Homoclinic Solutions to Infinity and Oscillatory Motions in the Restricted Planar Circular Three Body Problem

Abstract

The circular restricted three body problem models the motion of a massless body under the influence of the Newtonian gravitational force caused by two other bodies, the primaries, which describe circular planar Keplerian orbits. The system has a first integral, the Jacobi constant. The existence of oscillatory motions for the restricted planar circular three body problem, that is, of orbits which leave every bounded region but which return infinitely often to some fixed bounded region, was proved by Llibre and Simo [18] in 1980. However, their proof only provides such orbits for values of the ratio between the masses of the two primaries exponentially small with respect to the Jacobi constant. In the present work, we extend their result proving the existence of oscillatory motions for any value of the mass ratio. The existence of these motions is a consequence of the transversal intersection between the stable and unstable manifolds of infinity, which guarantee the existence of a symbolic dynamics that creates the oscillatory orbits. We show that this intersection does happen for any value of the mass ratio and for big values of the Jacobi constant. We remark that, since in our setting the mass ratio is no longer small, this transversality cannot be checked by means of classical perturbation theory respect to the mass ratio. Furthermore, since our method is valid for all values of mass ratio, we are able to detect a curve in the parameter space, formed by the mass ratio and the Jacobi constant, where cubic homoclinic tangencies between the invariant manifolds of infinity appear.