Chaotic dynamics analysis and control of iterative procedure of capacity spectrum method

The capacity spectrum method (CSM), capable of predicting the demands of forces and deformations of the inelastic system, has been applied in the ATC and FEMA guidelines. The deformation of an inelastic system is solved iteratively by using the equivalent linearization for CSM, which actually forms a nonlinear map or discrete dynamical system. However, the iterative procedure of CSM did not converge for some inelastic systems, and the complicated dynamical phenomena for the solutions such as the periodic oscillation, period-doubling bifurcation and chaos may occur, which were shown in the bifurcation plots of iterative map of the simplified CSM in ATC40 and FEMA440. This paper presents a novel method to analyze and control the non-convergence of the iterative procedure of CSM from the perspective of chaotic dynamics. The Lyapunov exponent of the dynamical system is employed to identify the evolutional state and stability of solutions. Finally, the stability transformation method as a simple, versatile and effective chaos feedback control approach is applied to control the convergent failure of CSM in ATC40 and FEMA440. The numerical results illustrate that the stability transformation method can capture the desired fixed points of the dynamical system and obtain the stable convergent solutions of CSM.     

几何缺陷对拱结构动力稳定性的影响

分析了外激励下几何缺陷对拱结构动力稳定性的影响。推导了拱结构边界确定而结构本身节点坐标偏差随机且指数相关时的条件相关矩阵,分解得到几何缺陷的分布方式和大小。从非线性运动方程出发,分别得出了周期荷载作用下非线性刚度矩阵可线性化,非周期荷载作用下同时考虑几何、材料非线性的Lyapunov指数计算方法。最后以一圆弧拱为例分别对周期荷载、阶跃荷载、脉冲荷载及地震荷载作用下几何缺陷的影响进行了数值分析。结果表明周期激励作用下拱结构存在动力失稳频域;在不同分布方式几何缺陷中动力稳定性对与屈曲模态相似的缺陷最为敏感。     

Experimental test of asymmetrical cable-stayed bridges using MR-damper for vibration control

In this paper, a semi-active control by MR-damper is researched; its purpose is to effectively control vibration of asymmetrical cable-stayed bridges when earthquake is loaded on the type of bridge. For an experimental study, a model of 10.2 m high and 28 m long asymmetrical cable-stayed bridge structure was built being similar to a real one in size and function. A MR damper was also designed in proper size suitable for the control of the model. The experiment was performed in the way in which three piers were fixed on three shaking tables with 30% of El-centro earthquake wave, and a control device was placed on the lower part of its upper deck for horizontal control. As for control algorithms, Lyapunov and Clipped-optimal control algorithms were applied. The effectiveness of the semi-active control with MR damper for the asymmetrical cable-stayed bridge was measured under five control conditions: Un-control, Passive-off, Passive-on, Lyapunov Control, Clipped-optimal Control. The experiment showed that the semi-active control applying Lyapunov and Clipped-optimal algorithms effectively increased controllability almost in double, and decreased displacement 75% compared with the condition of passive-off. Therefore the semi-active method suggested in this paper is proven effective in controlling asymmetrical cable-stayed bridges.     

岩石声发射混沌特征分析

岩石等脆性材料在加载过程中,随着荷载的增加,材料内部的微裂纹产生、扩展并伴随着声发射现象的发生。声发射是研究脆性材料破坏的良好工具。对砂岩、细砂岩和高丽山砂岩3类岩石进行了声发射试验,记录了加载及破坏过程中产生的声发射信号,并且采用混沌动力学理论研究了3类岩石的声发射活动规律,计算了岩石的关联维数和最大Lyapunov指数。研究结果表明,岩石加载及破坏过程具有混沌特征,用相空间重构法可以较好地揭示岩石破坏过程的动力学特征,这为混沌理论在岩石、岩体声发射其他领域的研究及应用奠定了基础。     

