Determination of parameters of elements and coupling architecture in ensembles of coupled time-delay systems from their time series

We propose a method for reconstructing model differential equations with time delay for ensembles of coupled time-delay systems from their time series. The method has made it possible to recover the parameters of elements of the ensemble as well as the architecture and strength of couplings in ensembles of nonidentical systems with delay with an arbitrary number of unidirectional and bidirectional couplings between them. The effectiveness of the method is demonstrated for chaotic and periodic time series of model equations for ensembles of diffusively coupled systems with time delay in the presence of noise, as well as experimental time series for resistively coupled radiotechnical oscillators with delayed feedback.     

混沌时间序列的时滞判定

运用重构混沌吸引子的嵌入技术，开发了提取混沌时间序列信息的宏观信息熵方法，用以判定时滞的存在性和时滞的大小．仿真实验说明了本文方法的有效性 关键词：     

A Hybrid Method Using HAVOK Analysis and Machine Learning for Predicting Chaotic Time Series

The prediction of chaotic time series systems has remained a challenging problem in recent decades. A hybrid method using Hankel Alternative View Of Koopman (HAVOK) analysis and machine learning (HAVOK-ML) is developed to predict chaotic time series. HAVOK-ML simulates the time series by reconstructing a closed linear model so as to achieve the purpose of prediction. It decomposes chaotic dynamics into intermittently forced linear systems by HAVOK analysis and estimates the external intermittently forcing term using machine learning. The prediction performance evaluations confirm that the proposed method has superior forecasting skills compared with existing prediction methods.     

Modeling global vector fields of chaotic systems from noisy time series with the aid of structure-selection techniques

We address the problem of reconstructing a set of nonlinear differential equations from chaotic time series. A method that combines the implicit Adams integration and the structure-selection technique of an error reduction ratio is proposed for system identification and corresponding parameter estimation of the model. The structure-selection technique identifies the significant terms from a pool of candidates of functional basis and determines the optimal model through orthogonal characteristics on data. The technique with the Adams integration algorithm makes the reconstruction available to data sampled with large time intervals. Numerical experiment on Lorenz and Rossler systems shows that the proposed strategy is effective in global vector field reconstruction from noisy time series.     

时滞复Lorenz混沌系统特性及其自时滞同步

自时滞同步是指保持混沌系统结构和参数不变的情况下, 使时滞系统和原系统同步, 从而避免了现实中因为时滞而产生的各种问题. 本文以时滞复Lorenz系统为例, 研究其动态特性及时滞因数的影响, 并提出了一种非线性反馈控制器实现了复Lorenz系统的自时滞混沌同步. 数值仿真结果验证了该控制器的有效性. 该控制器只对部分状态进行控制, 实现了所有状态的同步, 原理简单, 易于工程实现. 关键词： 时滞系统 复混沌系统 自时滞同步 Lyapunov指数     

Recovery of delay time from time series based on the nearest neighbor method

We propose a method for the recovery of delay time from time series of time-delay systems. The method is based on the nearest neighbor analysis. The method allows one to reconstruct delays in various classes of time-delay systems including systems of high order, systems with several coexisting delays, and nonscalar time-delay systems. It can be applied to time series heavily corrupted by additive and dynamical noise.     

基于状态观测器的分数阶时滞混沌系统同步研究

研究分数阶时滞混沌系统同步问题,基于状态观测器方法和分数阶系统稳定性理论,设计分数阶时滞混沌系统同步控制器,使得分数阶时滞混沌系统达到同步,同时给出了数学证明过程.该同步控制器采用驱动系统和响应系统的输出变量进行设计,无需驱动系统和响应系统的状态变量,简化了控制器的设计,提高了控制器的实用性.利用Lyapunov稳定性理论和分数阶线性矩阵不等式,研究并给出了同步控制器参数的选择条件.以分数阶时滞Chen混沌系统为例,设计基于状态观测器的同步控制器,实现了分数阶时滞Chen混沌系统同步,并将其应用于保密通信系统中.仿真结果证明了该同步方法的有效性.     

Local filtering of noisy nonlinear time series

We investigate the use of different local nonlinear modelling and nonlinear filtering techniques to clean a noisy time series obtained from a deterministic chaotic systems. The methods are tested on data from the Ikeda map and the Mackey-Glass delay differential equation. We test the results of the filtered times series using the correlation dimension statistic and SNR gain. In all cases we see that local filtering has produced a new time series which is more consistent with the original clean time series.     

Time-delayed chameleon: Analysis,synchronization and FPGA implementation

In this paper we report a time-delayed chameleon-like chaotic system which can belong to different families of chaotic attractors depending on the choices of parameters. Such a characteristic of self-excited and hidden chaotic flows in a simple 3D system with time delay has not been reported earlier. Dynamic analysis of the proposed time-delayed systems are analysed in time-delay space and parameter space. A novel adaptive modified functional projective lag synchronization algorithm is derived for synchronizing identical time-delayed chameleon systems with uncertain parameters. The proposed time-delayed systems and the synchronization algorithm with controllers and parameter estimates are then implemented in FPGA using hardware–software co-simulation and the results are presented.     

