Linear Stability Analysis of Resonant Periodic Motions in the Restricted Three-Body Problem

The equations of the restricted three-body problem describe the motion of a massless particle under the influence of two primaries of masses 1 −μ and μ, 0≤μ≤ 1/2, that circle each other with period equal to 2π. When μ=0, the problem admits orbits for the massless particle that are ellipses of eccentricity e with the primary of mass 1 located at one of the focii. If the period is a rational multiple of 2π, denoted 2π p/q, some of these orbits perturb to periodic motions for μ > 0. For typical values of e and p/q, two resonant periodic motions are obtained for μ > 0. We show that the characteristic multipliers of both these motions are given by expressions of the form in the limit μ→ 0. The coefficient C(e,p,q) is analytic in e at e=0 and C(e,p,q)=O(e^|p-q|). The coefficients in front of e^|p-q|, obtained when C(e,p,q) is expanded in powers of e for the two resonant periodic motions, sum to zero. Typically, if one of the two resonant periodic motions is of elliptic type the other is of hyperbolic type. We give similar results for retrograde periodic motions and discuss periodic motions that nearly collide with the primary of mass 1 −μ.     

Symbolic Dynamic Patterns of Written Exchanges: Hierarchical Structures in an Electronic Problem Solving Group

A spontaneous scientific creative problem solving discussion, which had transpired across a listserver, was captured and content-analyzed according to two protocols. One protocol centered on social interaction and the second pertained to problem completion behavior. Data were analyzed for topological entropy and dimensional complexity with a hierarchical variation of the symbolic dynamics procedure that was first introduced by Guastello, Hyde, and Odak (1998). The principal findings were that (1) There was a greater prevalence of clarification and initiating (particularly creative) behaviors compared to a real-time creative problem solving episode; they were presumably liberated by the unblocking effect that tends to characterize computer-assisted group discussions. (2) There were complex responses containing up to five conversational elements as identified by a Social Interaction scoring system. (3) The two different scoring protocols produced different indicators of string length and entropy. The content of those patterns, which are, in principle, decomposed orbits, is discussed.     

Plane Collision Problem for Two Identical Elastic Parabolic Bodies—Direct Central Impact

A direct central collision of two identical infinite cylindrical bodies is studied. A nonstationary plane elastic problem is solved. The variable boundary of the contact area is determined. A mixed boundary problem is formulated. Its solution is represented by Fourier series. An infinite system of Volterra equations of the second kind for the unknown expansion coefficients is derived by satisfying boundary conditions. The basic characteristics of the collision process are determined numerically depending on the curvature of the frontal surface of the bodies     

Isoenergetic Families of Planar Orbits Generated by Homogeneous Potentials

In the light of the inverse problem of dynamics, we study in some detail the monoparametric isoenergetic families of planar orbits f (x, y) = c, created by homogeneous potentials V (x, y). For any preassigned family of orbits and for any degree of homogeneity m of the potential, we offer the criteria which the family has to satisfy so that it can result from such a potential. When the criteria are fulfilled, Szebehelys first order partial differential equation for the unknown potential V (x, y) is substantially simplified and, in most of the cases, can be solved to completion and uniqueness.     

Axisymmetric Collision Problem for Two Identical Elastic Solids of Revolution

A direct central collision of two identical bodies of revolution is studied. A nonstationary mixed boundary-value problem with an unknown moving boundary is formulated. Its solution is represented by a series in term of Bessel functions. An infinite system of Volterra equations of the second kind for the unknown expansion coefficients is derived by satisfying the boundary conditions. The basic characteristics of the collision process are determined depending on the curvature of the frontal surface of the bodies     

The equations of external impacted dynamics between multi-rigidbody systems

I.Intr0ducti0nTheimpactproblemofmulti-rigidbodysystemisanimportantresearchingtaskirithedomainofmodernengineering.Uptonow,afewscholarshavestudiedthisproblem.Startingfromthewewton-Eu1erdynamicequation,paperIl]discussesthegeneralsituationofimpactdynamicsofmu…     

The Complexity of Artificial Grammars

In experimental psychology, artificial grammars, generated by directed graphs, are used to test the ability of subjects to implicitly learn the structure of complex rules. We introduce the necessary notation and mathematics to view an artificial grammar as the sequence space of a dynamical system. The complexity of the artificial grammar is equated with the topological entropy of the dynamical system and is computed by finding the largest eigenvalue of an associated transition matrix. We develop the necessary mathematics and include relevant examples (one from the implicit learning literature) to show that topological entropy is easy to compute, well defined, and intuitive and, thereby, provides a quantitative measure of complexity that can be used to compare data across different implicit learning experiments.     

一维正方准晶非周期平面弹性有限摩擦接触问题

利用平面弹性复变方法和解析函数边值理论,讨论了一维正方准晶材料非周期平面上的有限摩擦接触问题.基于一维正方准晶非周期平面内应力和位移分量的广义复变函数表示,有限摩擦接触问题被转换为Riemann-Hilbert边值问题.对于平底压头,得到了压头下方接触应力的显式表达式,并用数值算例分析了摩擦系数和声子场-相位子场耦合系数对压头下方接触应力分布的影响.结果表明:(1)摩擦系数对接触应力大小和分布规律的影响几乎可以忽略;(2)耦合系数对接触应力的分布规律无影响,对声子场接触应力值的影响微乎其微,对相位子场接触应力大小的影响比较明显.特殊条件下,本文所得结论可退化为一维四方准晶和一维六方准晶非周期平面内有限摩擦接触问题的解.当摩擦系数等于0时,还可以得到对应无摩擦接触问题的解.     

