Chen’s混沌吸引子及其特征量

计算了Chen’s混沌吸引子的Lyapunov指数, 关联维数, Kolmogorov熵等混沌特征量. 采用两种不同的计算方法得到了一致的Lyapunov指数. 这从数值实验上有力地证实了Chen’s吸引子是一个与Lorenz系统和Rossler系统均不拓扑等价的新的混沌吸引子.     

受控Rabinovich系统的超混沌系统

在三维Rabinovich系统的基础上,通过引入一个线性状态反馈控制器构建一个新的四维超混沌系统,分析其基本动力学行为.在保证系统有界的前提下,通过计算Lyapunov指数值和研究其分岔的途径,证实其超混沌的特性.还给出了四维超混沌系统的指数吸引域和正向不变集等.同时也设计了实现四维Rabinovich超混沌吸引子的实际电路,验证了理论分析的结果.     

只含一个非线性项的超混沌系统及其电路实现

构造出了一个新的只含有一个非线性项的四维非线性动力学系统,研究了该系统的超混沌吸引子、Lyapunov指数、庞加莱映射图等复杂动力学特征,然后以Multisim电路仿真软件为平台进行了仿真实验,验证了超混沌系统的混沌现象.     

一个新超混沌系统及其线性反馈控制

为提高混沌吸引子的拓扑结构的复杂性,构造了一个新的四维超混沌系统.用数值模拟的方法研究了该系统的超混沌吸引子的相图、系统的分岔图、Lyapunov指数谱图和Lyapunov维数等.分析结果表明,新的四维系统当参数满足一定条件时,具有两个正的Lyapunov指数,是一个超混沌系统,系统的分岔图与Lyapunov指数谱是完全吻合的,随着参数变化呈现周期、混沌及超混沌动力学行为.利用线性反馈控制法镇定了超混沌系统的不稳定平衡点,数值模拟结果表明该方法的可行性和有效性.     

一个三维非线性系统的混沌运动及其控制

通过数值计算、理论推导分析了一个三维类Lorenz混沌系统的基本动力学特性, 并通过数值仿真、相图、Poincare截面图和功率谱研究了这个系统的混沌行为. 然后, 构建一个受控系统并利用Lyapunov指数谱、分叉图分析了该系统混沌吸引子的形成机制, 通过对控制参数的改变, 系统的混沌运动可以得到有效控制.     

新混沌系统及其特性分析

提出了一个新的不同于Lorenz系统和Chen系统的三维自治混沌系统。这个混沌系统的奇怪吸引子与Lorenz、Chen系统、Ln系统以及Liu系统不同,存在六个平衡点,其拓扑结构与它们的拓扑结构也完全不同。该系统含有五个参数,其中三个方程中均含有非线性顷。分析了该新系统的混沌吸引子相图、平衡点及其性质、Lyapunov指数和分形维数等非线性动力学特性。     

超混沌Lorenz系统的电路实现与应用

在经典三维Lorenz系统的基础上,增加一个非线性控制器,构造了一个新的四维超混沌Lorenz系统。通过数值计算,模拟分析了新系统的分岔图,Lyapunov指数随控制参数的变化,超混沌吸引子的相图,求出系统的Lyapunov指数及其吸引子分形维数。结果显示,通过改变新引入的非线性控制器的控制参数,可以使超混沌Lorenz系统分别呈现超混沌、混沌以及周期、拟周期等动力学行为。根据新Lorenz系统的状态方程,设计了与之相对应的实验电路,并在示波器中观察到电路系统的动力学行为,该结果与数值仿真结果基本吻合。将系统应用于图像加密,模拟实验结果表明,该系统能产生具有良好密码学特性的伪随机序列。     

超混沌系统自适应广义投影同步及电路仿真

通过引入反馈控制,将三维Lü系统扩展为四维,构造了一个新的超混沌系统.分析了该系统的平衡点稳定性、超混沌吸引子、Lyapunov指数、系统功率谱等特性.基于自适应线性反馈控制方法,运用恰当的控制函数和控制参数实现了广义投影同步,并通过Multisim电路仿真软件设计构造了一种实现超混沌的电路,验证了该超混沌系统复杂动力学、同步特性以及电路实现的可行性.     

具有共存吸引子的混沌系统及其分数阶系统的镇定

研究一个具有共存吸引子的混沌系统及对应分数阶系统的镇定问题.提出了一个新的具有双翼与四翼吸引子共存的混沌系统,利用Lyapunov指数谱和分岔图对系统的性质进行了分析.借助于拓扑马蹄理论和数值计算,找到了系统的拓扑马蹄,并获得了拓扑熵.构造了相应的分数阶混沌系统,此系统亦存在两个孤立的双翼吸引子以及四翼吸引子且共存的双翼吸引子之间没有重叠.设计了线性反馈标量控制器,此控制器用于分数阶混沌系统的镇定.在控制过程中并未删除系统的非线性项,理论分析与仿真计算表明了该方法的有效性.     

