Coorbital homoclinic and chaotic dynamics in the Restricted 3-Body Problem

Abstract: The description of unstable motions in the Restricted Planar Circular 3-Body Problem, modeling the dynamics of a Sun-Planet-Asteriod system, is one of the fundamental problems in Celestial Mechanics. The goal of this paper is to analyze homoclinic and instability phenomena at coorbital motions, that is when the negligible mass Asteroid is at 1:1 mean motion resonance with the Planet (i.e. nearly equal periods) and performs close to circular motions. Several bodies in our Solar system belong to such regimes. In this paper, we obtain the following results. First, we prove that, for a sequence of ratios between the masses of the Planet and the Sun going to 0, there exist a 2-round homoclinic orbit to the Lagrange point L3, i.e. homoclinic orbits that approach the critical point twice. Second, we construct chaotic motions (hyperbolic sets with symbolic dynamics) as a consequence of the existence of transverse homoclinic orbits to Lyapunov periodic orbits associated to L3. Finally, we prove that the RPC3BP possesses Newhouse domains by proving that the energy level unfolds generically a quadratic homoclinic tangency to a periodic orbit.