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The Restricted Three-Body Problem as a Perturbed Duffing Equation

Rongchang Liu, Qiudong Wang

TL;DR

The paper analyzes transversal homoclinic intersections in the restricted circular planar three-body problem by recasting the dynamics near parabolic orbits as a perturbed Duffing equation. It develops integral equations for primary stable and unstable manifolds via a CW-type framework and computes a Poincaré–Melnikov integral using the explicit unperturbed Duffing homoclinic solution, establishing analytic dependence on the Jacobi constant and mass ratio. The main result shows that for a fixed large negative Jacobi constant $J$, there exists a $J_0<0$ such that for all $J<J_0$ and all but finitely many mass ratios, the parabolic invariant surface splits to yield transversal homoclinic intersections; crucially, the required $|J_0|$ is uniform across mass ratios. The approach is self-contained and yields a simpler, direct argument compared to prior work, producing uniform results across mass ratios and avoiding reliance on McGehee's analysis.

Abstract

This paper investigates the restricted circular planar three-body problem. We prove that for every negative Jacobi constant of sufficiently large magnitude, the surface of unperturbed parabolic solutions breaks to induce homoclinic tangle for all but at most finitely many mass ratios of primaries. This result is not covered by \cite{G} as the required large magnitude of the Jacobi constant is uniform across the mass ratios. Our approach consists of three main ingredients. First, by introducing new coordinate transformations, we reformulate the restricted three-body problem as a perturbed Duffing equation. Second, we adopt the method recently introduced in \cite{CW} to derive integral equations for the primary stable and unstable solutions. This enables us to effectively capture the order of the singularities involved and to further establish the existence and analytic dependence of the invariant manifolds on the mass ratio of the primaries and the Jacobi constant. Third, in evaluating the Poincaré-Melnikov integral, we take advantage of the explicit homoclinic solution of the unperturbed Duffing equation. Compared to existing works, our proof is significantly simpler and more direct. Moreover, the paper is self-contained: we do not rely on McGehee's analysis in \cite{Mc} to justify the applicability of the Poincaré-Melnikov method.

The Restricted Three-Body Problem as a Perturbed Duffing Equation

TL;DR

The paper analyzes transversal homoclinic intersections in the restricted circular planar three-body problem by recasting the dynamics near parabolic orbits as a perturbed Duffing equation. It develops integral equations for primary stable and unstable manifolds via a CW-type framework and computes a Poincaré–Melnikov integral using the explicit unperturbed Duffing homoclinic solution, establishing analytic dependence on the Jacobi constant and mass ratio. The main result shows that for a fixed large negative Jacobi constant , there exists a such that for all and all but finitely many mass ratios, the parabolic invariant surface splits to yield transversal homoclinic intersections; crucially, the required is uniform across mass ratios. The approach is self-contained and yields a simpler, direct argument compared to prior work, producing uniform results across mass ratios and avoiding reliance on McGehee's analysis.

Abstract

This paper investigates the restricted circular planar three-body problem. We prove that for every negative Jacobi constant of sufficiently large magnitude, the surface of unperturbed parabolic solutions breaks to induce homoclinic tangle for all but at most finitely many mass ratios of primaries. This result is not covered by \cite{G} as the required large magnitude of the Jacobi constant is uniform across the mass ratios. Our approach consists of three main ingredients. First, by introducing new coordinate transformations, we reformulate the restricted three-body problem as a perturbed Duffing equation. Second, we adopt the method recently introduced in \cite{CW} to derive integral equations for the primary stable and unstable solutions. This enables us to effectively capture the order of the singularities involved and to further establish the existence and analytic dependence of the invariant manifolds on the mass ratio of the primaries and the Jacobi constant. Third, in evaluating the Poincaré-Melnikov integral, we take advantage of the explicit homoclinic solution of the unperturbed Duffing equation. Compared to existing works, our proof is significantly simpler and more direct. Moreover, the paper is self-contained: we do not rely on McGehee's analysis in \cite{Mc} to justify the applicability of the Poincaré-Melnikov method.
Paper Structure (17 sections, 20 theorems, 219 equations)

This paper contains 17 sections, 20 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Derivation of Equations Around Parabolic Solution
  3. Equations for the Restricted Three-body Problem
  4. Reduction to a Perturbed Duffing Equation
  5. Equations Around Unperturbed Parabolic Solution
  6. Existence and Analyticity of Invariant Manifolds
  7. Canonical Coordinates and Integral Equations
  8. Existence and Analytical Dependence on $\theta_0, \rho, \varepsilon$
  9. Existence of Transversal Homoclinic Intersections
  10. Splitting Distance and Its First Approximation
  11. Fourier Expansion of ${\mathbb D}_0(\theta_0)$
  12. Estimates On $I^{(j)}_{n,m,\mathrm{odd}}$, $I^{(j)}_{n,m,\mathrm{even}}$ and $J_{n,m,\mathrm{odd}}$, $J_{n,m,\mathrm{even}}$
  13. Expansions in $\varepsilon$
  14. Expansions of $R_{13}$ and $R_{23}$
  15. Expansions of $U,F,G$
  16. ...and 2 more sections

Key Result

Lemma 2.1

For all solutions of equation diff-eqn2, is an integral constant, which is commonly referred to as the Jacobi constant.

Theorems & Definitions (39)

  • Lemma 2.1
  • Definition 3.1
  • Proposition 3.2
  • Remark 1
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Remark 2
  • Proposition 3.5
  • ...and 29 more