双状态切换下BVP振子的复杂行为分析

非线性切换系统具有广泛的工程背景,而传统的非线性理论不能直接用来解决其中的问题,因而成为当前国内外热点和前沿课题之一. 目前相关工作大都是围绕固定时间或单状态切换开展的,而实际工程系统大都属于多状态切换问题,同时多状态切换涉及到更为丰富的动力学行为. 本文基于两广义BVP 振子,通过引入双向切换开关,构建了双状态切换下的非线性动力学模型,进而研究状态切换导致的各种运动模式及其相应的产生机制. 应用非光滑系统的Poincaré映射理论,推导了双状态切换下的Lyapunov 指数的计算公式,结合子系统的分岔分析,得到了切换系统随分岔参数变化的动力学演化过程及其相应的最大Lyapunov 指数的变化情况. 得到了双状态切换条件下系统存在着各种形式的振荡行为,分析了诸如周期突变等现象及通往混沌的倍周期分岔道路,揭示了不同运动模式的产生机制及倍周期序列的本质. 与固定时间切换和单状态切换系统不同,双临界状态切换系统存在着更为丰富的非线性现象,其主要原因在于双状态切换会产生更多的切换点,且切换点的位置更加多变. 同时切换系统的倍周期分岔序列与光滑系统中的倍周期分岔序列不同,切换系统的倍周期分岔序列只对应于切换点数目的成倍增加,而其相应的周期一般不对应于严格的周期倍化过程.      

混沌振动压实动力学的仿真研究——（Ⅰ）混沌识别

本文提出了四自由度混沌振动压路机“机架-振动轮-土”系统的力学模型;建立了其数学模型;对数学模型进行了数值仿真;根据振动轮的运动,利用混沌识别的定性方法(相轨图、功率谱图和Poincare图)与定量方法(最大Lyapunov指数),对系统的混沌特征进行了识别．结果表明：系统的运动是混沌的．     

非光滑动力系统Lyapunov指数谱的计算方法

对 n 维非光滑(刚性约束和分段光滑)动力系统引进局部映射，利用 Poincaré映射分析方法得出了非光滑系统 Lyapunov 指数谱的通用计算方法．以一类刚性约束的非线性动力系统为例，给出了 Lyapunov 指数谱随参数大范围变化的规律，并与相应的 Poincaré映射分岔图进行对照，验证了上述通用计算方法的正确性和有效性．     

某型发动机试验数据的Lyapunov指数研究

研究某型发动机的八级轴流压气机级间压力信号的最大Lyapunov指数,推断出失速和喘振的起始点具有相同的动力学特性:都为某一极限环边界上的点。提出了一种新的压气机最先失速级检测方法:将最大Lyapunov指数的零点作为失速的判据,最大Lyapunov指数最先达到零点的那一级就是最先失速级。计算结果表明该方法可以在压气机深度失速前0.2s检测到失速起始信号,且最先失速级为第三级。     

基于奇异值分解的航空发动机转子碰摩故障特征提取方法

提出了利用奇异值分解(SVD)提取航空发动机转子碰摩故障特征信号的方法,通过数值仿真得到,奇异值分解方法可以非常有效地将各种不同特征信号完全提取分离,论证了该方法的可行性。然后以某型国产航空发动机在试车台开车过程中某稳定状态的转子振动测量数据为基础,利用奇异值分解的优选差分谱理论对振动信号进行了降噪处理;根据傅里叶变换频谱图中各频率点峰值与奇异值分解差分谱峰值序列相对应的特点,对降噪后振动信号进行了特征提取,实现了转子系统的碰摩故障特征信号的提取。实际结果表明,该方法能够有效地诊断转子系统碰摩故障及提取相应的故障特征信号。     

控制系统中的混沌现象研究

本文介绍了实际控制系统中的混沌现象,并应用相轨迹、功率谱、Lyapunov指数等方法,证明了混沌现象在实际自动控制系统中的存在,并分析了产生混沌现象的原因．     

快速预测混沌控制效果的一个量化指标

本文对Pyragas的时滞自反馈控制方法作了深入详细的研究，并从瞬态Lyapunov特性指数（ILCE）和短期平均Lyapunov特性指数（SLCE）的角度进一步揭示了该方法的控制机理，提出了一个利用瞬态Lyapunov特性指数实时反映混沌控制效果的量化指标P。     

一种快速产生1/fγ分形信号的新方法

分形噪声是光纤陀螺的主要噪声源.为了能快速仿真产生常用谱指数范围的分形信号,提出了一种分数阶差分方法.该方法首先基于分数布朗运动模型,推导出其分数微分形式;接着用分数差分算子替代微分算子将其转换到离散域,那么分形信号就可以表示为高斯白噪声分数阶差分形式.这样只要指定谱指数γ,就可以构造分数差分矩阵,与高斯白噪声序列做卷积,结果就是所求的分形序列.最后与小波变换和FFT组合法生成的分形序列进行仿真性能对比,结果表明了本方法的可行性和优越性,而且分数阶差分本质上是序列的卷积运算,计算简单,能够满足分形信号快速仿真生成需求.     

