化问题转化为具有等式约束的优化问题进行求解. 利用稀疏贝叶斯学习算法进一步修正误差, 得到更准确的重构信号. 在信道含有加性高斯白噪声的条件下对二进制信号进行了压缩测量与重构的数值仿真, 仿真结果表明:基于Cat 序列的循环测量矩阵的压缩测量效果明显优于传统的高斯随机矩阵; 平滑函数逼近法对二进制信号的重构性能明显优于经典的l₁最小化方法.     

Reconstruction evaluation of intensity ratio distribution for extraction of absorber information in homogeneous scattering medium

To extract effectively the absorber information embedded in a homogeneous scattering medium in the transmission geometry, a method of taking the ratio of the output intensity in a measured medium to that in a reference medium is evaluated. The reference medium is a virtual one with the same scattering coefficient distribution as that of the measured medium, but with a uniform absorption coefficient distribution. Numerical results show that the proposed method can enhance the output signal by extracting the ballistic-like component. We also apply a backprojection method to reconstruct the single absorber using many pairs of input and output intensity ratio distributions. The reconstruction position error and the quantitative measurement of absorption coefficient are discussed.     

Comparison study of typical algorithms for reconstructing time series from the recurrence plot of dynamical systems

The three most widely used methods for reconstructing the underlying time series via the recurrence plots (RPs) of a dynamical system are compared with each other in this paper. We aim to reconstruct a toy series, a periodical series, a random series, and a chaotic series to compare the effectiveness of the most widely used typical methods in terms of signal correlation analysis. The application of the most effective algorithm to the typical chaotic Lorenz system verifies the correctness of such an effective algorithm. It is verified that, based on the unthresholded RPs, one can reconstruct the original attractor by choosing different RP thresholds based on the Hirata algorithm. It is shown that, in real applications, it is possible to reconstruct the underlying dynamics by using quite little information from observations of real dynamical systems. Moreover, rules of the threshold chosen in the algorithm are also suggested.     

一种混沌海杂波背景下的微弱信号检测方法

基于复杂非线性系统的相空间重构理论,提出了一种基于遗传算法的支持向量机预测方法.利用改进的自相关法和饱和关联维数法确定混沌信号的时间延迟和嵌入维,从而实现相空间重构.通过遗传算法优化支持向量机中的惩罚系数和核函数参数,并结合支持向量机建立混沌序列的单步预测模型,从预测误差中检测出淹没在混沌背景中的微弱信号(包括瞬态信号和周期信号).以Lorenz系统和加拿大McMaster大学利用IPIX雷达实测得到的海杂波数据作为混沌背景噪声进行仿真实验,结果表明该方法能够有效地从混沌背景噪声中检测出微弱目标信号,所得的均方根误差为0.00049521(信噪比为-89.7704 dB),这比传统支持向量机方法的均方根误差(0.049,信噪比为-54.60 dB)降低了两个数量级.     

基于指数加权-核在线序列极限学习机的混沌系统动态重构研究

利用指数加权在线核序列极限学习机(exponential weighted online sequential extreme learning machine with kernel, EW-KOSELM)辨识算法,开展了针对混沌动力学系统的动态重构研究. EW-KOSELM算法将核递归最小二乘(kernel recursive least squares, KRLS)算法直接延伸至在线ELM (extreme learning machine)框架中,通过引入遗忘因子削弱了旧数据的影响,并基于"固定预算(fixed-budget, FB)"内存技术,应对在线核学习算法所固有的规模不断增长的计算困难.将所提辨识算法应用于Duffing-Ueda振子的混沌动力学系统数值仿真实例中,对基于FB-EW-KOSELM的辨识模型与原系统的动态性能进行了定性与定量的分析校验,定性校验准则是基于对比辨识模型与原系统吸引子(轨迹嵌入)、庞加莱映射、分岔图、极限环完成的,定量校验准则包括对比辨识模型与原系统的李雅普诺夫指数与关联维.进一步将其分别应用于来自测量蔡氏电路产生双涡卷吸引子与螺旋吸引子的实测数据实验及某一实际混沌电路所产生的时间序列中,对于具有低信噪比的实测电压或电流数据还需进行了小波降噪预处理.通过分析辨识模型重构吸引子,实验结果表明,FB-EW-KOSELM算法具有良好的动态重构性能,能精确地再生出展示混沌动态行为的过程非线性模型,且具有与原混沌系统非常接近的动态不变性指标.     

混沌背景中微弱信号检测的神经网络方法

基于复杂非线性系统相空间重构理论，提出了混沌背景中微弱信号检测的神经网络方法，利用神经网络强大的学习和非线性处理能力，建立了混沌背景噪声的一步预测模型，从预测误差中检测淹没在混沌背景噪声中的微弱目标信号(包括周期信号和瞬态信号)，研究了混沌背景中存在白噪声时该方法的检测能力，指出了目标信号为瞬态信号和周期信号时检测原理的异同点，最后以Lorenz系统作为混沌背景噪声进行了仿真实验，实验表明该方法能有效地将混沌背景中极其微弱的信号检测出来. 关键词： 混沌 神经网络 信号检测     

Application of local polynomial estimation in suppressing strong chaotic noise

In this paper, we propose a new method that combines chaotic series phase space reconstruction and local polynomial estimation to solve the problem of suppressing strong chaotic noise. First, chaotic noise time series are reconstructed to obtain multivariate time series according to Takens delay embedding theorem. Then the chaotic noise is estimated accurately using local polynomial estimation method. After chaotic noise is separated from observation signal, we can get the estimation of the useful signal. This local polynomial estimation method can combine the advantages of local and global law. Finally, it makes the estimation more exactly and we can calculate the formula of mean square error theoretically. The simulation results show that the method is effective for the suppression of strong chaotic noise when the signal to interference ratio is low.     

Spiking dynamics of interacting oscillatory neurons

Spiking sequences emerging from dynamical interaction in a pair of oscillatory neurons are investigated theoretically and experimentally. The model comprises two unidirectionally coupled FitzHugh-Nagumo units with modified excitability (MFHN). The first (master) unit exhibits a periodic spike sequence with a certain frequency. The second (slave) unit is in its excitable mode and responds on the input signal with a complex (chaotic) spike trains. We analyze the dynamic mechanisms underlying different response behavior depending on interaction strength. Spiking phase maps describing the response dynamics are obtained. Complex phase locking and chaotic sequences are investigated. We show how the response spike trains can be effectively controlled by the interaction parameter and discuss the problem of neuronal information encoding.     

含噪混沌时间序列预测误差分析

对于含噪混沌时间序列预测问题, 传统方法存在较大的经验性, 对预测误差的构成分析不足, 因而忽略了混沌动态重建与预测模型建立之间的差异性. 本文将实际预测误差分解为预测器偏差和输入扰动误差, 并对整体最小二乘和正则化两种全局预测方法进行分析比较, 进而说明整体最小二乘适用于混沌动态的重建, 对预测器偏差影响较大, 而正则化方法能够改善预测器敏感性, 对输入扰动误差影响较大. 通过两个仿真实例, 展示了混沌动态重建与预测模型建立之间的差异, 在对比最小二乘和正则化方法的同时验证了实际预测误差受预测器偏差和输入扰动误差共同作用. 并指出, 在实际操作时应在二者间寻求平衡, 以便使模型预测精度达到最优. 关键词： 混沌时间序列预测 噪声 整体最小二乘 正则化