Cantor Set Structure of the Weak Stability Boundary for Infinitely Many Cycles in the Restricted Three-Body Problem

TL;DR

This paper demonstrates that the weak stability boundary in the planar restricted three-body problem, for infinitely many cycles, has a fractal structure composed of Cantor sets, sharing properties with the Mandelbrot set and relating to the non-existence of KAM tori.

Contribution

It extends the understanding of the weak stability boundary to infinitely many cycles, revealing its fractal Cantor set structure and properties similar to the Mandelbrot set.

Findings

Boundary consists of infinitely many Cantor sets

Region with bounded cycling motion is bounded by this fractal boundary

Shares properties with the Mandelbrot set and relates to non-existence of KAM tori

Abstract

The geometry of the weak stability boundary region for the planar restricted three-body problem about the secondary mass point has been an open problem. Previous studies have conjectured that it may have a fractal structure. In this paper, this region is studied for infinitely many cycles about the secondary mass point, instead of a finite number studied previously. It is shown that in this case the boundary consists of a family of infinitely many Cantor sets and is thus fractal in nature. It is also shown that on two-dimensional surfaces of section, it is the boundary of a region only having bounded cycling motion for infinitely many cycles, while the complement of this region generally has unbounded motion. It is shown that that this shares many properties of a Mandelbrot set. Its relationship to the non-existence of KAM tori is described, among many other properties. Applications are discussed.