新混沌系统的全局指数吸引集及其数值模拟

混沌系统的全局指数吸引集在混沌的控制和同步之中起着非常重要的作用。给出了一个三维混沌系统的动力系统模型,借助一个适当的Lyapunov函数和最优化理论,研究了这个新混沌系统的全局指数吸引集,得到了它的全局指数吸引集估计。通过计算机模拟,数值模拟验证了计算理论的可行性。     

一种Lyapunov指数算法及其实现

提出了一种利用周期轨道不同权重计算Lyapunov指数的算法。对混沌序列的周期轨道进行统计，并计算不同的周期轨道的Lyapunov指数，依据周期轨道的权重加权求和得到整个混沌吸引子的平均Lyapunov指数。深入讨论了初始值等对平均Lyapunov指数的影响。该算法不用舍去开始迭代点，适用于复杂混沌系统。     

A chaotic limit cycle paradox

Duffing's equation with sinusoidal forcing produces chaos for certain combinations of the forcing amplitude and frequency. To determine the most chaotic response achieveable for given bounds on the input force, an optimal control problem was investigated to maximize the largest Lyapunov exponent, which in this case also corresponds to maximizing the Kaplan-Yorke Lyapunov fractal dimension. The resulting bang-bang optimal feedback controller yielded a bounded attractor with a positive largest Lyapunov exponent and a fractional Lyapunov dimension, indicating a chaotic strange attractor. Indeed, the largest Lyapunov exponent was approximately twice as large as that achieved with sinusoidal forcing at the same amplitude. However, the resulting attractor is just a stable limit cycle and is not chaotic or fractal at all! this contradicts the basic idea that a bounded attractor with at least one positive Lyapunov exponent must be chaotic and fractal.This article provides details of this chaotic limit cycle paradox and the resolution of the paradox. In particular for systems of differential equations with only piecewise differentiable right-hand sides, a jump discontinuity condition must be imposed on the state perturbations in order to compute correct Lyapunov exponents.Presently at Department of Mechanical Design, College of Engineering, Kei Myung University, Dal Seo Gu, Shin Dang Dong 100, Taegu 704-701, Korea.Webster's Dictionary defines paradox as contrary to expectation; a tenet contrary to received opinion; a statement that is seemingly contradictory and yet is perhaps true.     

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.     

一种带绝对值项系统的分岔、激变与混沌

研究了一种含有绝对值项的三维微分动力系统,用李雅普诺夫方法得到了系统发生第一次Hopf分岔的条件.利用相轨迹图、分岔图、最大李雅普诺夫指数谱等非线性动力学分析方法,分析了该系统从规则运动转化到混沌运动的规律.该系统是按照Feigenbaum途径(倍周期分岔)通向混沌的,在混沌区域存在周期窗口.当参数达到激变临界点时,混沌吸引子和不稳周期轨道在吸引子边界上碰撞,发生边界激变,激变临界值的领域内还存在相对长时间的瞬态混沌过程.     

一个新的多翅膀混沌系统分析与同步

提出了一个新的三维自治混沌系统,通过分析系统的李亚普诺夫指数谱和分岔图可得新系统具有如下性质： 双恒李亚普诺夫指数谱混沌锁定,锁定后的混沌系统的幅值和相位可调节．根据该混沌系统的平衡点和吸引子的拓扑结构,通过构造偶对称多分段平方函数族,可实现在某一方向上扩展指标2的鞍焦平衡点,从而实现多翅膀的扩展．设计了混沌电路实现,验证了电路实现与仿真结果的一致性．最后针对扩展后的多翅膀混沌系统,通过选取合适的驱动信号,达到响应系统与驱动系统混沌同步,并通过仿真验证了所得的结果．     

Non-linear analysis of EEG signals at various sleep stages

Application of non-linear dynamics methods to the physiological sciences demonstrated that non-linear models are useful for understanding complex physiological phenomena such as abrupt transitions and chaotic behavior. Sleep stages and sustained fluctuations of autonomic functions such as temperature, blood pressure, electroencephalogram (EEG), etc., can be described as a chaotic process. The EEG signals are highly subjective and the information about the various states may appear at random in the time scale. Therefore, EEG signal parameters, extracted and analyzed using computers, are highly useful in diagnostics. The sleep data analysis is carried out using non-linear parameters: correlation dimension, fractal dimension, largest Lyapunov entropy, approximate entropy, Hurst exponent, phase space plot and recurrence plots. These non-linear parameters quantify the cortical function at different sleep stages and the results are tabulated.     

Chaotic oscillations of a shallow cylindrical shell with a concentrated mass under periodic excitation

This paper presents effects of a concentrated mass on chaotic oscillations of a shallow cylindrical shell under gravity and periodic acceleration. The rectangular shell is simply supported and is elastically in-plane constrained. Assuming mode functions, the Donnell equation with inertia force is reduced to non-linear coupled differential equations by the Galerkin method. The chaotic response is calculated numerically and is examined by the maximum Lyapunov exponent. Dominant chaotic responses are generated within restricted frequency regions of sub-harmonic resonance of 1/2 order. As the concentrated mass increases, the chaotic response is shifted to the lower frequency region. The increment of the concentrated mass decreases the maximum Lyapunov exponent.     

计算球面对称动力系统Lyapunov指数

通过跟踪球面上相距很近的两个点的球面轨道并计算球面点之间的球内弦长,提出计算球面动力系统轨道的平均Lyapunov指数的计算公式.采用该方法,实现了随机搜索参数并自动计算相应动力系统的Lyapunov指数.当Lyapunov指数大于0时,可得到一个构造球面混沌吸引子的动力系统;当Lyapunov指数小于0时,可得到一个构造球面充满Julia集的动力系统.文中提出的方法可用于随机搜索参数进而生成球面上的对称混沌吸引子和充满Julia集图形.