Periodic motion around libration points in the Elliptic Restricted Three-Body Problem

Periodicity of motion around the collinear libration point associated with the Elliptic Restricted Three-Body Problem is studied. A survey of periodic solutions in the Circular Restricted Three-Body Problem is presented considering both Sun–Earth and Earth–Moon systems. Halo, Lyapunov and Vertical families around L1, L2 and L3 points are investigated, and their orbital period ranges through the entire family are reported. Resonant motions within the orbit families in the circular problem are identified and selected as suitable initial guess to find periodic orbits in the elliptic problem, which are targeted using a differential correction algorithm. Periodic solutions found are cataloged depending on the number of revolutions around libration points. Geometry, dynamical behavior and stability properties of single-revolution orbits are shown, as well as double-, triple- and quadruple-revolution solutions.     

Cross Sections in the Three-Body Problem

In Dynamical Systems, Birkhoff gave a clear formulation of a cross section, suggested a possible generalization to cross sections with boundary, and raised the question of whether or not such cross sections exist in the three-body problem. In this work, we explicitly develop Birkhoff's notion of a generalized cross section, formulate homological necessary conditions for the existence of a cross section or generalized cross section, and show that these conditions are not satisfied in the three-body problem.     

Periodic Solutions of the Elliptic Isosceles Restricted Three-body Problem with Collision

The elliptic isosceles restricted three-body problem with collision, is a restricted three-body problem where the primaries move having consecutive elliptic collisions and the infinitesimal mass is moving in the plane perpendicular to the primaries motion that passes through the center of mass of the primary system. Our purpose in this paper is to prove the existence of many families of periodic solutions using Continuation’s method, where the perturbing parameter is related with the energy of the primaries. This work is merely analytic and uses symmetry conditions and appropriate coordinates. Partially supported by Dirección de Investigación UBB, 064608 3/RS.     

Stability of triangular libration points in a planar restricted elliptic three body problem in cases of double resonances

Stability of triangular libration points in a planar restricted elliptic three-body problem is investigated for two sets of parameters corresponding to the cases of double resonances. These positions are shown to be formally stable in the case of the fourth-order double resonance. When a weak third-order and a strong fourth-order resonances occur in the system, any motion of an approximate autonomous system is found to be bounded if initial conditions are sufficiently close to the unperturbed motion. Boundedness domain estimate is obtained.     

Linear Stability of Elliptic Lagrangian Solutions of the Planar Three-Body Problem via Index Theory

It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter \({\beta=27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2 \in [0, 9]}\) and the eccentricity \({e \in [0, 1)}\) . We are not aware of any existing analytical method which relates the linear stability of these solutions to the two parameters directly in the full rectangle [0, 9] × [0, 1), aside from perturbation methods for e > 0 small enough, blow-up techniques for e sufficiently close to 1, and numerical studies. In this paper, we introduce a new rigorous analytical method to study the linear stability of these solutions in terms of the two parameters in the full (β, e) range [0, 9] × [0, 1) via the ω-index theory of symplectic paths for ω belonging to the unit circle of the complex plane, and the theory of linear operators. After establishing the ω-index decreasing property of the solutions in β for fixed \({e\in [0, 1)}\) , we prove the existence of three curves located from left to right in the rectangle [0, 9] × [0, 1), among which two are ?1 degeneracy curves and the third one is the right envelope curve of the ω-degeneracy curves, and show that the linear stability pattern of such elliptic Lagrangian solutions changes if and only if the parameter (β, e) passes through each of these three curves. Interesting symmetries of these curves are also observed. The linear stability of the singular case when the eccentricity e approaches 1 is also analyzed in detail.     

Stability of Singular Periodic Motions in a Vibro-Impact Oscillator

A single-degree-of-freedom vibro-impact oscillator is considered. Forsome values of parameters, a non-differentiable fixed point of thePoincaré map exists: a local expansion of the Poincaré map around such apoint is given, including a square root term on the impact side. Fromthis approximate map, the stability of the fixed point can beinvestigated, and it is shown that the periodic solution is stable whenthe Floquet multipliers are real.     

Birth of periodic and artificial halo orbits in the restricted three-body problem

We investigate the bifurcation of artificial halo orbits from the Lyapunov planar family of periodic orbits around the collinear libration points of the circular, spatial, restricted three-body problem. Beside the gravitational forces, our model includes also the effect of the Solar Radiation Pressure (SRP) and this motivates the use of the term ‘artificial’ halo orbits. Indeed, as a typical problem, one may think of a solar sail, which is characterized by a performance parameter measuring the strength of the effect of the SRP on the spacecraft.To settle the model, we determine the position of the collinear points as a function of the mass and performance parameters and the energy values at which Hill׳s surfaces allow for transit orbits between the primaries. To analyze the dynamics we use a consolidated procedure which consists in the computation of a resonant normal form, allowing the reduction to the center manifold and providing an integrable approximation of the Hamiltonian dynamical system. Finally, we compute the bifurcation thresholds of the 1:1 resonant periodic orbit families (which have the standard ‘halo’ orbits as their first member) as a function of the performance and mass parameters.The results show that SRP is indeed a relevant ingredient for new dynamical features and must definitely be considered when planning a mission of a solar sail with trajectories in the neighborhoods of collinear points.     

