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

Edward Belbruno

TL;DR

This work investigates the fractal geometry of the weak stability boundary W' for infinite cycling about the secondary body in the planar circular restricted three-body problem. By constructing stable/unstable cycling sets and their limits on Poincaré sections via a line L(θ) from P2 and a real-analytic map Φ, it shows that W' is an infinite union of Cantor sets across energy slices, while the interior Ŝ corresponds to trajectories cycling infinitely many times in a stable manner. The boundary of the invariant region M^* equals W', and M^* exhibits Mandelbrot-like fractal properties, with a rigorous NoKAM result stating that points on ∂M^* cannot lie on KAM tori. These findings illuminate how invariant-manifold tangles and fractal boundaries govern long-term dynamics in gravitational systems and suggest potential applications to low-energy permanent capture trajectories.

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.

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

TL;DR

This work investigates the fractal geometry of the weak stability boundary W' for infinite cycling about the secondary body in the planar circular restricted three-body problem. By constructing stable/unstable cycling sets and their limits on Poincaré sections via a line L(θ) from P2 and a real-analytic map Φ, it shows that W' is an infinite union of Cantor sets across energy slices, while the interior Ŝ corresponds to trajectories cycling infinitely many times in a stable manner. The boundary of the invariant region M^* equals W', and M^* exhibits Mandelbrot-like fractal properties, with a rigorous NoKAM result stating that points on ∂M^* cannot lie on KAM tori. These findings illuminate how invariant-manifold tangles and fractal boundaries govern long-term dynamics in gravitational systems and suggest potential applications to low-energy permanent capture trajectories.

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.
Paper Structure (16 sections, 12 theorems, 27 equations, 5 figures, 1 table)

This paper contains 16 sections, 12 theorems, 27 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Restricted Three-Body Problem
  3. Weak Stability Boundary and Related Sets
  4. Basic Definitions
  5. Sets $W$, $S$, $\hat{S}$, $W'$ and their Properties
  6. $\bm{W_n(\theta)}$, $\bm{W'(\theta)}$, Invariant Manifolds, Cantor Sets
  7. An Invariant Set for $\bm{\Phi}$
  8. Mappings and Properties
  9. Bounded Regions Defined by $\bm{W'}, \bm{\hat{S}}$
  10. Non-existence of KAM tori
  11. Discussion of Results
  12. Supporting Calculations
  13. Kepler Energy, $\bm{E_2}$
  14. $\bm{E_2 \geq 0}$ When $\bm{P}$ Cycles About $\bm{P_1}$
  15. Definition of $\bm{S_{\theta_0}}$
  16. ...and 1 more sections

Key Result

Theorem 3.5

If $C$ is sufficiently large ($C \in C_L$) then $\hat{S}(\theta)$ is non-empty.

Figures (5)

  • Figure 1: Rotating coordinate system $(y_1,y_2)$.
  • Figure 2: $W^{s,u}(\gamma_i)$, $i=1, 2$, and the flow of the trajectories towards or away from $\gamma_i$, $C \lessapprox C_2$. $P$ is shown moving between $H_1, H_2, H_O$ on transit orbits. This is an illustrative figure.
  • Figure 3: Line $L(\theta)$ emanating from $P_2$ making an angle $\theta$ with the $Y_1$-axis. 1-stable and 1-unstable motion for $n=1$ cycles for $P$ on the trajectory ${\bf{Y}}(t)$. This is an illustrative figure.
  • Figure 4: $\hat{S}(\theta), \ S^*(\theta), \ W'(\theta), \ \mathcal{C}.$ This is a rough sketch.
  • Figure 5: $W', \ \hat{S}$ on $P_{Y, \Lambda}$. This is a rough sketch.

Theorems & Definitions (12)

  • Theorem 3.5
  • Lemma 3.7
  • Lemma 3.12
  • Corollary 3.13
  • Lemma 3.14
  • Lemma 3.15
  • Lemma 3.16
  • Lemma 3.17
  • Lemma 4.1
  • Lemma 5.1
  • ...and 2 more