Lagrangian-based investigation of multiphase flows by finite-timeLyapunov exponents

Multiphase flows are ubiquitous in our daily life and engineering applications. It is important to investigate the flow structures to predict their dynamical behaviors ef- fectively. Lagrangian coherent structures （LCS） defined by the ridges of the finite-time Lyapunov exponent （FTLE） is utilized in this study to elucidate the multiphase interactions in gaseous jets injected into water and time-dependent turbu- lent cavitation under the framework of Navier-Stokes flow computations. For the gaseous jets injected into water, the highlighted phenomena of the jet transportation can be observed by the LCS method, including expansion, bulge, necking/breaking, and back-attack. Besides, the observation of the LCS reveals that the back-attack phenomenon arises from the fact that the injected gas has difficulties to move toward downstream re- gion after the necking/breaking. For the turbulent cavitating flow, the ridge of the FTLE field can form a LCS to capture the front and boundary of the re-entraint jet when the ad- verse pressure gradient is strong enough. It represents a bar- rier between particles trapped inside the circulation region and those moving downstream. The results indicate that the FFLE field has the potential to identify the structures of mul- tiphase flows, and the LCS can capture the interface/barrier or the vortex/circulation region.

New prediction of chaotic time series based on local Lyapunov exponent

A new method of predicting chaotic time series is presented based on a local Lyapunov exponent, by quantitatively measuring the exponential rate of separation or attraction of two infinitely close trajectories in state space. After recon- structing state space from one-dimensional chaotic time series, neighboring multiple-state vectors of the predicting point are selected to deduce the prediction formula by using the definition of the locaI Lyapunov exponent. Numerical simulations are carded out to test its effectiveness and verify its higher precision over two older methods. The effects of the number of referential state vectors and added noise on forecasting accuracy are also studied numerically.

Hurst Exponent Analysis of Financial Time Series

Statistical properties of stock market time series and the implication of their Hurst exponents are discussed. Hurst exponents of DJIA (Dow Jones Industrial Average) components are tested using re scaled range analysis. In addition to the original stock return series, the linear prediction errors of the daily returns are also tested. Numerical results show that the Hurst exponent analysis can provide some information about the statistical properties of the financial time series.

Relationships of exponents in multifractal detrended fluctuation analysis and conventional multifractal analysis

Multifractal detrended fluctuation analysis （MF-DFA） is a relatively new method of multifractal analysis. It is extended from detrended fluctuation analysis （DFA）, which was developed for detecting the long-range correlation and the fractal properties in stationary and non-stationary time series. Although MF-DFA has become a widely used method, some relationships among the exponents established in the original paper seem to be incorrect under the general situation. In this paper, we theoretically and experimentally demonstrate the invalidity of the expression r（q） = qh（q） - 1 stipulating the relationship between the multifractal exponent T（q） and the generalized Hurst exponent h（q）. As a replacement, a general relationship is established on the basis of the universal multifractal formalism for the stationary series as .t-（q） = qh（q） - qH - 1, where H is the nonconservation parameter in the universal multifractal formalism. The singular spectra, a and f（a）, are also derived according to this new relationship.

Robust H∞ Control with Exponent Stability for Time-Delay Uncertain Systems

This paper is concerned with the robust H ∞ control with exponent stability for a class of time delay uncertain systems. Attention is focused on the design of controllers such that the resulting closed loop system not only is exponentially stable but also satisfies, H ∞ disturbance attenuance via memoryless state feedback control. Sufficient conditions for feasibility are obtained in terms of LMIs. Moreover, optimization of LMI is considered such that the controller with low gain parameters is formulated.

Coherent structures over riblets in turbulent boundary layer studied by combining time-resolved particle image velocimetry(TRPIV),proper orthogonal decomposition(POD),and finite-time Lyapunov exponent(FTLE)

Time-resolved particle image velocimetry（TRPIV） experiments are performed to investigate the coherent structure＇s performance of riblets in a turbulent boundary layer（TBL） at a friction Reynolds number of 185. To visualize the energetic large-scale coherent structures（CSs） over a smooth surface and riblets, the proper orthogonal decomposition（POD） and finite-time Lyapunov exponent（FTLE） are used to identify the CSs in the TBL. Spatial-temporal correlation is implemented to obtain the characters and transport properties of typical CSs in the FTLE fields. The results demonstrate that the generic flow structures, such as hairpin-like vortices, are also observed in the boundary layer flow over the riblets, consistent with its smooth counterpart. Low-order POD modes are more sensitive to the riblets in comparison with the high-order ones,and the wall-normal movement of the most energy-containing structures are suppressed over riblets. The spatial correlation analysis of the FTLE fields indicates that the evolution process of the hairpin vortex over riblets are inhibited. An apparent decrease of the convection velocity over riblets is noted, which is believed to reduce the ejection/sweep motions associated with high shear stress from the viscous sublayer. These reductions exhibit inhibition of momentum transfer among the structures near the wall in the TBL flows.

