离散动力系统无退化-配置N个正Lyapunov指数

针对离散时间混沌动力学系统,该文提出一种基于矩阵特征值以及特征向量配置Lyapunov指数为正的新算法。计算离散受控矩阵的特征值以及特征向量,设计一类具有正Lyapunov指数的通用控制器,理论证明系统轨道的有界性和Lyapunov指数的有限性。对线性反馈算子以及微扰反馈算子进行数值仿真分析,验证了算法的正确性、通用性和有效性。性能评估表明,与Chen-Lai算法相比,该方法可以构建较低计算复杂度的混沌系统,并且运行时间较短,其输出序列也具有较强的随机性,实现了无退化、无兼并的离散混沌系统。     

简并光学参量振荡器的同步研究

通过计算简并光学参量振荡器系统的李雅普诺夫指数,证明了该系统存在混沌和超混沌吸引子,即简并光学参量振荡器是混沌系统,并得到了系统处于混沌状态时的参数取值范围.进一步,利用反馈技术实现了简并光学参量振荡器的混沌同步和周期同步,给出了反馈控制的具体形式并进行了仿真计算,理论结果与数值仿真结果一致.     

条件Lyapunov指数和时间τ-条件Lyapunov指数的研究

本文研究条件Ｌｙａｐｕｎｏｖ指数与τ指数的定义、求解技术及其应用。两种指数从不同角度对系统本质特性进行刻划,条件Ｌｙａｐｕｎｏｖ指数在混沌同步中有重要应用,近来它还衩用来进行相空间重构问题的研究。时间τ－条件Ｌｙａｐｕｎｏｖ指数是一类利用状态变量的离散采样作驱动信号的脉冲方式的重要定量指标。本文提出一种简便的求解技术,在Ｗｏｌｆ求解Ｌｙａｐｕｎｏｖ指数谱程序的基础上,稍加改动即可使其适用于Ｌｙａｐ     

一类混沌电路的跃变参数识别与同步*

基于李亚普诺夫稳定性理论和矩阵理论,用两种方法对一类混沌电路系统参数发生跃变情况下的参数识别与同步控制进行了理论分析和数值计算机模拟。第一种方法是通过负反馈将系统镇定到某个稳定态来识别系统的跃变参数（系统参数突然发生阶跃性变化）,通过计算李亚普诺夫指数获得反馈系数临界值。第二种方法是基于李亚普诺夫稳定性理论得到的参数观测器包含了可调节的增益系数,当两个混沌系统达到完全同步时驱动系统的5个未知参数在阶跃变化情况下也可以被准确识别。最后对两种方法的优缺点进行了比较和分析。     

Three-Dimensional Chaotic Autonomous System with a Circular Equilibrium: Analysis,Circuit Implementation and Its Fractional-Order Form

A three-dimensional autonomous chaotic system with a circular equilibrium is investigated in this paper. Some dynamical properties and behaviors of this system are described in terms of equilibria, eigenvalue structures, bifurcation diagrams, Lyapunov exponents, time series and phase portraits. For specific parameters, the system displays periodic and chaotic attractors. The physical existence of the chaotic behavior found in the proposed system is verified by using the Orcad-PSpice software and experimental verification. A good qualitative agreement is shown between the experimental results, PSpice and numerical simulations. Furthermore, the commensurate fractional-order version of the system with a circular equilibrium is numerically studied. It is found that chaos exists in this system with order less than three. By tuning the commensurate fractional order, the system with a circular equilibrium displays chaotic and periodic attractors, respectively. Finally, chaos synchronization of identical fractional-order chaotic systems with a circular equilibrium is achieved by using the unidirectional linear error feedback coupling. It is shown that the fractional-order chaotic system can achieve synchronization for appropriate coupling strength.     

基于反正切的网格混沌吸引子及其保密通信

提出了一种用处处光滑的反正切函数序列生成多维多涡卷网格混沌吸引子的方法,可以生成一维n涡卷、二维n×m网格多涡卷和三维n×m×l网格多涡卷混沌吸引子。平衡点分析、数值仿真、Lyapunov指数谱、分岔图和Poincaré映像都表明系统是混沌的。用简单的线性反馈控制实现了同结构网格多涡卷混沌系统之间的同步,可应用于保密通信。简单的理论分析和数值仿真证明了该方法的有效性。     

Chaos: A tutorial for engineers

This tutorial presents an in-depth introduction to chaos in dynamical systems, and presents several practical techniques for recognizing and classifying chaotic behavior. These techniques include the poincaré map, Lyapunov exponents, capacity, information dimension, correlation dimension, Lyapunov dimension, and the reconstruction of attractors from a single time series.     

