Arnold diffusion in the elliptic Hill four-body problem: geometric method and numerical verification

Abstract

We present a mechanism for Arnold diffusion in energy in a model of the elliptic Hill four-body problem. Our model is expressed as a small perturbation of the circular Hill four-body problem, with the small parameter being the eccentricity of the orbits of the primaries. The mechanism relies on the existence of two normally hyperbolic invariant manifolds (NHIMs), and on the corresponding homoclinic and heteroclinic connections. The dynamics along homoclinic/heteroclinic orbits is encoded via scattering maps, which we compute numerically. We provide two arguments for diffusion. In the first argument, we successively apply a single scattering map and use Birkhoff's Ergodic Theorem to obtain pseudo-orbits that, on average, gain energy. In the second argument, we use two scattering maps and, at each step, select one that increases energy. Either argument yields pseudo-orbits of scattering maps along which the energy grows by an amount independent of the small parameter. A shadowing lemma concludes the existence of diffusing orbits.