A research on the largest Lyapunov exponent of BAEP time series

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Regular nonlinear response of the driven Duffng oscillator to chaotic time series

Nonlinear response of the driven Duffing oscillator to periodic or quasi-periodic signals has been well studied. In this paper, we investigate the nonlinear response of the driven Duffing oscillator to non-periodic, more specifically, chaotic time series. Through numerical simulations, we find that the driven Duffing oscillator can also show regular nonlinear response to the chaotic time series with different degree of chaos as generated by the same chaotic series generating model, and there exists a relationship between the state of the driven Duffing oscillator and the chaoticity of the input signal of the driven Duffing oscillator. One real-world and two artificial chaotic time series are used to verify the new feature of Duffing oscillator. A potential application of the new feature of Duffing oscillator is also indicated.

Characterization of chaotic and periodic dynamics in a time series

In order to characterize synthesizing periodic and chaotic component dynamics in a time series, classical chaotic system-logistic map and Duffing system was examined by time dependent exponent (TDE) of the direct dynamical test. The simulative calculation results with the logistic map and Duffing system showed that periods and chaotic dynamics can be readily characterized by computing a series of TDE curves. The method was also applied to study power short-term load time series as well as measured a time series. Their dynamic characteristics of chaotic component and period were conveniently found.

Chaotic phenomenon and the maximum predictable time scale of observation series of urban hourly water consumption

The chaotic characteristics and maximum predictable time scale of the observation series of hourly water consumption in Hangzhou were investigated using the advanced algorithm presented here is based on the conventional Wolf's algorithm for the largest Lyapunov exponent. For comparison, the largest Lyapunov exponents of water consumption series with one-hour and 24-hour intervals were calculated respectively. The results indicated that chaotic characteristics obviously exist in the hourly water consumption system; and that observation series with 24-hour interval have longer maximum predictable scale than hourly series. These findings could have significant practical application for better prediction of urban hourly water consumption.

A time fractional model to represent rainfall process

This paper deals with a stochastic representation of the rainfall process. The analysis of a rainfall time series shows that cumulative representation of a rainfall time series can be modeled as a non-Gaussian random walk with a log-normal jump distribution and a time-waiting distribution following a tempered a-stable probability law. Based on the random walk model, a fractional Fokker-Planck equation （FFPE） with tempered a-stable waiting times was obtained. Through the comparison of observed data and simulated results from the random walk model and FFPE model with tempered a-stable waiting times, it can be concluded that the behavior of the rainfall process is globally reproduced, and the FFPE model with tempered a-stable waiting times is more efficient in reproducing the observed behavior.

Quantitative Comparison of Predictabilities of Warm and Cold Events Using the Backward Nonlinear Local Lyapunov Exponent Method

The backward nonlinear local Lyapunov exponent method(BNLLE)is applied to quantify the predictability of warm and cold events in the Lorenz model.Results show that the maximum prediction lead times of warm and cold events present obvious layered structures in phase space.The maximum prediction lead times of each warm(cold)event on individual circles concentric with the distribution of warm(cold)regime events are roughly the same,whereas the maximum prediction lead time of events on other circles are different.Statistical results show that warm events are more predictable than cold events.

Chaos in Time Series of Sakarya River Daily Flow Rate

In this study, possible low dimensional chaotic behavior of Sakarya river flow rates is investigated via nonlinear time series techniques. To reveal the chaotic dynamics, the maximal positive Lyapunov exponent is calculated from the reconstructed phase space, which is obtained using the phase space reconstruction method. The method reconstructs a phase space from the scalar time series, which depicts the real system’s invariants Positive values, because the Lyapunov exponent values calculated using the appropriate software program indicate possibility of chaotic behavior. Analyzed data involve the monthly average flow rates of eleven main branches of Sakarya River through the years 1960-2000.

