变质量非线性非完整系统相对运动动力学方程的积分方法

本文给出积分变质量非线性非完整系统相对于非惯性系动力学方程的梯度法,单分量法和场方法。首先,将这类问题的动力学方程表示为正则形式和场方程形式;然后,分别用梯度法,单分量法和场方法积分相应常质量完整系统相对于惯性系的动力学方程,并加上非完整约束对初始条件的限制而得到变质量非线性非完整系统相对于非惯性系动力学方程的解。     

Optimal control with connected initial and terminal conditions

An optimal control problem with linear dynamics is considered on a fixed time interval. The ends of the interval correspond to terminal spaces, and a finite-dimensional optimization problem is formulated on the Cartesian product of these spaces. Two components of the solution of this problem define the initial and terminal conditions for the controlled dynamics. The dynamics in the optimal control problem is treated as an equality constraint. The controls are assumed to be bounded in the norm of L₂. A saddle-point method is proposed to solve the problem. The method is based on finding saddle points of the Lagrangian. The weak convergence of the method in controls and its strong convergence in state trajectories, dual trajectories, and terminal variables are proved.     

专利申请与授权量的时间序列分析

专利的申请和授权量动态分析与预测是建立专利预警机制、设计与制定相关政策和战略的基础,具有重要的理论意义。起伏型时间序列法是一种新的时间序列分析法,提出用起伏型时间序列法对专利申请与授权数据进行动态分析。以2005年国内专利申请与授权数量月动态为研究数据,对专利申请、授权及发明专利申请数量的月动态进行建模模拟,结果令人满意,说明起伏型时间序列分析方法可应用于专利申请与授权动态模拟,从而丰富了专利申请与授权数据动态分析方法。     

Coupled problems in transient fluid and structural dynamics in nuclear engineering: Part 1: Safety problems solved by flow singularity methods

Some important problems in coupled fluid-structural dynamics which occur in safety investigations of liquid metal fast breeder reactors (LMFBR), light water reactors and nuclear reprocessing plants are discussed and a classification of solution methods is introduced. A distinction is made between the step by step solution procedure, where available computer codes in fluid and structural dynamics are coupled, and advanced simultaneous solution methods, where the coupling is carried out at the level of the fundamental equations. Results presented include the transient deformation of a two-row pin bundle surrounded by an infinite fluid field, vapour explosions in a fluid container and containment distortions due to bubble collapse in the pressure suppression system of a boiling water reactor. A recently developed simultaneous solution method is presented in detail. Here the fluid dynamics (inviscid, incompressible fluid) is described by a singularity method which reduces the three-dimensional fluid dynamics problem to a two-dimensional formulation. In this way the three-dimensional fluid dynamics as well as the structural (shell) dynamics can be described essentially by common unknowns at the fluid-structural interface. The resulting equations for the coupled fluid-structural dynamics are analogous to the equations of motion of the structural dynamics alone.     

An optimal data assimilation method and its application to the numerical simulation of the ocean dynamics

An original data assimilation (DA) scheme with a general dynamics model is considered. It is shown that this scheme can be approximated by the stochastic diffusion process. The sufficient conditions to provide this approximation are formulated. Based on this algorithm a new DA method is developed. The method combines variational and statistical approaches commonly used in DA theory and minimizes the variance of the trajectory of a diffusion process in conjunction with a dynamics numerical model. In this sense the method is optimal in contrast to other DA approaches. The proposed scheme takes the model dynamics into account and in this way it differs from the well-known Kalman filter. Furthermore, the derived DA method can be applied to a very wide field of dynamical systems, for example, gas dynamics, fluid dynamics and other disciplines. However, the current study deals with oceanography and DA in oceanography specifically. Then the method is applied to the HYbrid Coordinate Ocean Model and assimilates satellite sea level anomaly data from the Archiving, Validating and Interpolating Satellite Oceanography Data over the Atlantic Ocean to correct the model state. Several numerical experiments have been performed. The experiments show that the method substantially changes the synoptic and mesoscale structure of ocean dynamics. Also, the distribution of the obtained result is estimated through the solution of the Fokker–Planck–Kolmogorov equation.     

Lotka-Volterra's model and migrations: Breaking of the well-known center

This paper is devoted to the study of the effect of individual behavior on the Lotka-Volterra predation. We assume that the individuals have many activities in a day for example. Each population is subdivided into subpopulations corresponding to different activities. In order to be clear, I have chosen the case of two activities for each population. We assume that the activities change is faster than the other processes (reproduction, mortality, predation…). This means that we consider population in which the individuals change their activities many times in a day while the reproduction and the predation effects are sensible after about ten days, for example. We use the aggregation method developed in [1] to obtain the global dynamics. Indeed, we start with a micro-model governing the micro-variables, which are the subpopulation densities; the aggregation method permits us to obtain a simpler system governing the macro-variables, which are the global population densities. Furthermore, this method allows us to observe emergence of the dynamics. Indeed, the method implies that the dynamics of the micro-system is close to an invariant manifold after a short time. We show that the dynamics on this manifold is a perturbation of the well-known center of the Lotka-Volterra model. Finally, we prove that a weak change of behavior can lead to a subcritical Hopf bifurcation in the global dynamics.     

