Order and chaos in the planar isosceles three-body problem

Abstract

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.