Conditional Entropy

In this paper we show that the Conditional Entropy of nearby orbits may be a useful tool to explore the phase space associated to a given Hamiltonian. The arc length parameter along the orbits, instead of the time, is used as a random variable to compute the entropy. In the first part of this work we summarise the main analytical results to support this tool while, in the second part, we present numerical evidence that this technique is able to localise (stable) periodic and quasiperiodic orbits, 'aperiodic' orbits (chaotic motion) and unstable periodic orbits (the 'source' of chaotic motion). Besides, we show that this technique provides a measure of chaos which is similar to that given by the largest Lyapunov Characteristic Number. It is important to remark that this method is very simple to compute and does not require long time integrations, just realistic physical times. This revised version was published online in July 2006 with corrections to the Cover Date.     

Symplectic Tangent map for Planetary Motions

Equations are presented for the computation of tangent maps for use in nearly Keplerian motion, approximated by use of a symplectic leapfrog map. The resulting algorithms constitute more accurate and efficient methods to obtain the Liapunov exponents and the state transition matrix, and can be used to study chaos in planetary motions, as well as in orbit determination procedures from observations. Applications include planetary systems, satellite motions and hierarchical, nearly Keplerian systems in general. This revised version was published online in July 2006 with corrections to the Cover Date.     

一个正压和斜压不稳定的充要条件

提出一个新的李雅普诺夫函数，首次得到了正压和斜压不稳定的充要条件，大大改善了热知的Ｒａｙｌｅｉｇｈ－郭晓岚定理和Ｆｊｏｒｔｏｆｔ定理，解决了Ｒａｙｌｅｉｇｈ（１８８０）提出的无粘切变流正压稳定性问题。     

Lyapunov exponents of a simple stochastic model of the thermally and wind-driven ocean circulation

A reformulation of the simple model of the thermally and wind-driven ocean circulation introduced by Maas [Tellus 46A (1994) 671] is considered. Under a realistic range of forcing parameters, this model displays multiple attractors, corresponding to thermally direct and indirect circulations. The fixed point associated with the thermally direct circulation is unstable for a broad range of parameters, leading to limit cycles and chaotic behaviour. It is demonstrated that if weather variability is parameterised as stochastic perturbations to the mechanical and buoyancy fluxes, then the leading Lyapunov exponent of the circulation can become positive for sufficiently strong fluctuations in parameter ranges where it is deterministically zero. If the fluctuations are sufficiently small that the stochastic trajectories are not too far from the deterministic attractor, it is demonstrated that the sign of the leading Lyapunov exponent can have a substantial effect on the predictability of the system.     

数值天气预报和气候预测可预报性研究的若干动力学方法

简要回顾了数值天气预报和气候预测可预报性研究的若干动力学方法,包括用于研究第一类可预报性问题的线性奇异向量（LSV）和条件非线性最优初始扰动（CNOP-I）方法,以及Lyapunov指数和非线性局部Lyapunov指数方法。前两种方法用于研究预报或预测的预报误差问题,可以用于估计天气预报和气候预测的最大预报误差,而且根据导致最大预报误差的初始误差结构的信息,这两种方法可以用于确定预报或预测的初值敏感区。应该指出的是,LSV是基于线性化模式,对于描述非线性大气和海洋的运动具有局限性。因而,对于非线性模式,应该选择使用CNOP-I估计最大预报误差。Lyapunov指数和非线性局部Lyapunov指数可以用于研究第一类可预报性问题中的预报时限问题,前者是基于线性模式,不能解释非线性对预报时限的影响,而非线性局部Lyapunov指数方法则考虑了非线性的影响,能够较好地估计实际天气和气候的预报时限。第二类可预报性问题的研究方法相对较少,本文仅介绍了由我国科学家提出的关于模式参数扰动的条件非线性最优参数扰动（CNOP-P）方法,该方法可以用于寻找到对预报有最大影响的参数扰动,并可以进一步确定哪些参数最应该利用观测资料进行校准。另一方面,通过对比CNOP-I和CNOP-P对预报误差的影响,可以判断导致预报不确定性的主要误差因子,进而指导人们着力改进模式或者初始场。     

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.