Non-zero Lyapunov exponents and Lyapunov's coefficient of regularity for flows

We obtain sharp lower and upper bounds for the Lyapunov coefficient of regularity defined by the solutions of a non-autonomous linear equation. We also show that the dynamics can always be reduced to one defined by an upper triangular coefficient matrix with the property that the canonical basis is normal. The bounds are then used to show in a simple manner that the coefficient of regularity is related to the notion of non-uniform hyperbolicity, more precisely to the non-uniform part of a non-uniform exponential contraction or dichotomy. As an application, we show that for almost all trajectories of a measure-preserving flow with negative Lyapunov exponents, the non-uniformity can be made arbitrarily small.     

Lyapunov exponents for systems with unbounded coefficients

In this article we propose a generalization of the concept of Lyapunov exponents for discrete linear systems which may be used in the case of unbounded coefficients. We show some simplest properties of this generalization and apply it to define a generalization of regular system. Finally, we discuss the problem of stability by linear approximation.     

基于Lyapunov指数的管道超声导波小缺陷定位实验研究

为了提高超声导波检测长距离管道中小缺陷的检测灵敏度,提出了基于杜芬方程最大Lyapunov指数的超声导波损伤定位方法。依据管道超声导波实测信号的采样频率、中心频率以及杜芬方程特性,设置检测系统参数,将待测信号作为杜芬方程外策动力扰动项输入杜芬方程中。通过比较杜芬系统在无信号输入和输入实测导波信号后,最大Lyapunov指数随策动力幅值F的变化,确定可用于识别导波信号的F值。利用移动窗函数给出了缺陷超声导波的波到时刻识别方法,从而给出了缺陷定位方法。实验研究表明,利用最大Lyapunov指数可有效提高超声导波的检测灵敏度。     

两自由度碰撞振动系统的Lyapunov指数谱分析

摘要 针对一类两自由度碰撞振动系统,取碰撞前瞬时的定相位面为Poincaré截面,引入局部映射,构造Poincaré映射并给出其Jacobi矩阵.利用Gram-Schmidt正交化、范数归一化和迭代的方法,得出两自由度碰振系统Lyapunov指数谱的计算方法.利用数值模拟,讨论系统周期吸引子和混沌吸引子的Lyapunov指数谱的收敛序列,序列的收敛性很好. 为了验证该计算方法的正确性和有效性,分析当系统参数在大范围内变化时,相应的最大Lyapunov指数的变化规律.     

On the Lyapunov exponents of a class of second-order discrete time linear systems with bounded perturbations

In this note we describe the maximal and the minimal values of Lyapunov exponents for second-order discrete time-invariant linear system perturbed by time-varying bounded perturbations. An interpretation of the results in terms of generalized spectral radius is given. An application of obtained formulas to the robust stability problem is demonstrated on a numerical example.     

Lyapunov指数的算法改进与加权预测

对在计算最大Lyapunov指数方面获得广泛应用的Rosenstein方法作出了改进。Rosenstein方法在计算时,只选取一个最近邻点,容易受到噪声的影响。Kantz提出用固定邻域内多点平均的方法来减少噪声的影响,但固定邻域在实际计算时存在困难。针对这个问题,提出了实用的可变邻域的取法。并且提出了将最大Lyapunov指数用于对预测模型进行加权,仿真表明预测效果得到了明显改善。     

Covering relations and Lyapunov condition for topological conjugacy

In this paper, we study stability of non-autonomous discrete dynamical systems. For a two-sided non-autonomous systems with covering relations determined by a transition matrix A, we show that any small C⁰ perturbed system has a sequence of compact invariant sets restricted to which the system is topologically semi-conjugate to σ_A, the two-sided subshift of finite type induced by A. Together with Lyapunov condition of good rate, the semi-conjugacy will become conjugacy. Moreover, if the Lyapunov condition is strict and has perfect rate, then any small C¹ perturbed systems is topological conjugate to σ_A. We also study topological chaos of one-sided systems and systems with limit functions. Lack of hyperbolicity and the time dependence of the rate prevent us from applying classical hyperbolic results or earlier works for autonomous systems: cone condition and Lyapunov function. Two examples are provided to demonstrate the existence of a non-trivial, non-hyperbolic invariant and essential of controlling rate.     