不可压缩Navier-Stokes方程最优动力系统建模和分析

研究了同时满足任意速度边界条件和速度不可压条件的Navier-Stokes方程最优动力系统的建模方法.通过对方柱绕流问题的最优动力系统的建模与分析,发现该最优动力系统的动力学特性为极限环.同时,该最优动力系统仅使用了三个最优基函数就很好地描述了所有主要的流场特征和该问题的动力学特性,故满足任意速度边界条件和速度不可压条件Navier-Stokes方程最优动力系统的建模方法,能够用最少的基函数最大限度地描述复杂流体问题及其动力学特性.     

Nonlinear vibration and characteristics of flexible plate-strips with non-symmetric boundary conditions

Regular and chaotic vibrations together with bifurcations of flexible plate-strips with non-symmetric boundary conditions, are investigated through the Bubnov–Galerkin method and a finite difference method of error O(h⁴). Particular attention is paid to non-symmetric boundary conditions. Lyapunov exponents are estimated via Bennetin’s method. Some new examples of routes from regular to chaotic dynamics, and within chaotic dynamics are illustrated and discussed. The phase transitions from chaos to hyperchaos, and a novel phenomenon of a shift from hyperchaos to hyperhyper chaos is also reported.     

种群模型在生物进化研究中的应用

介绍了进化动力学的基本知识和研究现状,把表型特征引入种群动力学模型,进而推导出进化适应动力学模型;总结了如何建立适应度函数以及分析研究进化动力学行为的一般理论和方法,并列举实例,模拟分析验证前面所陈述的理论方法,模拟结果说明收获对生物进化产生重要影响,并有效解释了物种多样性。     

递推牛顿——欧拉动力学算法的推广及其应用

将固定底座的串联机器人的递推牛顿——欧拉动力学算法推广到运动底座 ,给出了不考虑连杆重力和考虑连杆重力两种情况下的递推牛顿——欧拉动力学算法 .然后用数学方法证明了后一种情况下的递推牛顿——欧拉动力学算法是由前一种情况下的递推牛顿——欧拉动力学算法通过改动初始值得到的 .这种通过改动初始值而得到的算法具有较好的计算效率 ,将此算法应用到由清华大学设计的一种新型混联机床的动力学分析上 .     

Stationarity and stochastic differential equations

A linear differential equation of order N with stochastic process coefficients and excitation are studied. The objective of this paper is to demonstrate that, by using an expansion method, when the coefficients and excitation are strict sense stationary processes, the response is also a strict sense stationary process. Such problems occur frequently in the engineering sciences and are very important. Applications include parametric random vibrations, turbulent environment rotorcraft dynamics, and dynamics of axially loaded structural members, among others. An example application is provided.     

一阶线性和拟线性双曲型方程的格点模型

一阶线性和拟线性双曲型方程的格点模型李元香（武汉大学软件工程国家重点实验室）黄樟灿（武汉工学院）ＬＡＴＴＩＣＥＭＯＤＥＬＳＦＯＲＦＩＲＳＴＯＲＤＥＲＬＩＮＥＡＲＡＮＤＱＵＡＳＩ－ＬＩＮＥＡＲＨＹＰＥＢＯＬＩＣＥＱＵＡＴＩＯＮＳ￥ＬｉＹｕａｎ－ｘｉａｎ...     

Generalized synchronization of chaotic systems by pure error dynamics and elaborate Lyapunov function

The generalized synchronization is studied by applying pure error dynamics and elaborate Lyapunov function in this paper. Generalized synchronization can be obtained by pure error dynamics without auxiliary numerical simulation, instead of current mixed error dynamics in which master state variables and slave state variables are presented. The elaborate Lyapunov function is applied rather than the current plain square sum Lyapunov function, deeply weakening the power of Lyapunov direct method. The scheme is successfully applied to both autonomous and nonautonomous double Mathieu systems with numerical simulations.     

