多元多项式混沌法在随机方腔流动模拟中的应用

本文介绍了多元多项式混沌方法,采用此方法对随机层流Navier-Stokes方程进行求解,模拟了同时存在多个不确定因素影响的二维方腔驱动流。研究了当上下边界驱动速度和流体黏性系数为服从高斯分布的随机变量时所引起的流动结构的不确定性,着重分析了流场速度的统计特性,并与蒙特卡洛方法的计算结果进行了对比,对多项式混沌方法的结果进行了验证和确认。研究结果表明多项式混沌方法可以准确高效地模拟多个不确定性在流场中的传播,与蒙特卡洛方法相比体现出明显的优势。     

求解Kuramoto-Sivashinsky方程的平移基无单元Galerkin方法

Kuramoto-Sivashinsky方程是一种可以描述复杂混沌现象的高阶非线性演化方程.方程中高阶导数项的存在,使得传统无单元Galerkin方法采用高次多项式基函数构造形函数时,形函数违背了一致性条件.因此,本文提出了一种采用平移多项式基函数的无单元Galerkin方法.与传统无单元Galerkin方法相比,该方法在方程离散时依然采用Galerkin进行离散,但形函数的构造采用了基于平移多项式基函数的移动最小二乘近似.通过对具有行波解和混沌现象的Kuramoto-Sivashinsky方程的数值模拟,验证了本文方法的有效性.     

从有序到混沌的计算机模拟——逻辑斯蒂方程的演示

介绍利用计算机ＩＢＭ３８６模拟逻辑斯蒂方程从有序到混沌的方法和结果。     

时空混沌系统参量辨识律的设计与投影同步研究

研究了相互耦合的时空混沌系统的参量辨识与投影同步问题. 依据Lyapunov定理, 设计了参量辨识律和表征耦合强度的待定函数的自适应律, 对响应系统中的未知参量进行了有效辨识, 并完成了时空混沌系统的投影同步研究. 采用具有时空混沌行为的Burgers方程作为实例进行了仿真分析.     

单模激光混沌特性的理论研究

在光学中，激光是产生双稳态、振荡、混沌等多种自组织现象的典型例子。本文采用非线性理论分析了单模激光的混沌特性，对描述单模激光的非线性动力学方程进行了稳定性分析，从而确定了系统失稳后进入混沌态的分歧点，并利用计算机进行了数值计算，给出了系统呈现混沌态的模拟结果。计算机数值模拟的结果与理论分析的结果相吻合。     

一种简易的自组混沌摆装置

混沌现象是非线性系统的经典行为,其相关研究至今仍属于前沿领域。本文设计并实现了一套可变参数的外力驱动的自组混沌摆实验装置。相关研究包括：建立此实验装置的简化动力学方程,改变实验参数并通过图像识别的方法进行实验数据处理及绘制相图,最后将动力学方程的数值模拟结果与实验数据进行比较,验证了装置的可靠性。此外,实验过程中进一步探究与分析混沌现象。该装置提供了一种低成本的对混沌现象的直观化研究方法,可以作为本科生的非线性物理或计算物理课程的实践内容,加深学生对混沌现象及其原理的认识。     

利用速度反馈方法控制时空混沌

以空间一维的复Ginzburg-Landau方程为模型,研究了利用系统变量随时间的变化率作为反馈控制信号控制偏微分方程系统中时空混沌的可能性.通过理论分析和数值模拟方法讨论了控制参数与可控性所满足的关系,解释了不同目标态时临界控制参数的尺寸效应. 关键词： 混沌控制 时空混沌 速度反馈 复Ginzburg-Landau方程     

基于区间的传输线感应电压不确定性分析方法

为分析多导体传输线耦合情况下线缆结构参数的不确定性对终端电压的影响,引入了一种基于区间分析的切比雪夫(Chebyshev)多项式逼近方法。该方法首先将传输线电报方程转换为常微分方程求解;其次采用Chebyshev多项式求得电报方程的扩张函数,进而获得终端电压的波动范围。相比于混沌多项式方法和蒙特卡罗（MC）法,此方法只需要输入随机参数的波动范围。针对电磁脉冲辐照下高度和间距随机变动的多导体线束进行仿真,仿真结果表明,间距基本不影响终端电压,终端电压对高度更为敏感。在计算结果基本一致的情况下,Chebyshev多项式逼近方法的计算耗时远小于MC方法。     

一维元胞自动机随机交通流模型的宏观方程分析

在一维局部作用元胞自动机交通流模型中引入刹车噪声与产生、消失概率,得到一完全随机的元胞自动机交通流模型.利用玻耳兹曼近似,建立了该模型的宏观动力学方程,并对宏观方程在特殊条件下的解进行了理论分析和计算机模拟. 关键词： 元胞自动机 交通流 Burgers方程 激波     

爆炸波中的混合不确定度量化方法

将认知不确定作为外层,偶然不确定作为内层,利用混合不确定理论处理爆炸波中的不确定度问题.分别使用非嵌入多项式混沌方法（NIPC）和Dempster-Shafer理论处理偶然不确定和认知不确定,用迎风格式求解确定系统.结果表明：NIPC和Dempster-Shafer结构为模型输入参数和初边值不确定性对输出响应量的影响提供一种有效方法,对各种建模与模拟不确定性评估和确认建模与模拟具有很好的参考价值.     