心率变异的非线性特征参数估算

心率变异（Heart Rate Variability,HRV）是指人的心脏节律的微小变动量，与人的健康状态和精神状态直接相关，具有明显的非线性特征。在本文中，对HRV时间序列的几个非线性特征参数进行估算．从而对心脏健康状态（心率正常）与非健康状态（心率变异）HRV之间的差别进行比较。首先利用小波变换技术对心电信号（ECG）数据进行R波的准确定位，经过重采样得到HRV序列。关联维的计算结果表明，健康状态和非健康状态HRV时间序列具有不同的分形结构，在相空间重构的基础上对HRV进行最大李雅普诺夫指数的估算。结果表明，健康状态和非健康状态HRV时间序列的最大李雅普诺夫指数均为正值，但处于心率不齐状态的节律的混沌程度明显低于健康状态，健康状态HRV的复杂度要高于非健康状态HRV的复杂度，近似嫡和复杂度的分析结果基本相似，健康状态HRV的近似熵要高于非健康状态HRV的近似熵。利用这些非线性特征参数对健康状态和非健康状态的HRV进行比较分析，可以为诊断提供依据，并对深入了解生命规律有潜在价值     

利用Lyapunov指数分析一种非稳态油膜轴承单盘柔性转子的分岔和混沌特性

从一个由三个函数确定的非稳态油膜力模型出发，以短轴承支撑的不平衡弹性转子系统为研究对象，利用Lyapunov指数对该系统进行了一些分岔和混沌的研究。     

A method for calculating the lyapunov exponent spectrum of a periodically excited non-autonomous dynamical system

The relation between the Lyapunov exponent spectrum of a periodically excited non-autonomous dynamical system and the Lyapunov exponent spectrum of the corresponding autonomous system is given and the validity of the relation is verified theoretically and computationally. A direct method for calculating the Lyapunov exponent spectrum of non-autonomous dynamical systems is suggested in this paper, which makes it more convenient to calculate the Lyapunov exponent spectrum of the dynamical system periodically excited. Following the definition of the Lyapunov dimensionD _L ^(A) of the autonomous system, the definition of the Lyapunov dimensionD _L of the non-autonomous dynamical system is also given, and the difference between them is the integer 1, namely,D _L ^(A) −D_L=1. For a quasi-periodically excited dynamical system, similar conclusions are formed. Project supported by the National Natural Science Foundation of China (No. 19772027), the Science Foundation of Shanghai Municipal Commission of Education (99A01) and the Science Foundation of Shanghai Municipal Commission of Science and Technology (No. 98JC14032).     

Chaotic Vibrations of a Cylindrical Shell-Panel with an In-Plane Elastic-Support at Boundary

This paper presents numerical results on chaotic vibrations of a shallow cylindrical shell-panel under harmonic lateral excitation. The shell, with a rectangular boundary, is simply supported for deflection and the shell is constrained elastically in an in-plane direction. Using the Donnell--Mushtari--Vlasov equation, modified with an inertia force, the basic equation is reduced to a nonlinear differential equation of a multiple-degree-of-freedom system by the Galerkin procedure. To estimate regions of the chaos, first, nonlinear responses of steady state vibration are calculated by the harmonic balance method. Next, time progresses of the chaotic response are obtained numerically by the Runge--Kutta--Gill method. The chaos accompanied with a dynamic snap-through of the shell is identified both by the Lyapunov exponent and the Poincaré projection onto the phase space. The Lyapunov dimension is carefully examined by increasing the assumed modes of vibration. The effects of the in-plane elastic constraint on the chaos of the shell are discussed.     

