From the circular to the spatial elliptic restricted three-body problem

We deal with the study of the spatial restricted three-body problem in the case where the small particle is far from the primaries, that is, the so-called comet case. We consider the circular problem, apply double averaging and compute the relative equilibria of the reduced system. It appears that, in the circular problem, we find not only part of the equilibria existing in the elliptic case, but also new ones. These critical points are in correspondence with periodic and quasiperiodic orbits and invariant tori of the non-averaged Hamiltonian. We explain carefully the transition between the circular and the elliptic problems. Moreover, from the relative equilibria of elliptic type, we obtain invariant 3-tori of the original system.     

Effect of oblateness on the non-linear stability of the triangular liberation points of the restricted three-body problem in the presence of resonances

The non-linear stability of the triangular libration points of the restricted three-body problem is studied under the presence of third and fourth order resonance's, when the more massive primary is an oblate spheroid. In this study Markeev's theorem are utilised with the help of KAM theorem. It is found that the stability of the triangular libration points are unstable in the third order resonance case and stable in the fourth order resonance case, for all the values of oblateness factor A₁. This revised version was published online in August 2006 with corrections to the Cover Date.     

我国网络经济的发展及趋势分析

网络经济是区别于自然经济、工业经济的人类历史发展的新阶段,狭义上说,它应该是IT产业的总称.我国网络经济正处在大发展阶段的前夜;网络技术的运用需要解决网络安全问题,网络经济的发展需要解决相应的制度创新问题;梯度演进与梯度发展将是我国网络经济发展的新格局.     

Symplectic Mapping for Trojan-Type Motion in the Elliptic Restricted Three-Body Problem

A symplectic mapping for Trojan-type motion has been developed in the secularly changing elliptic restricted three-body problem. The mapping describes well the characteristics of Trojan-type dynamics at small eccentricities. By using this mapping the boundary of the stability region has been studied for different values of the initial eccentricities of hypothetical Jupiter's Trojans. It has been found that in the secularly changing elliptic case the chaotic diffusion at the border of the stability region is stronger than simply in the elliptic case. An explanation of this observation might be the destruction of the chain of islands of the 13:1 secondary resonance between the short and long period component of the Trojan-like motion, caused possibly by the indirect perturbations of Saturn.     

一个二阶奇异初值问题的正解

利用不等式和对角序列，研究了一个显含一阶导数的二阶奇异初值问题非正解的存在性。     

The scattering map in the planar restricted three body problem

We study homoclinic transport to Lyapunov orbits around a collinear libration point in the planar restricted three body problem. A method to compute homoclinic orbits is first described. Then we introduce the scattering map for this problem (defined on a suitable normally hyperbolic invariant manifold) and we show how to compute it using the information already obtained for the homoclinic orbits. An example application to Astrodynamics is also proposed.     

On some exceptional cases in the integrability of the three-body problem

We consider the Newtonian planar three-body problem with positive masses m ₁, m ₂, m ₃. We prove that it does not have an additional first integral meromorphic in the complex neighborhood of the parabolic Lagrangian orbit besides three exceptional cases ∑m _i m _j /(∑m _k )² = 1/3, 2³/3³, 2/3² where the linearized equations are shown to be partially integrable. This result completes the non-integrability analysis of the three-body problem started in papers [Tsygvintsev, A.: Journal für die reine und angewandte Mathematik N 537, 127–149 (2001a); Celest. Mech. Dyn. Astron. 86(3), 237–247 (2003)] and based on the Morales–Ramis–Ziglin approach.     

Critical Periodic Orbits in the Restricted Three–Body Problem With Oblateness

In a recent paper in this journal series of horizontally critical symmetric periodic orbits of the six basic families of the photogravitational restricted three–body problem were computed (Perdios, 2003). In this paper, such series are determined in the framework of the restricted three–body problem when the more massive primary is an oblate spheroid. The vertical stability of the horizontally critical orbits is also computed.     

关于二阶非线性奇异边值问题的正解

提供二阶奇异边值问题：x″＋f（t,x,x′）=0,0＜f＜1,x（0）=x′（1）=0正解的一些不存在结果.     

Computing with certainty individual members of families of periodic orbits of a given period

The accurate computation of families of periodic orbits is very important in the analysis of various celestial mechanics systems. The main difficulty for the computation of a family of periodic orbits of a given period is the determination within a given region of an individual member of this family which corresponds to a periodic orbit. To compute with certainty accurate individual members of a specific family we apply an efficient method using the Poincaré map on a surface of section of the considered problem. This method converges rapidly, within relatively large regions of the initial conditions. It is also independent of the local dynamics near periodic orbits which is especially useful in the case of conservative dynamical systems that possess many periodic orbits, often of the same period, close to each other in phase space. The only computable information required by this method is the signs of various function evaluations carried out during the integration of the equations of motion. This method can be applied to any system of celestial mechanics. In this contribution we apply it to the photogravitational problem.