Melnikov方法和圆型平面限制性三体问题的横截同宿研究<sup>*</sup>

Melnikov方法和圆型平面限制性三体问题的横截同宿研究^*

基金项目: * 国家自然科学基金资助项目

1.
LNM, Institute of Mechanics, Academia Sinica, Beijing 100080, P. R. China;
2.
Department of Mechanical Engineering, Yale University, New Haven, CT 06520, U. S. A

摘要: 本文对由两自由度近可积哈密顿系统经非正则变换而得到的,具有高阶不动点的非哈密顿系统给出了判别横截同宿轨和横截异宿轨存在性的两条判据。对原二体质量比很小时近可积圆型平面限制性三体问题,采用本文判据证明存在横截同宿轨,从而存在横截同宿穿插现象;还在一定假设下证明了存在横截异宿轨;并给出了全局定性相图。
- 限制性三体问题 /
- 近可积哈密顿系统 /
- 高阶不动点 /
- Melnikov方法 /
- 横截同宿(异宿)轨
Abstract: Non-Hamiltonian systems containing degenerate fixed points obtained from twodegrees of freedom near-integrable Hamiltonian systems through non-canonicaltransformations are dealt with in this paper. Two criteria.for determining theexistence of transversal homoclinic and heteroclinic orbits are presented. By exploitingthese criteria the existence of the transversal homoclinic orbits and so, of thetransversal homoclinic tangle.phenomenon in the near-integrable circular planarrestricted three-body problem with sufficiently small mass ratio of the two primaries isproven. Under some assumptions, the existence of the transversal heleroclinic orbits isproven. The global qualitative phase diagram is also illustrated.
- restricted three-body problem /
- near integrable Hamiltoniansystem /
- degenerate fixed point /
- Melnikov method /
- transversalhomoclinic(heteroclinic)orbit

[1]	A.Wintner,The Analytical Foundations of Celestial Mechanics,Princeton Univ.Press(1947).
[2]	T.M.Cherry,On Poincare's theorem of "the non-existence of uniform integrals of dynamical equation",Proc.Cambridge Phil.Soc.,22(1924).
[3]	V.Szebehely,Long Time Prediction in Dynamics,Ed.by W.Horton and L.Richl,Wiley,Austin(1981).
[4]	V.Szebehely,Is celestial mechanics deterministicϒ Applications of Modern Dynamics to Celestial Mechanics and Astrodynamics,Ed.by Szebehely,D.Reidel Pub.Company(1982),321-324.
[5]	R.Easton and R.McGehee,Homoclinic phenomena for orbits doubly asymptotic to an invariant three-sphere,Indiana U.Math.J.,28(1979),211-240.
[6]	R.Easton,Capture orbits and Melnikov integrals in the planar three-body problem,Cele.Mech.and Dyn.Astro.,50(1991),283-297.
[7]	R.McGehee,A stable manifold theorem for degenerate fixed points with application to celestial mechanics,J.DiJJ.Eq.,14(1973),70-88.
[8]	J.Guckenheimer and P.Holmes,Nonlinear Oscillations,Dynamical Systems,and Bifurcations of Vector Fields,Springer-Verlag(1983).
[9]	S.Wiggins,Global Bifurcations and Chaos,Springer-Verlag(1988).