伪抛物型p-Kirchhoff方程的爆破解

该文研究了一类具有变指数的伪抛物型p-Kirchhoff方程的齐次Dirichlet初边值问题.首先,采用改进的势阱法和辅助函数法,对初始能量进行最优分类,得到解的爆破和整体存在的判据.其次,分别得到了解的爆破时间估计和整体解的大时间性态.

水下航行体边界层转捩流动结构实验研究

采用激光诱导荧光(Laser Induced Fluorescence,LIF)和粒子图像测速(Particle Image Velocimetry,PIV)技术,对自由湍流条件下SUBOFF模型转捩边界层中的流动结构进行精细测量。实验在北京航空航天大学大型低速回流水洞中进行,SUBOFF模型长度为1.436 m,基于模型长度和来流速度的实验雷诺数为3.35×10^(5)。采用流动显示灰度场和有限时间李雅普诺夫指数(Finite-Time Lyapunov Exponents,FTLEs)对转捩边界层中的涡结构进行识别,并对转捩过程中发卡涡、二次涡等典型拟序结构的生成演化过程进行分析。采用两点相关方法提取转捩区拟序结构,同时采用椭圆拟合方法计算相干结构倾角,计算结果表明,结构倾角沿法向先增大后减小,在边界层附近达到最大值。为深入研究转捩流动结构特性,发展了基于流动显示的湍流/非湍流界面(T/NT)识别方法,并对界面几何特性进行了研究。研究结果表明,在转捩过程中,界面的法向高度和分形维数沿程增长。

Two Times Spline Interpolant in Power Exponent Form under the First Boundary Condition

In this paper the concept of first boundary condition （i）（i = 0, 1, 2,…, n） is proposed based on [1], the existence of two times spline interpolant under first boundary condition is proved using constructivity method and the uniqueness of the two times spline interpolant under first boundary condition（n） is proved too.

A SPECTRUM OF LYAPUNOV EXPONENTS OBTAINED FROM A CHAOTIC TIME SERIES

A complete spectrum of Lyapunov exponents (LEs) is obtained from 1970— 1985 daily mean pressure measurements at Shanghai by means of a correlation matrix analysis technique and it is found that there exist LEs≥0, and <0. with their sum <zero (∑λ_1<0), thus showing the evolution of the climate-weather system represented by the series to be chaotic. The sum of positive LE is known to represent the bodily divergence of the system and the sum of these positive LEs is theoretically found to be Kolmogorov entropy of the system. This paper shows that with the time lag τ=5, the parameter m=2 and the dimensionality d_M=9, the sum of the positive LEs sum fromλ_i>0λ_i=K=0.110405 whereupon T=1 /K =9 is obtained as the predictable time scale, a result close to that acquired by the dynamic-statistical approach in early days and also in agreement with that present by the authors themselves(1991).

基于行波时频特征的单端故障测距方法

行波波速度的确定及故障行波波头到达时刻的标定是影响行波测距精度的主要因素。在考虑线路参数频变特性对行波传播特性的影响的基础上,提出基于行波时频特征的单端测距方法:利用行波信号的利普希茨(Lipschitz)指数将行波信号的时频特性联系起来,运用最小二乘法拟合检测到的第2个行波波头的Lipschitz指数,并据此确定故障行波的频率成分及其到达时刻,计算故障距离。EMTDC仿真实验结果表明:所提方法测距精度较传统的行波测距方法有较明显的提高,且不受故障距离、故障电阻、故障初相角和故障类型的影响,能很好地适应完全换位和非完全换位线路。

基于 Lyapunov 指数的电力系统短期负荷预测

采用非线性系统理论对电力系统历史负荷数据序列进行了特征分析，计算出Ｌｙａｐｕｎｏｖ指数，并利用该Ｌｙａ－ｐｕｎｏｖ指数模式进行短期负荷预测，进而提出短期负荷预测的时间尺度的概念。这种方法不利用气候和气温等数据，只利用电力系统一维峰值负荷历史数据计算出过去的变动模式进行负荷预测，就可以得到较高的预测精度。对东北电网实际负荷数据进行了预测，结果令人满意，从而为电力系统短期负荷预测提供了一种新的预测方法。

水文时序趋势与变异点的R/S分析法

简要介绍了分形理论中的R/S分析法 ,给出了水文时间序列变异点的定义 ,研究了基于R/S分析的水文时序趋势分析方法和变异点诊断方法 ,并将该方法应用于华北某流域水文时序变异点的识别中 ,实证结果表明该方法具有实际意义 .