Multiplicity of Forced Oscillations on Manifolds and Applications to Motion Problems with One-Dimensional Constraints

This paper illustrates how the notion of an ejecting set recently introduced by the authors can be used to get multiplicity results for forced oscillations. The motion problem of a mass point constrained to a one-dimensional differentiable manifold and acted on by a periodic force is treated in detail.     

Characterizations of optimality in convex programming: the nondifferentiable case

Primal, dual and saddle-point characterizations of optimality are given for convex programming in the general case (nondifferentiable functions and no constraint qualification).     

Runge-Kutta Methods Adapted to Manifolds and Based on Rigid Frames

We consider numerical integration methods for differentiable manifolds as proposed by Crouch and Grossman. The differential system is phrased by means of a system of frame vector fields E ₁, ... , E _n on the manifold. The numerical approximation is obtained by composing flows of certain vector fields in the linear span of E ₁, ... , E _n that are tangent to the differential system at various points. The methods reduce to traditional Runge-Kutta methods if the frame vector fields are chosen as the standard basis of euclidean ⁿ . A complete theory for the order conditions involving ordered rooted trees is developed. Examples of explicit and diagonal implicit methods are presented, along with some numerical results.     

泛函微分方程的全局稳定周期解

本文综合运用相空间理论,泛函分析方法和李雅普诺夫第二方法,研究具无限时滞的泛函微分方程周期解的存在性,唯一性和稳定性问题,得到了新的结果．     

Almost-periodic solutions to systems of differential equations with fast and slow time in the degenerate case

We consider a system of ordinary first-order differential equations. The right-hand sides of the system are proportional to a small parameter and depend almost periodically on fast time and periodically on slow time. With this system, we associate the system averaged over fast time. We assume that the averaged system has a structurally unstable periodic solution. We prove a theorem on the existence and stability of almost periodic solutions of the original system. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 451–456, March, 1998.     

On elliptic solutions of nonlinear ordinary differential equations

The problem of constructing and classifying exact elliptic solutions of autonomous nonlinear ordinary differential equations is studied. An algorithm for finding elliptic solutions in explicit form is presented.     

On the asymptotic behavior of nonoscillatory solutions of second order quasilinear ordinary differential equations

In this paper second order quasilinear ordinary differential equations are considered, and a necessary and sufficient condition for the existence of a slowly growing positive solution is established. Moreover, the precise asymptotic forms as t→∞ of slowly growing positive solutions and slowly decaying positive solutions are obtained.     

A deficient spline function approximation to fourth-order differential equations

An approximate spline solution is developed for the initial value problem of a fourth-order ordinary differential equation. The approximation is based on deficient spline polynomials of degree M = 8 and deficiency 4. The existence and uniqueness of the solution, which satisfies a Lipschitz condition, are proved. The consistency, stability, and consequently convergence of the solution are established. Furthermore, the method is proved to be of order 9, and the errors are limited by the relation S⁽ⁱ⁾(x) − y⁽ⁱ⁾(x) = O(h^{9 − i}), I = 0(1)8.     

某常微分方程教材中的一个错例

针对东北师范大学微分方程教研室所编教材《常微分方程》中的一道例题,经过实际验算发现,原题所给向量函数并非如其所称是其所给微分方程组的基本解组;随后通过两种方法对该道例题给出正确求解．     

On the hyperbolicity of rapidly oscillating periodic solutions of functional differential equations

We consider periodic solutions of nonlinear functional differential equations with rational periods less than 2. We study the spectral properties of monodromy operators and state a hyperbolicity criterion for such solutions.__________Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 39, No. 1, pp. 82–85, 2005Original Russian Text Copyright © by A. L. Skubachevskii and H.-O. WaltherThe first authors research was supported by the Mercator-Programm of the Deutsche Forschungsgemeinschaft, RFBR grant No. 04-01-00256, and Russian Ministry of Education and Science grant No. E02-1.0-131.Translated by A. L. Skubachevskii and H.-O. Walther     

Special symmetric periodic solutions of differential systems with distributed delay

We discuss the existence of periodic solutions to a system of differential equations with distributed delay which shows a certain type of symmetry. For this, such solutions are related to the solutions of a system of second-order ordinary differential equations.     

抽象空间常微分方程弱解的存在和唯一性

本文研究了抽象空间中初值问题ｘ′＝ ｆ（ｔ,ｘ）,ｘ（ｔ０）＝ ｘ０ 的弱解存在和唯一性,并推广了文［１］、［２］中的有关结果     

A Note on the Construction of Crouch-Grossman Methods

Numerical integration methods based on rigid frames were introduced by Crouch and Grossman. The order theory of these methods were later analyzed by Owren and Marthinsen. The resulting order conditions are difficult to solve due to nonlinear relations on the weights of the methods. In this paper we propose a variant of the Crouch-Grossman method that uses modified vector fields so that the order conditions of this new method coincide with the classical order conditions for Runge-Kutta methods.