Multipulse Orbits in the Motion of Flexible Spinning Discs

Summary. The global dynamics of flexible spinning discs are studied. The discs studied are parametrically excited in their spin rate, and have imperfections that cause symmetry-breaking. After determining the equations of motion in a suitable form, the energy-phase method is employed to show the existence of chaotic dynamics by identifying multipulse jumping orbits in the perturbed phase space. We provide restrictions on the damping, forcing, and symmetry-breaking parameters in order for these complicated dynamics to occur. The dissipative version of the energy-phase method predicts a wider range of values for which chaotic dynamics occurs than the traditional Melnikov method. The results are then discussed in terms of the physical motion of the spinning disc system. The multipulse orbits are manifested in the physical system as a shifting between two different nodal configurations of the disc. When the motion is chaotic, an observer will see a random jumping between the two nodal configurations of the disc. Received February 7, 2000; accepted November 18, 2001     

Modified iteration method for solving fractional gas dynamics equation

This paper is devoted to the time‐fractional gas dynamics equation with Caputo derivative. Fractional operators are very natural tools to model memory‐dependent phenomena. Modified iteration method is proposed to obtain the approximate and analytical solution of the fractional gas dynamics equation. This method is a combined form of the new iteration method and Laplace transform. Modified iteration method really is powerful and simple method compared with other methods. Existence and uniqueness of solution are proven. Numerical results for different cases of the equation are obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonlinear generalized synchronization of chaotic systems by pure error dynamics and elaborate nondiagonal Lyapunov function

By applying pure error dynamics and elaborate nondiagonal Lyapunov function, the nonlinear generalized synchronization is studied in this paper. Instead of current mixed error dynamics in which master state variables and slave state variables are presented, the nonlinear generalized synchronization can be obtained by pure error dynamics without auxiliary numerical simulation. The elaborate nondiagonal Lyapunov function is applied rather than current monotonous square sum Lyapunov function deeply weakening the powerfulness of Lyapunov direct method. Both autonomous and nonautonomous double Mathieu systems are used as examples with numerical simulations.     

弹、粘性体动力学变分原理的Laplace变换形式、有限元构式及数值方法

作者在[1]文中提出了弹、粘动力学变分原理的谱分解形式,本文将其推广到Laplace变换形式,具体写出了薄壳动力学的混合变分原理以及弹-粘-孔隙介质力学的变分原理,并对后者作出了有限元构式. Laplace变换形式的变分原理具有简洁形式,为便于有限元法计算,当已知Laplace变换式的有限个值时,需求原时间函数的有限个值,对此当前尚无成熟方法,本文提供了求原函数的数值方法.从例题可见,这种数值方法是有效的. 结合以上两种理论:从变分原理进行有限元构式以及求Laplace反变换的数值方法,可以使相当广的一类固体动力学问题能够用电子计算机进行求解.     

基于Lie群的刚体动力学建模及数值计算方法研究

基于Lie群和Lie代数之间的指数映射等价关系,推导了基于Lie群的自由刚体连续动力学方程.结合离散变分原理,推导了其Lie群离散变分积分子.通过证明可知连续和离散动力学系统都具有动量守恒性.对连续动力学方程进行同维化处理,使其变为常规非线性方程组的形式,利用Runge-Kutta法进行求解;基于Runge-Kutta基本理论,推导了直接用于Lie群的Runge-Kutta法,从而使Runge-Kutta法可用于求解变维非线性方程组;通过Lie代数变换,利用Kelly变换和Newton迭代对Lie群离散变分积分子进行求解.仿真对比结果表明,3种算法下的计算结果高度吻合,且能高精度地保持系统的结构守恒和动量守恒性.     

Qualitative and quantitative combined nonlinear dynamics model and its application in analysis of price,supply–demand ratio and selling rate

The qualitative and quantitative combined nonlinear dynamics model proposed in this paper fill the gap in nonlinear dynamics model in terms of qualitative and quantitative combined methods, allowing the qualitative model and quantitative model to perfectly combine and overcome their weaknesses by learning from each other. These two types of models use their strengths to make up for the other’s deficiencies. The qualitative and quantitative combined models can surmount the weakness that the qualitative model cannot be applied and verified in a quantitative manner, and the high costs and long time of multiple construction as well as verification of the quantitative model. The combined model is more practical and efficient, which is of great significance for nonlinear dynamics. The qualitative and quantitative combined modeling and model analytical method raised in this paper is not only applied to nonlinear dynamics, but can be adopted and drawn on in the modeling and model analysis of other fields. Additionally, the analytical method of qualitative and quantitative combined nonlinear dynamics model proposed in this paper can satisfactorily resolve the problems with the price system’s existing nonlinear dynamics model analytical method. The three-dimensional dynamics model of price, supply–demand ratio and selling rate established in this paper make estimates about the best commodity prices using the model results, thereby providing a theoretical basis for the government’s macro-control of price. Meanwhile, this model also offer theoretical guidance to how to enhance people’s purchasing power and consumption levels through price regulation and hence to improve people’s living standards.     

非线性非完整系统Vacco动力学方程的积分方法*

本文给出积分非线性非完整系统Vacco动力学方程的积分方法.首先,将Vacco动力学方程表示为正则形式和场方程形式;然后,分别用梯度法,单分量法和场方法积分相应完整系统的动力学方程,并加上非完整约束对初条件的限制而得到非线性非完整系统Vacco动力学方程的解.