Melnikov方法和圆型平面限制性三体问题的横截同宿研究

Studies of Melnikov Method and Transversal Homoclinic Orbits in the Circular Planar Restricted Three-Body Problem

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。

Non-Hamiltonian systems containing degenerate fixed points obtained from two degrees of freedom near-integrable Hamiltonian systems through non-canonical transformations are dealt with in this paper. Two criteria for determining the existence of transversal homoclinic and heteroclinic orbits are presented. By exploiting these criteria the existence of the transversal homoclinic orbits and so, of the transversal homoclinic tangle phenomenon in the near-integrable circular planar restricted three-body problem with sufficiently small mass ratio of the two primaries is proven. Under some assumptions, the existence of the transversal heteroclinic orbits is proven. The global qualitative phase diagram is also illustrated.