The Existence of Transverse Homoclinic Points in the Sitnikov Problem

Using Melnikov's method we are able to prove the existence of transverse homoclinic orbits and therefore the existence of a horseshoe in a special restricted three-body problem. This analysis is an alternative to the one described by Moser (“Stable and Random Motions in Dynamical Systems,„ Princeton Univ. Press, Princeton, NJ, 1973), based on Sitnikov’s original work (Dokl. Akad. Nauk. USSR 133, No. 2 (1960), 303-306), where the task is accomplished using a more direct construction of the horseshoe.