Robust time delay estimation of bioelectric signals using least absolute deviation neural network

The time delay estimation (TDE) is an important issue in modern signal processing and it has found extensive applications in the spatial propagation feature extraction of biomedical signals as well. Due to the extreme complexity and variability of the underlying systems, biomedical signals are usually nonstationary, unstable and even chaotic. Furthermore, due to the limitations of the measurement environments, biomedical signals are often noise-contaminated. Therefore, the TDE of biomedical signals is a challenging issue. A new TDE algorithm based on the least absolute deviation neural network (LADNN) and its application experiments are presented in this paper. The LADNN is the neural implementation of the least absolute deviation (LAD) optimization model, also called unconstrained minimum L1-norm model, with a theoretically proven global convergence. In the proposed LADNN-based TDE algorithm, a given signal is modeled using the moving average (MA) model. The MA parameters are estimated by using the LADNN and the time delay corresponds to the time index at which the MA coefficients have a peak. Due to the excellent features of L1-norm model superior to Lp-norm (p > 1) models in non-Gaussian noise environments or even in chaos, especially for signals that contain sharp transitions (such as biomedical signals with spiky series or motion artifacts) or chaotic dynamic processes, the LADNN-based TDE is more robust than the existing TDE algorithms based on wavelet-domain correlation and those based on higher-order spectra (HOS). Unlike these conventional methods, especially the current state-of-the-art HOS-based TDE, the LADNN-based method is free of the assumption that the signal is non-Gaussian and the noises are Gaussian and, thus, it is more applicable in real situations. Simulation experiments under three different noise environments, Gaussian, non-Gaussian and chaotic, are conducted to compare the proposed TDE method with the existing HOS-based method. Real application experiment is conducted to extract time delay information between every two adjacent channels of gastric myoelectrical activity (GMA) to assess the spatial propagation characteristics of GMA during different phases of the migrating myoelectrical complex (MMC).     

Characterizing chaos through Lyapunov metrics

Science, engineering, medicine, biology, and many other areas deal with signals acquired in the form of time series from different dynamical systems for the purpose of analysis, diagnosis, and control of the systems. The signals are often mixed with noise. Separating the noise from the signal may be very difficult if both the signal and the noise are broadband. The problem becomes inherently difficult when the signal is chaotic because its power spectrum is indistinguishable from broadband noise. This paper describes how to measure and analyze chaos using Lyapunov metrics. The principle of characterizing strange attractors by the divergence and folding of trajectories is studied. A practical approach to evaluating the largest local and global Lyapunov exponents by rescaling and renormalization leads to calculating the m Lyapunov exponents for m-dimensional strange attractors either modeled explicitly (analytically) or reconstructed from experimental time-series data. Several practical algorithms for calculating Lyapunov exponents are summarized. Extensions of the Lyapunov exponent approach to studying chaos are also described briefly as they are capable of dealing with the multiscale nature of chaotic signals. The extensions include the Lyapunov fractal dimension, the Kolmogorov--Sinai and Re/spl acute/nyi entropies, as well as the Re/spl acute/nyi fractal dimension spectrum and the Mandelbrot fractal singularity spectrum.     

用李雅谱诺夫指数配置法控制统一混沌系统

解析设计控制器，通过配置相应的李雅谱诺夫指数，使统一混沌系统趋于预期点．由于受控后系统李雅谱诺夫指数具有先知性，因此，我们可以根据需要改变李雅谱诺夫指数的大小来控制系统收敛速度．设计还表明，受控的统一混沌系统的收敛情况与参数无关。     

超声键合换能系统中劈刀的振动特性分析

阐述了Lyapunov指数作为振动信号的混沌判断原理,并对换能系统的振动信号进行分析与研究,计算了换能系统在不同加载压力下劈刀的振动时间序列的Lyapunov指数.结果表明振动信号中的最大Lyapunov指数均大于0,根据混沌理论可以判定换能系统振动信号中存在混沌现象,应该采用混沌的方法研究与分析换能系统振动信号.通过不同加载压力下的振动速度信号的RMS值与最大Lyapunov指数的比较,发现两者随压力变化趋势相同,为进一步研究超声引线键合换能系统的振动特性提供了有价值的参考.     

Chaos Synchronization on Parameters Adaptive Control for Chen Chaotic System

1　IntroductionThesynchronizationoftwononlineardynamicalsystemsisaphenomenonofimportanceinawiderangofappliedsciences,andhasbeenpaidmuchat tentiontoandinvestigatedinthe pasttwentyyears[1～ 1 3] .Recentlysecurecommunicationusingchaossynchronizationhasbeenpaidmuchattentiontoasoneoftheimportantpracticalapplicationsofchaos[1 4 ] .Themainideaisthatwhenaninforma tionsignalismaskedbythechaoticsignal,itcanbeeasilyrecoveredatthereceiverbymeansofchaossynchronization ,whileexternalattackisalmostim possibl…     

一种新的混沌神经网络及其应用

提出了一种新的混沌神经元模型,该神经元的激励函数由Gauss和Sigmoid两种函数加和组成的非单调函数构成.通过分岔图及Lyapunov指数的计算,分析了其动力学特性.基于该模型,构造了一种新的具有暂态混沌特性的神经网络,该网络在寻优过程中同时衰减两种退火机制实现混沌退火.通过将特征点匹配问题转化为优化问题,该网络可以实现对目标识别问题的求解.仿真实验验证了该算法的有用性和有效性.     