Twist Character of the Fourth Order Resonant Periodic Solution

In this paper, we will give, for the periodic solution of the scalar Newtonian equation, some twist criteria which can deal with the fourth order resonant case. These are established by developing some new estimates for the periodic solution of the Ermakov–Pinney equation, for which the associated Hill equation may across the fourth order resonances. As a concrete example, the least amplitude periodic solution of the forced pendulum is proved to be twist even when the frequency acroses the fourth order resonances. This improves the results in Lei et al. (2003). Twist character of the least amplitude periodic solution of the forced pendulm. SIAM J. Math. Anal. 35, 844–867.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.     

Spatial Stability for the Quasi-static Problem of Thermoelasticity

This paper is concerned with the quasi-static problem of thermoelasticity. The classical system of equations of thermoelasticity is a coupling of an elliptic equation with a parabolic equation. It poses some new mathematical difficulties. Here we study the exponential spatial decay of solutions. An upper bound for the amplitude in terms of the boundary and initial conditions is obtained. The extension of the spatial stability results to thermoelasticity of type III is also treated. Dedicated to C.O. Horgan on the occasion of his 60th birthdayMathematics Subject Classifications (2000) 74F05, 74G50.     

Solar sail dynamics in the three-body problem: Homoclinic paths of points and orbits

In this paper we consider the orbital dynamics of a solar sail in the Earth-Sun circular restricted three-body problem. The equations of motion of the sail are given by a set of non-linear autonomous ordinary differential equations, which are non-conservative due to the non-central nature of the force on the sail. We consider first the equilibria and linearisation of the system, then examine the non-linear system paying particular attention to its periodic solutions and invariant manifolds. Interestingly, we find there are equilibria admitting homoclinic paths where the stable and unstable invariant manifolds are identical. What is more, we find that periodic orbits about these equilibria also admit homoclinic paths; in fact the entire unstable invariant manifold winds off the periodic orbit, only to wind back onto it in the future. This unexpected result shows that periodic orbits may inherit the homoclinic nature of the point about which they are described.     

二维弹性接触问题中随机边界元法与可靠性分析

发展了二维弹性接触问题中的随机边界元法，推导并建立了相应的随机边界元基本方程，并将所发展的方法用于静强度的可靠性分析，讨论了其数值解技术。通过算例分析表明，本文发展的方法是可行的。     

Convergence of a Modified SLP Algorithm for the Extended Linear Complementarity Problem

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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.     

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     

系泊海洋平台周期运动倍周期分岔的胞映射分析

应用胞映射方法研究了系泊海洋生产平台的周期运动及其倍周期分岔。系泊运动的数学模型是一个具有指数回复力特性的非线性强迫振子 ,以波浪作用力为外激励。将波浪激励周期作为分岔控制参数 ,研究了周期系泊运动的倍周期分岔。胞映射方法用于寻找系统的稳定吸引子并确定其吸引域。时间历程、相图、功率谱和Poincar啨映射用于确定吸引子的具体类型芯糠⑾?,分岔参数处于不同的区域时 ,系统存在着相异的倍周期分岔特性。观察到了倍周期分岔的产生和突然消失 ,也找到了一个趋于吸引子的倍周期分岔序列。根据吸引域的胞映射分析结果解释了上述不同的倍周期分岔特征。发现其原因在于倍周期序列中的每个吸引子是否具有全局吸引性。     

Some Properties of the Boundary Value Problem of Linear Elasticity in Terms of Stresses

The relation between the classical formulation of linear elastic problems in displacements and the stress formulation proposed by Pobedria is studied. It is shown that if the Navier and Pobedria differential operators are elliptic then corresponding boundary value problems are equivalent. The values of parameters for which Pobedria's boundary value problem has the Fredholm property are found. The homogeneous Pobedria's system is considered as a spectral problem with Poisson's ratio as a spectral parameter. The points of the essential spectrum are found and classified. The example of solving Pobedria's system for the Lamé problem for a spherical shell is presented.     

一般实矩阵特征值问题的多维灵敏度分析

应用Kronecker代数和矩阵微分理论系统发展了一般实矩阵特征值问题的矩阵灵敏度分析方法，结构系统特征值和特征向量的导数很方便地排列成二维矩阵，所得的结果很容易形成计算机程序。     

阻尼与纵向共振对结构动力稳定的影响

推导了有阻尼与发生纵向共振时柱状结构的动力稳定计算公式 ,计算了水中平台动力失稳区间 ,探讨了阻尼与纵向共振的影响 ,得出了一些有应用价值的结果     

二阶线性随机微分方程的渐近稳定性

对刚度系数是遍历过程的二阶线性随机微分方程，本文研究了其平凡解几乎处处渐近稳定性问题。利用刚度系数导数过程的性质，给出了平凡解几乎处处渐近稳定的充分条件。当刚度系数是遍历高斯过程或周期过程时，还具体计算了其渐进稳定区域。结果表明，本文结果改进了目前有关的渐近稳定性的条件。     

一种调整线性振动系统特征参数的方法

在设计一个线性振动系统时,为了获得某些希求的特征参数;或者为了调整数学模型的特征值,使能与试验值更好地符合,往往需要对原数学模型作些修改。利用线性规划,本文给出了一种求解逆特征值问题的迭代方法。通过算例说明本文方法是对线性振动系统进行再设计的一种有效和可靠的方法。