Zare K., Chesley S.
Department of Aerospace Engineering and Engineering Mechanics, University of Texas at Austin, Austin, Texas 78712.
Chaos. 1998 Jun;8(2):475-494. doi: 10.1063/1.166329.
The planar isosceles three-body problem has been reduced to a two-dimensional area preserving Poincare map f. Using certain symmetry properties of the underlying differential equations and numerical integration, we offer a global description of f in the case of three equal masses. This description, which is based on the mapping of areas, immediately leads to the existence of various types of motion such as capture-escape, permanent capture, ejection-collision, etc., and their corresponding measures in the map domain. Moreover, this technique readily allows one to distinguish between so-called "fast" and "chaotic" scattering. Although capture-escape is the subset with the highest measure, there exist two important distinct invariant subsets under f where the solutions neither are captured nor lead to escape. The first set is a Cantor set which has zero measure and it is the outcome of the fact that f acts similar to the Smale horseshoe map in part of the domain. On this subset the action of f is chaotic. The second subset is an invariant region with positive measure surrounding an elliptic fixed point. In this region f acts essentially as a perturbed twist mapping where the iterates of f for the points in a large subset move on invariant curves in an orderly manner. In an appendix we cast our results in the framework of the widely studied isosceles triple collision manifold. (c) 1998 American Institute of Physics.
平面等腰三体问题已被简化为一个二维保面积的庞加莱映射f。利用基础微分方程的某些对称性以及数值积分，我们给出了三个等质量情况下f的全局描述。这种基于面积映射的描述直接导致了各种类型运动的存在，如捕获 - 逃逸、永久捕获、弹射 - 碰撞等，以及它们在映射域中的相应测度。此外，该技术很容易让人区分所谓的“快速”散射和“混沌”散射。尽管捕获 - 逃逸是测度最高的子集，但在f作用下存在两个重要的不同不变子集，其解既不被捕获也不导致逃逸。第一个集合是一个测度为零的康托集，它是f在部分域中行为类似于斯梅尔马蹄映射的结果。在这个子集中，f的作用是混沌的。第二个子集是围绕一个椭圆不动点的具有正测度的不变区域。在这个区域中，f本质上作为一个受扰扭转映射，对于一个大子集中的点，f的迭代在不变曲线上有序移动。在附录中，我们将我们的结果置于广泛研究的等腰三体碰撞流形的框架中。（c）1998美国物理研究所。