Simple 4D chaotic oscillator

An extremely simple 4D chaotic oscillator is presented. It contains a single op amp, two LC circuits and a diode used as a nonlinear device. The chaotic oscillations have been characterised using the correlation dimension of the strange attractor, the Lyapunov exponents and the Lyapunov dimension. The dimensions are found to be >3     

Generating Multi-Scroll Chaotic Attractors via?Switched Fractional Systems

A method is presented to generate multi-scroll chaotic attractors by switched fractional systems. Two sub-models in this structure are constructed by different piecewise-linear functions. It is proved that the switched systems can demonstrate multi-scroll chaotic attractors with suitable parameters and switching rule. Moreover, in order to obtain the multi-scroll chaotic attractors with the lowest system order, the relationship between Lyapunov exponents and parameters is studied. It is found that the multi-scroll chaotic attractors can be presented with a system order larger than 0.9. Numerical examples are provided to illustrate the effectiveness of the proposed scheme.     

A comparison of five methods for studying a hyperchaotic circuit

A hyperchaotic oscillator based on an LC oscillator and the Deliyannis single amplifier biquad (SAB), coupled by means of a diode, is studied. The inductor of the resonant circuit is realized in practice by using a generalized impedance converter (GIC). Various methods for detecting chaos, such as the Lyapunov exponents, the Lyapunov dimension, the Poincaré map, the spectrum and the phase portraits, are used, in order to confirm the chaotic behaviour of the chaotic circuit. Experimental results fully coincide with theoretical and simulation results. The paper also attempts to answer the question ‘which of the above methods is the most preferable, and why?'. It is concluded that the Lyapunov exponents, Lyapunov dimension and Poincaré map constitute the more reliable methods of confirming chaos.     

Chaos from second-order PLL in presence of CW interference

The chaotic behaviour of a second-order phase-locked loop (PLL) in the presence of continuous wave (CW) interference is examined. The Lyapunov exponents and dimension of the system are calculated to confirm the chaotic phenomenon. A range of interfering signal power and frequency has been found where chaos could occur     

Chaotic electronic oscillator from single amplifier biquad

The present letter reports a simple chaotic electronic oscillator. A single amplifier biquad (SAB) based active high-Q Band Pass Filter (BPF) is converted into a chaotic oscillator by introducing a single passive nonlinear element in the form of a general purpose pn junction diode, and a storage element in the form of an inductor. The chaotic circuit is mathematically modeled, which is a set of four coupled first-order autonomous nonlinear differential equations. The behavior of the proposed circuit is investigated through numerical simulations and electronic hardware experiments. It is found that the circuit shows complex behaviors, like, bifurcations and chaos, for a certain range of circuit parameters. The chaotic behavior of the circuit is ensured qualitatively by bifurcation diagram, phase plane plot and experimentally obtained power spectrum, and quantitatively by Lyapunov exponents and Kaplan–York dimensions.     

一个具有无限多吸引子共存的新混沌系统的动力学分析及其电路实现

提出了一个新的具有超多稳定性的三维连续自治混沌系统,该系统仅有2个非线性项.对系统的耗散性、平衡点的稳定性进行了定量分析,并利用分岔图、Lyapunov指数、彭加莱截面和吸引子相图分析了系统参数以及初值对其动力学行为的影响.在参数固定的情况下,分析初值变化的分岔图,得到了无限多种共存吸引子.通过采用模拟开关电子元件设计实现了该系统,同时运用Multisim软件仿真了该系统的混沌电路.结论证明电路仿真与数值仿真结果一致.     

Integrated Circuits of Map Chaos Generators

A chaotic noise is one of the most important implements for information processing such as neural networks. It has been suggested that chaotic neural networks have high performance ability for information processing. In this paper, we report two designs of a compact chaotic noise generator for large integration circuits using CMOS technology. The chaotic noise is generated using map chaos. We design both of the logistic map type and the tent map type circuits. These chaotic noise generators are compact as compared with the other circuits. The results show that the successful chaotic operations of the circuits because of the positive Lyapunov number. We calculate the Lyapunov exponents to certify the results of the chaotic operations. However, it is hard to estimate its accurate number for noisy data using the conventional method. And hence, we propose the modified calculation of the Lyapunov exponent for noisy data. These two circuits are expected to be utilized for various applications.