On stability analysis via Lyapunov exponents calculated from a time series using nonlinear mapping—a case study

The concept of Lyapunov exponents has been mainly used for analyzing chaotic systems, where at least one exponent is positive. The methods for calculating Lyapunov exponents based on a time series have been considered not reliable for computing negative and zero exponents, which prohibits their applications to potentially stable systems. It is believed that the local linear mapping leads to inaccurate matrices which prevent them from calculating negative exponents. In this work, the nonlinear approximation of the local neighborhood-to-neighborhood mapping is derived for constructing more accurate matrices. To illustrate the approach, the Lyapunov exponents for a stable balancing control system of a bipedal robot during standing are calculated. The time series is generated by computer simulations. Nonlinear mapping is constructed for calculating the whole spectrum of Lyapunov exponents. It is shown that, as compared with those from the linear mapping, (1) the accuracy of the negative exponents calculated using the nonlinear mapping is significantly improved; (2) their sensitivity to the time lag and the evolution time is significantly reduced; and (3) no spurious Lyapunov exponent is generated if the dimension of the state space is known. Thus, the work can contribute significantly to stability analysis of robotic control systems. Issues on extending the concept of Lyapunov exponents to analyzing stable systems are also addressed.     

Principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation

The principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation is studied by using the method of multiple scales and numerical simulations. The first-order approximations of the solution, together with the modulation equations of both amplitude and phase, are derived. The effects of the frequency detuning, the deterministic amplitude, the intensity of the random excitation and the time delay on the dynamical behaviors, such as stability and bifurcation, are studied through the largest Lyapunov exponent. Moreover, the appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. It is found that the appropriate choice of the time delay can broaden the stable region of the trivial steady-state solution and enhance the control performance. The theoretical results are well verified through numerical simulations.     

Moment Lyapunov exponent for three-dimensional system under real noise excitation

The pth moment Lyapunov exponent of a two-codimension bifurcation system excited parametrically by a real noise is investigated. By a linear stochastic transformation, the differential operator of the system is obtained. In order to evaluate the asymptotic expansion of the moment Lyapunov exponent, via a perturbation method, a ralevant eigenvalue problem is obtained. The eigenvalue problem is then solved by a Fourier cosine series expansion, and an infinite matrix is thus obtained, whose leading eigenvalue is the second-order of the asymptotic expansion of the moment Lyapunov exponent. Finally, the convergence of procedure is numerically illustrated, and the effects of the system and the noise parameters on the moment Lyapunov exponent are discussed.     

宽带噪声作用下黏弹性板的矩Lyapunov指数

主要研究了在超音速流中受宽带噪声作用的黏弹性板随机振动系统的矩Lyapunov指数.首先, 采用vonKarman板弯理论, 活塞理论以及Galerkin近似法建立了两个自由度耦合的系统运动的随机微分方程. 其次, 应用随机平均法将四维系统降为二维系统. 接着, 对系统采用极坐标变换,通过Girsanov定理和Feynmann-Kac公式得到后向微分算子. 通过对特征函数进行正交Fourier余弦级数展开得到系统矩Lyapunov指数的近似解析式. 并通过MonteCarle仿真得到系统矩Lyapunov指数的数值解验证了近似解析式的可信性. 最后研究了系统参数、气动力参数以及随机噪声谱密度对黏弹性板稳定性的影响.      

The matric algorithm of Lyapunov exponent for the experimental data obtained in dynamic analysis

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Invariant Measures and Lyapunov Exponents for Stochastic Mathieu System

The principal resonance of the stochastic Mathieu oscillator to randomparametric excitation is investigated. The method of multiple scales isused to determine the equations of modulation of amplitude and phase.The behavior, stability and bifurcation of steady state response arestudied by means of qualitative analyses. The effects of damping,detuning, bandwidth, and magnitudes of random excitation are analyzed.The explicit asymptotic formulas for the maximum Lyapunov exponent areobtained. The almost-sure stability or instability of the stochasticMathieu system depends on the sign of the maximum Lyapunov exponent.     

Lyapunov exponent and stochastic stability of quasi-non-integrable Hamiltonian systems

An n degree-of-freedom (DOF) non-integrable Hamiltonian system subject to light damping and weak stochastic excitation is called quasi-non-integrable Hamiltonian system. In the present paper, the stochastic averaging of quasi-non-integrable Hamiltonian systems is briefly reviewed. A new norm in terms of the square root of Hamiltonian is introduced in the definitions of stochastic stability and Lyapunov exponent and the formulas for the Lyapunov exponent are derived from the averaged Itô equations of the Hamiltonian and of the square root of Hamiltonian. It is inferred that the Lyapunov exponent so obtained is the first approximation of the largest Lyapunov exponent of the original quasi-non-integrable Hamiltonian systems and the necessary and sufficient condition for the asymptotic stability with probability one of the trivial solution of the original systems can be obtained approximately by letting the Lyapunov exponent to be negative. This inference is confirmed by comparing the stability conditions obtained from negative Lyapunov exponent and by examining the sample behaviors of averaged Hamiltonian or the square root of averaged Hamiltonian at trivial boundary for two examples. It is also verified by the largest Lyapunov exponent obtained using small noise expansion for the second example.