The dynamics of coupled planar rigid bodies. II. Bifurcations,periodic solutions,and chaos

We give a complete bifurcation and stability analysis for the relative equilibria of the dynamics of three coupled planar rigid bodies. We also use the equivariant Weinstein-Moser theorem to show the existence of two periodic orbits distinguished by symmetry type near the stable equilibrium. Finally we prove that the dynamics is chaotic in the sense of Poincaré-Birkhoff-Smale horseshoes using the version of Melnikov's method suitable for systems with symmetry due to Holmes and Marsden.     

THE NUMERICAL SOLUTION OF A SINGULARLY PERTURBED PROBLEM FOR SEMILINEAR PARABOLIC DIFFERENTIAL EQUATION

The numerical solution of a singularly perturbed problem for the semilinear parabolicdifferential equation with parabolic boundary layers is discussed.A nonlinear two-leveldifference scheme is constructed on the special non-uniform grids.The uniform convergenceof this scheme is proved and some numerical examples are given.     

压电介质平面问题的基本解

应用复变函数的方法,对于压电介质平面问题,分析导出了无限介质或半无限介质受任意 集中载荷作用时的复势函数基本解;这些结果可作为边界元法的基本解,以求解具有复杂边界压电体的平面问题。     

复合材料条形域问题混合状态Hamiltonian元的半解析解

给出了复合材料条形域问题的混合状态Ｈａｍｉｌｔｏｎ正则方程及其有效的半解析法。该方法不同于通常的半解析法，需先给出满足规则几何形状和简单边界条件的解析函数，利用Ｈａｍｉｌｔｏｎ矩阵的正交性质，采用控制论中的理论与方法后给出复杂几何形状和边界条件的解析函数，这样沿板厚方向就不需引入任何有关位移和应力的人为假设，从而引入了复合材料计算中剪切效应的影响，且发挥了Ｈ型方程的传递矩阵法优点，保证了层间位移和应刀的连续，建立了条形梁上下表面相变量之间的关系式，然后利用打靶法进行求解。     

The Local Conditional Stability Estimate for an Inverse Contact Problem in the Elasticity

In this paper, we consider an inverse problem in elasticity of determining stress on a contact domain from measurements outside the contact domain. The local conditional stability estimate for determining the stress vector is obtained. This conditional stability can give the convergence rate for Tikhonov's regularized solutions. This revised version was published online in July 2006 with corrections to the Cover Date.     

The Finite Element Analysis of Interaction Problem Through Water,Soil, Balloon and Pile

This paper presents the finite element analysis of an interaction problem involving water, soil, balloon and pile. A building with friction piles is considered, and balloons are introduced to the top of piles. To control the vertical displacement of the building, water is injected into or removed from the balloons. The two-dimensional incompressible Navier-Stokes equation is introduced, and the ALE (Arbitrary Lagrangian Eulerian) method is applied to the water flow analysis. The FS (Fractional Step) method is also applied in the finite element formulation. The soil, which is assumed as a linear elastic body, is subjected to the deformation analysis. The balloon and pile are assumed as a linear elastic truss and a rigid frame, and the deformation analysis is also performed. All the components are discretized by the finite element method in space and are interactively solved by taking into account continuity conditions of traction and displacement.     

Global Constraints for Stress Constrained Materials: The Problem of Saint Venant

Based on the global constraint principle of Antman and Marlow, a new solution of Saint Venant's problem is proposed. The solutions for the six fundamental cases of loading in terms of stress are obtained with relative ease and converge to the classical Saint Venant's solution as the length of the beam is increased. It is also shown that the assumptions of a special technical rod theory are coherent with the requirements of the global constraint theory for the Saint Venant cylinder.     

含共线刚性线夹杂各向异性体的平面问题

应用复变函数方法,给出了含共线刚性线夹杂各向异性体平面问题的一般解;对于一个或二个夹杂的情形,给出了封闭形式的应力奇异性系数解;结果表明,应力奇异性系数与材料常数和ε∞ｘ 有关,这里ε∞ｘ 为无限远处ｘ 方向的线应变     

双材料反平面问题界面端奇异应力场分析

利用位移函数的级数展开,对任意角度的反平面问题界面端的应力场进行了分析研究,得到了全场解。研究一阶场后发现,奇异规律与一般平面问题界面端有显著区别,在界面端关于界面对称的情况下,平角界面端(θ1 = θ2 = θ = 90°) 应力场没有奇异性,其它形状的界面端随着角度θ 从90°到180°,奇异指数也从0到0.5。当界面端是非对称时,平角界面端(θ1 θ2 = 180°)、直角界面端(θ1 = 90°,θ2 = 180°)以及其它形状界面端的奇异指数是一个与两相材料常数比Γ有关的常数。以上两种情况下的应力强度因子完全类似单相材料中裂纹尖端附近应力强度因子,故可根据定义得到