Breakdown of homoclinic orbits to L3: Nonvanishing of the Stokes constant

Abstract

The Restricted Planar Circular 3-Body Problem models the mo- tion of a body of negligible mass under the gravitational influence of two mas- sive 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 L3 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 L3 in a transverse section for small values of the mass ratio 0 < μ ≪ 1. This distance is exponentially small with respect to μ 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.