嵌入随机多项式的抛物方程不确定声场快速算法

为了得到准确且高效计算起伏海洋介质中随机声场的算法,本文将随机多项式展开嵌入到宽角抛物方程声场计算模型(简称RAM模型)中,发展了一种不确定声场的快速算法.其计算结果比使用嵌入随机多项式的窄角抛物方程准确,计算时间小于作为参考的蒙特卡洛方法.在仿真算例中,随机多项式展开法对声强均值、方差、概率密度的计算准确;在一定的随...     

Numerical analysis of the Burgers’ equation in the presence of uncertainty

The Burgers’ equation with uncertain initial and boundary conditions is investigated using a polynomial chaos (PC) expansion approach where the solution is represented as a truncated series of stochastic, orthogonal polynomials.     

Time-dependent generalized polynomial chaos

Generalized polynomial chaos (gPC) has non-uniform convergence and tends to break down for long-time integration. The reason is that the probability density distribution (PDF) of the solution evolves as a function of time. The set of orthogonal polynomials associated with the initial distribution will therefore not be optimal at later times, thus causing the reduced efficiency of the method for long-time integration. Adaptation of the set of orthogonal polynomials with respect to the changing PDF removes the error with respect to long-time integration. In this method new stochastic variables and orthogonal polynomials are constructed as time progresses. In the new stochastic variable the solution can be represented exactly by linear functions. This allows the method to use only low order polynomial approximations with high accuracy. The method is illustrated with a simple decay model for which an analytic solution is available and subsequently applied to the three mode Kraichnan–Orszag problem with favorable results.     

基于小波变换的光混沌信号消噪与Lyapunov指数计算

针对动力学方程未知且信噪比小的光混沌信号,采用小波多分辨分解算法对其进行噪音消减.用Lorenz混沌信号对该算法的消噪效果进行了检验.提出利用互信息量法和Cao氏法来改进小数据量法在时间延迟和嵌入维数计算上存在的主观选择性,对经过噪音消减的Lorenz混沌信号利用此改进的小数据量法计算其最大Lyapunov指数.结果表明,信噪比可提高近10 dB左右,最大Lyapunov指数计算误差可减少近30%,并求得半导体放大器光混沌信号的最大Lyapunov指数为0.389 6.     

Continuous feedback control of KdV Burgers system

The chaos in the KdV Burgers equation describing a ferroelectric system has been successfully controlled by using a continuous feedback control. This system has two stationary points. In order to know whether the chaos is controlled or not, the instability of control equation has been analysed numerically. The numerical analysis shows that the chaos can be converted to one point by using one control signal, however, it can converted to the other point by using three control signals. The chaotic motion is converted to two desired stationary points and periodic orbits in numerical experiment separately.     

调制深度引导CO_2激光器到达混沌

从描述CO2激光器的非线性方程出发,构造一种具有调制损耗的新动力学系统,并从理论上分析了这种调制损耗的CO2激光器的动力学行为。结果表明,在调制深度的引导下,系统由单一稳定输出转变为不稳定运行,呈现2nP倍周期振荡态及混沌态。     

BECOOL: Ballooning eigensolver with COOL finite elements

An incompressible variational ideal ballooning mode equation is discretized with the COOL finite element discretization scheme using basis functions composed of variable order Legendre polynomials. This reduces the second order ordinary differential equation to a special block pentadiagonal matrix equation that is solved using an inverse vector iteration method. A benchmark test of BECOOL (Ballooning Eigensolver using COOL finite elements) with second order Legendre polynomials recovers precisely the eigenvalues computed by the VVBAL shooting code. Timing runs reveal the need to determine an optimal lower order case. Eigenvalue convergence runs show that cubic Legendre polynomials construct the optimal ballooning mode equation for intensive computations.     

On the mathematically reliable long-term simulation of chaotic solutions of Lorenz equation in the interval [0,10000]

Using 1200 CPUs of the National Supercomputer TH-A1 and a parallel integral algorithm based on the 3500th-order Taylor expansion and the 4180-digit multiple precision data,we have done a reliable simulation of chaotic solution of Lorenz equation in a rather long interval 0 t 10000 LTU(Lorenz time unit).Such a kind of mathematically reliable chaotic simulation has never been reported.It provides us a numerical benchmark for mathematically reliable long-term prediction of chaos.Besides,it also proposes a safe method for mathematically reliable simulations of chaos in a finite but long enough interval.In addition,our very fine simulations suggest that such a kind of mathematically reliable long-term prediction of chaotic solution might have no physical meanings,because the inherent physical micro-level uncertainty due to thermal fluctuation might quickly transfer into macroscopic uncertainty so that trajectories for a long enough time would be essentially uncertain in physics.     

Characterization and control of chaotic dynamics in a nerve conduction model equation

In this paper we consider the Bonhoeffer-van der Pol (BVP) equation which describes propagation of nerve pulses in a neural membrane, and characterize the chaotic attractor at various bifurcations, and the probability distribution associated with weak and strong chaos. We illustrate control of chaos in the BVP equation by the Ott-Grebogi-Yorke method as well as through a periodic instantaneous burst.