指标-2微分-代数系统的连续扩展Runge-Kutta迭代方法

1引言微分-代数系统包括具有约束条件的微分方程和奇异隐式微分方程,在实际应用中,如:约束力学系统、流体动力学、化学反应动力学、电子网络模拟、控制工程和机器人技术等领域就产生了诸多问题需要求解．近年来,微分-代数系统已极大地引起了许多工程师和数学工作者的关注,开展了众多相关问题的探讨,提出了许多新的算法理论[1-3]．在本文中我们对指标-2的微分-代数方程利用Runge-Kutta方法进行时间的离散和动力学迭代,研究离散迭代系统的收敛性．     

解分数阶微分代数系统的Adomian分解方法

研究了利用Adomian分解求解分数阶微分代数系统的方法.分析了代数约束对Adomian方法求解的影响,指出直接解出代数约束变量,将原系统转化为微分系统进行Adomian分解的困难.提出确定代数变量级数解各分量的新方法,据此进行Adomian分解,得到整个系统的级数解.特别研究了代数约束为线性的分数阶微分代数系统的Adomian解法,证明了各变量间的线性代数约束关系可以转化为相应级数解中各分量的线性关系,从而方便求解,并结合具体例子证明了该方法简便有效.     

一类随机惯性时滞神经网络的稳定性北大核心

研究一类随机惯性时滞神经网络稳定性问题.通过引入适当变量变换将二阶微分系统转换为一阶微分系统,利用同胚映射,Ito公式和微分算子,构造恰当的Lyapunov函数和采用递推归纳,给出其系统平衡点存在唯一及全局渐近稳定和解指数稳定判定的充分条件,最后通过数值模拟例子说明所得理论结果的正确性.     

无界时滞中立型微分大系统的不稳定性

本文采用Ｌｉａｐｕｎｏｖ函数法，研究了两类无界时滞的非线性中立型微分大系统的不稳定性，给出了一些简单、实用的代数判据，推广和改进了有关的研究。     

广义大系统的Lyapunov稳定性分析

广义大系统的稳定性是一个非常重要的问题 ,由于广义大系统的复杂性 ,对其稳定性的研究也是一件相当困难的事情 .本文利用 Lyapunov方程 ,应用 Lyapunov函数法 ,研究了广义线性大系统和广义非线性大系统的稳定性和不稳定性 ,得到了系统的关联稳定参数域和不稳定域 .给出例子说明该方法的可行性 .     

脉冲摄动微分系统的有界性

通过利用变分Lyapunov函数方法, 该文主要研究了脉冲摄动微分系统关于两个测度的有界性. 与以前结果相比, 不难发现变分Lyapunov函数方法是Lyapunov函数方法的推广 .     

微分代数系统的渐近性

从动力系统的角度研究微分代数系统,利用单调流理论中的结果和方法讨论微分代数系统渐近性态.首先,我们把所考察的系统嵌入到一族相关的系统,引进使得系统族中的每个系统生成单调流的相应偏序和条件.然后给出了若干关于解收敛于平衡点的一般性结果,并对一类微分代数系统的渐近性作了较为精细的讨论.我们的结果是Hirch等关于常微分方程的相关结果的推广和改进.     

关于代数体函数微分多项式的某些结果

本文应用代数体函数及其微分多项式的有关理论,将仪洪勋关于亚纯函数微分多项式的几个结果推广到代数体函数物微分多项式。     

具概率延迟反馈金融系统的脉冲控制

研究了概率时滞脉冲金融系统平衡点的全局渐近稳定性问题。首先,通过定义合适的时滞分段区间上的随机变量,给出了概率时滞的脉冲金融系统的数学模型,根据脉冲微分不等式特点构造了一个简便合适的Lyapunov函数利用脉冲微分不等式引理、控制脉冲间隔与脉冲量以及概率时滞分析技巧,获得了较大时滞允许范畴下的平衡点的全局指数稳定,并通过数值实例验证了方法的可行性以及概率时滞的优势。特别地,稳定性判定准则的时滞允许上限的增大,扩大了准则的实用性.     

代数体函数微分多项式的值分布

利用亚纯函数的Nevanlinna值分布理论和方法,研究了一类代数体函数微分多项式的值分布问题,推广和改进了相关文献的结论.     

A note on bifurcation points for index-1 differential-algebraic problems

The arclength continuation method is proposed for index-1 DAEs with singular points. In particular, the cusp points, the tangency points and equilibrium points are investigated. The numerical iterative matrix is studied at a singular point. The numerical examples are given to illustrate the robustness of the continuation method.     

Stability Issues in Regular and Noncritical Singular DAEs

This paper addresses equilibrium stability issues in both regular and singular differential-algebraic equations (DAEs). We present a survey of available results and discuss some commonly-used methods in the qualitative analysis of low-index autonomous systems. Additionally, we extend the use of matrix pencil theory to the stability study of singular problems, pointing out some interesting relations between regular and singular DAEs. This framework is applied to the qualitative analysis of singular equations arising in the context of the Singularity Induced Bifurcation theorem, and also to the stability study of stationary equilibria in singular DAEs.     

