Article Contents

2025, Volume 45, Issue 1: 56-88. Doi: 10.3934/dcds.2024086

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Breakdown of homoclinic orbits to $ L_3 $: Nonvanishing of the Stokes constant

1.
Departament de Matemàtiques & IMTECH, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain
2.
Faculty of Applied Mathematics, AGH University of Kraków, al. Mickiewicza 30, 30-059 Kraków, Poland
3.
IMCCE, CNRS, Observatoire de Paris, Université PSL, Sorbonne Université, 77 Avenue Denfert-Rochereau, 75014 Paris, France
4.
Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via, 585, 08007 Barcelona, Spain
5.
Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Barcelona, Spain

^*Corresponding author: Mar Giralt
^*Corresponding author: Mar Giralt

Received: December 2023

Revised: June 2024

Early access: July 2024

Published: January 2025

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Abstract

The Restricted Planar Circular 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies, called the primaries, which perform circular orbits coplanar with that of the massless body. In rotating coordinates, it can be modelled by a two degrees of freedom Hamiltonian system, which has five critical points called the Lagrange points. Among them, the point $ L_3 $ is a saddle-center which is collinear with the primaries and beyond the largest of the two. The papers [3,4] provide an asymptotic formula for the distance between the one dimensional stable and unstable manifolds of $ L_3 $ in a transverse section for small values of the mass ratio $ 0 < \mu\ll 1 $. This distance is exponentially small with respect to $ \mu $ and its first order depends on what is usually called a Stokes constant. The non-vanishing of this constant implies that the distance between the invariant manifolds at the section is not zero. In this paper, we prove that the Stokes constant is non-zero. The proof is computer assisted.

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Mathematics Subject Classification: 37J46, 37N05, 37M21.

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Figure 1. Projection onto the $ q $-plane of the Lagrange equilibrium points for the RPC3BP on rotating coordinates

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Figure 2. Projection onto the $ q $-plane of the unstable (red) and stable (green) manifolds of $ L_3 $, for $ {\mu = 0.0028} $

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Figure 3. The inner domain, $ \mathcal{D}_\kappa^{\mathrm{u}} $, for the unstable case (see (10))

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Figure 4. The bound on the domain on $ U $ within which our trajectory resides is depicted as the black box. The red line is $ \text{Im}\; U = -\rho_{0} $. In blue we have a non-rigorous plot of the trajectory, which is added to the figure as a point of reference

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Figure 5. A closeup of the crossing of the trajectory through the section $ \{\text{Re}\; U = 0\} $ projected onto $ U $ on the left (compare with Figure 4), and onto coordinates $ (\text{Re}\; U, \text{Re}\; Y $ on the right. In black we have the interval arithmetic bounds. In green, we have singled out the bounds on the trajectory for two disjoint time intervals, to demonstrate that it indeed does cross $ \{\text{Re}\; U = 0\} $. In blue we have a non-rigorous plot of the trajectory, which is added to the figure as a point of reference

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