二阶随机系统的Lyapunov指数与稳定性

利用线性变换方法研究了二阶系统在随机扰动下系统的运动稳定性及分叉问题。给出了线性化系统最大Ｌｙａｐｕｎｏｖ指数的计算公式，从而由其最大Ｌｙａｐｕｎｏｖ指数为零可求出线性化系统几乎必然稳定区域的边界。     

广义Lyapunov方程的一个性质及其应用

研究广义Lyapunov方程PA＋A'P ∑^ni=1C'iPCi=-Q在精确能观性的假设下，证明了该方程有正解等价于（A，∑^ni=1,Ci)的稳定性。最后给出了该性质的一个应用。     

Normal forms and nonlocal chaotic behavior in Sprott systems

The Sprott systems are used as benchmarks for investigating the applicability of the normal form transformation in estimating nonlocal properties of attractors such as positive and zero Liapunov exponents. Possibility of a relation between complex conjugate eigenvalue pairs and zero Liapunov exponents; conditions under which the normal form expansion can represent the attractor; an averaging relation for the largest Liapunov exponent based on this representation are studied. Nonlinear transformations that can change the order of a resonance are considered. In spite of their convergence problems, it is seen that the normal form approach can give reasonable estimates of nonlocal properties of attractors near Hopf bifurcations.     

Performance of Multivariate Process Capability Indices Under Normal and Non‐Normal Distributions

In the context of process capability analysis, the results of most processes are dominated by two or even more quality characteristics, so that the assessment of process capability requires that all of them are considered simultaneously. In recent years, many researchers have developed different alternatives of multivariate capability indices using different approaches of construction. In this paper, four of them are compared through the study of their ability to correctly distinguish capable processes from incapable processes under a diversity of simulated scenarios, defining suitable minimum desirable values that allow to decide whether the process meets or does not meet specifications. In this sense, properties analyzed can be seen as sensitivity and specificity, assuming that a measure is sensitive if it can detect the lack of capability when it actually exists and specific if it correctly identifies capable processes. Two indices based on ratios of regions and two based on the principal component analysis have been selected for the study. The scenarios take into account several joint distributions for the quality variables, normal and non‐normal, several numbers of variables, and different levels of correlation between them, covering a wide range of possible situations. The results showed that one of the indices has better properties across most scenarios, leading to right conclusions about the state of capability of processes and making it a recommendable option for its use in real‐world practice. Copyright © 2015 John Wiley & Sons, Ltd.     

液氧贮存输送系统的改造及稳定运行

叙述了液氧贮存输送系统存在的问题,分析原因,并加以改造解决;说明只有正确的设计、安装和合理的动力配置,才能确保液氧贮存输送系统的安全、稳定运行。     

基于变形Lorenz系统和李雅普诺夫指数的微弱周期信号检测

提出了一种变形的Lorenz混沌检测系统,此系统主要具有混沌态和类周期态2种状态。利用最大李雅普诺夫指数作为判断混沌系统由混沌态趋于类周期态变化的量化依据,自动识别混沌系统的临界状态,更准确地判断微弱信号的存在。将互相关检测方法与混沌检测方法相结合,实现强噪声背景下微弱周期信号检测。仿真结果表明,该方法能有效地检测出强噪声中的微弱周期信号。     

基于最大Lyapunov指数和小波变换的管道检测信号降噪技术

针对分布式光纤管道安全预警系统检测信号的混沌特性,最大Lyapunov指数（λm ax）被定义为降噪指标用于评价各种小波及其阈值组合的降噪性能.基于小波阈值降噪法,采用不同的小波族系和阈值选取及重调方法对现场实测人工挖掘信号进行降噪处理.现场实验数据的分析结果表明检测信号存在混沌特征.在此基础上,λm ax被用于评价小波降噪性能.最后,通过对比实验结果,sym2小波、Sqtwolog阈值选取规则和M ln阈值重调方法,可以有效地消除现场实测信号中的噪声,达到最优的降噪效果.     

空分设备长周期稳定运行分析探讨

仪征化纤动力生产中心E#KDONAr-5000/13000/155型空分设备投运两年来以来,故障停车20多次,运行期间产品纯度遭破坏超过10次。分析E#空分设备的生产条件,旨在通过检修、技术改造和操作水平的提高等措施,来保证空分设备长周期稳定运行。     

高寒地区制氧机冬季稳定运行的几个问题及解决办法

论述了制氧机在高寒地区由于设计及设备本身原因出现的一些影响机组稳定运行的问题 ,并针对这些问题提出解决方案 ,确保了制氧机在冬季稳定运行。     

新鲜果蔬常温充气保鲜性能的研究

研究了不同的充氮时间对于不同的新鲜果蔬常温保鲜性能的影响,并与自然气调的保鲜效果进行了比较.結果表明在相同的包装薄膜(尼龙/PET复合袋)内,相同的包装期限,不同的充氮时间,包装梨的袋内积聚的CO2气体浓度变化平缓,即梨常温充氮保鲜效果好.