不确定时滞混沌系统的自适应动态神经网络控制

研究了一类具有不确定时滞的非自治混沌系统的控制问题. 通过结合Lyapunov-Krasovskii函数和Lyapunov函数设计参数可调的不确定时滞补偿器,使得反馈控制输入信号不受时延的影响;同时引入动态结构自适应神经网络,以消除系统的不确定性,其隐层神经元的个数可以随着逼近误差的增大而自适应增加,改善了逼近速度与网络复杂度的关系;最后,用Duffing混沌系统的控制仿真示例表明该方法的有效性. 关键词： 混沌系统 自适应控制 不确定时滞 动态结构神经网络     

OPTIMAL CONTROL METHOD WITH TIME DELAY IN CONTROL

Optimal control method for active vibration control of linear time-delay systems is investigated in this paper. In terms of two cases that time delay is integer and non-integer times of sampling period, motion equation with time delay is transformed as standard discrete forms which contain no time delay by using zero order holder respectively. Discrete quadratic function is used as objective function in design of controller to guarantee good control efficiency on sampling points. In every step of computation of the deduced controller, it contains not only current step of state feedback but also linear combination of some former steps of control. Because the controller is deduced directly from time-delay differential equation, system stability can be guaranteed easily, thus this method is generally applicable to ordinary control systems. The performance of the control method proposed and system stability when using this method are all demonstrated by numerical simulation results. Simulation results demonstrate that the presented method is a viable and attractive control strategy for applications to active vibration control. Instability in responses occurs possibly if the systems with time delay are controlled using controller designed in case of no time delay.     

A NEW SLIDING MODE CONTROL FOR A CLASS OF UNCERTAIN TIME-DELAY CHAOTIC SYSTEMS

We propose a new sliding mode control scheme for a class of uncertain time-delay chaotic systems. It is shown that a linear time invariant system with the desired system dynamics is used as a reference model for the output of a time-delay chaotic system to track. A sliding mode controller is then designed to drive the output of the time-delay chaotic system to track the desired linear system. On the sliding mode, the output of the controlled time-delay chaotic system can behave like the desired linear system. A simulation example is given in support of the proposed control scheme.     

Characteristics and synchronization of time-delay systems driven by a common noise

We investigate the characteristics of time-delay systems in the presence of Gaussian noise. We show that the delay time embedded in the time series of time-delay system with constant delay cannot be estimated in the presence noise for appropriate values of noise intensity thereby forbidding any possibility of phase space reconstruction. We also demonstrate the existence of complete synchronization between two independent identical time-delay systems driven by a common noise without explicitly establishing any external coupling between them.     

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.     

分数阶Willis环脑迟发性动脉瘤时滞系统混沌分析

在脑血动脉瘤的临床研究中,Willis环脑动脉血管瘤系统(Willis aneurysm system,WAS)起着重要作用,分数阶WAS尽管能进一步加深该系统的机理刻画,但是不能描述原因不明的迟发性动脉瘤.鉴于此,本文提出分数阶Willis环脑迟发性动脉瘤时滞系统(fractional Willis aneurysm system with time-delay,FWASTD)并验证了其有效性;利用时间序列图、相图、Poincaré截面等证实了FWASTD的混沌特性;研究时滞对于系统的重要生理参量的影响,发现了血流阻力系数在时滞状态下对系统稳定的重要性;根据分数阶时滞系统的稳定性理论,设计相应线性控制器,对FWASTD进行了有效控制,同时也探讨了时滞系统的自同步控制.本文完善了脑动脉瘤系统的理论基础.     

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.     

一阶时滞混沌的参数辨识

估计混沌系统的未知参数是混沌控制与同步中必须解决的关键问题.针对两种不同类型的时滞混沌系统中的未知参数的辨识问题，将系统的未知参数看作系统的未知状态，利用非线性状态观测器理论进行参数估计，通过选取观测器中非线性增益函数，使得闭环误差系统指数或渐进稳定，从而给出参数估计器存在的充分条件.以典型的时滞Logistic系统为例进行了数值模拟，数值仿真结果表明，使用该方法可以对时滞混沌系统的未知参数进行有效地估计.     

Impulsive stabilization and synchronization of a class of chaotic delay systems

The problems of control and synchronization of a class of chaotic systems with time delay via the impulsive control approach are investigated. Based on the Lyapunov-like stability theory for impulsive functional differential equations, several sufficient conditions are derived to guarantee chaos control and synchronization. Furthermore, we address the chaos quasisynchronization in the presence of single-parameter mismatch. Several illustrated examples are also given to show the effectiveness of the proposed methods.     

多变量时间序列最大李雅普诺夫指数的计算

根据单变量时间序列计算最大Lyapunov指数的算法思想，本文提出了一种于多变量时间序列最大Lyapunov指数计算的方法.针对原有算法需要使用重构相空间的特点，推广算法给出了多变量时间序列相空间重构参数的选择方法，并采用多变量重构相空间进行最大Lyapunov指数计算.经耦合Rssler系统产生多变量时间序列的仿真计算，验证了该算法的有效性，推广算法的计算结果表明多变量时间序列的计算结果优于单变量的结果，且更加接近理论计算结果. 关键词： 多变量时间序列 相空间重构 Lyapunov指数     

Synchronization of time delayed semiconductor lasers and its applications in digital cryptography

This work aims to demonstrate the effect of synchronization phenomena in chaotic laser systems described by modified Lang-Kobayashi's (L-K) delay differential equations. The synchronized system considered for numerical simulations and for cryptography consists of identical semiconductor lasers operating in single longitudinal mode. The two lasers are bidirectionally coupled by linear optical feedback. As this is essential in simultaneous transmission of messages, we have applied the corresponding coupled chaotic dynamics to secure communications. An investigation of the system together with a novel scheme for digital cryptography and visual recurrence analysis (VRA) of the chaotic time series are presented. Extended statistical tests with the proposed two phase scheme demonstrate the efficiency of these infinite dimensional systems in being tolerant to different types of statistical attacks. The result emphasizes the merits of the uncertainty and high dimensionality of optical chaos system in duplex high speed secure communications.