Iterative Solution of SPARK Methods Applied to DAEs

In this article a broad class of systems of implicit differential–algebraic equations (DAEs) is considered, including the equations of mechanical systems with holonomic and nonholonomic constraints. Solutions to these DAEs can be approximated numerically by applying a class of super partitioned additive Runge–Kutta (SPARK) methods. Several properties of the SPARK coefficients, satisfied by the family of Lobatto IIIA-B-C-C^*-D coefficients, are crucial to deal properly with the presence of constraints and algebraic variables. A main difficulty for an efficient implementation of these methods lies in the numerical solution of the resulting systems of nonlinear equations. Inexact modified Newton iterations can be used to solve these systems. Linear systems of the modified Newton method can be solved approximately with a preconditioned linear iterative method. Preconditioners can be obtained after certain transformations to the systems of nonlinear and linear equations. These transformations rely heavily on specific properties of the SPARK coefficients. A new truly parallelizable preconditioner is presented.     

The function cascade synchronization method and applications

In this paper, a new function cascade synchronization method of chaos system is proposed to achieve generalized projective synchronization for chaotic systems. Based on Laypunov stability, the proposed synchronization technique is applied to three famous chaotic systems: the unified chaotic system, Liu system and Rössler system, which can make the states of two identical chaotic systems asymptotically synchronized by choosing different special suitable error functions. Numerical simulations are presented to show the effectiveness.     

Convergence of multistep discretizations of DAEs

Standard ODE methods such as linear multistep methods encounter difficulties when applied to differential-algebraic equations (DAEs) of index greater than 1. In particular, previous results for index 2 DAEs have practically ruled out the use of all explicit methods and of implicit multistep methods other than backward difference formulas (BDFs) because of stability considerations. In this paper we embed known results for semi-explicit index 1 and 2 DAEs in a more comprehensive theory based on compound multistep and one-leg discretizations. This explains and characterizes the necessary requirements that a method must fulfill in order to be applicable to semi-explicit DAEs. Thus we conclude that the most useful discretizations are those that avoid discretization of the constraint. A freer use of e.g. explicit methods for the non-stiff differential part of the DAE is then possible.Dedicated to Germund Dahlquist on the occasion of his 70th birthdayThis author thanks the Centro de Estadística y Software Matemático de la Universidad Simón Bolivar (CESMa) for permitting her free use of its research facilities.Partial support by the Swedish Research Council for Engineering Sciences TFR under contract no. 222/91-405.     

预测校正公式求解指标2微分代数方程

1．前言微分代数方程（EEES）是经常出现于实际问题中的一类方程．其数值求解已成为常微分方程数值求解领域十分活跃的一个方向．目前微分代数方程求解的数值方法主要是nunge－Kutta型方法及BDF方法．Runge－Kutta型方法在网,问中有详细的介绍．Hairer等人据此编制了软件RADAU,而目前使用最广泛的软件还是PetZold等编制的DASSL．DASSL使用的方法为BDF方法,它在微分代数方程中的应用最早可以追述到Gear的开创性工作问．BDF方法一个很大的优点是刚性稳定．然而对于非刚性的微分代数方程,刚性稳定已不是主要考虑的因素．因此…     

Comparing stability properties of three methods in DAEs or ODEs with invariants

Three approaches for solving ODEs with invariants or DAEs are the constrained least squares method, the coordinate projection method and the derivative projection method. The stability properties of these three methods are compared for linear ODEs with linear invariants. There exist examples where each of the three approaches is to prefer.     

Constraint preserving integrators for general nonlinear higher index DAEs

Summary. In the last few years there has been considerable research on numerical methods for differential algebraic equations (DAEs) where is identically singular. The index provides one measure of the singularity of a DAE. Most of the numerical analysis literature on DAEs to date has dealt with DAEs with indices no larger than three. Even in this case, the systems were often assumed to have a special structure. Recently a numerical method was proposed that could, in principle, be used to integrate general unstructured higher index solvable DAEs. However, that method did not preserve constraints. This paper will discuss a modification of that approach which can be used to design constraint preserving integrators for general nonlinear higher index DAEs. Received August 25, 1993 / Revised version received April 7, 1994     

A network model for incompressible two-fluid flow and its numerical solution

In this paper we consider an incompressible version of the two-fluid network model proposed by Porsching (Nu. Methods Part. Diff. Eq., 1 , 295–313 [1985]). The system of equations governing the model is a mixed system of differential and algebraic equations (DAEs). These DAEs are then recast, through proper transformation, into a system of ordinary differential equations on a submanifold of ?ⁿ, for which uniqueness, existence, and stability theorems are proved. Numerical simulations are presented.     

Convex and Concave Relaxations for the Parametric Solutions of Semi-explicit Index-One Differential-Algebraic Equations

A method is presented for computing convex and concave relaxations of the parametric solutions of nonlinear, semi-explicit, index-one differential-algebraic equations (DAEs). These relaxations are central to the development of a deterministic global optimization algorithm for problems with DAEs embedded. The proposed method uses relaxations of the DAE equations to derive an auxiliary system of DAEs, the solutions of which are proven to provide the desired relaxations. The entire procedure is fully automatable.