Otto Vaughn Osterman (University of Maryland)
We consider the planar circular restricted three-body problem, modeling the motion of a massless asteroid in the plane undergoing gravitational attraction toward two bodies, each of which moves in a circular path around their common center of mass. For small mass ratios, the motion of the asteroid is approximated by the Kepler problem when the asteroid is far from a collision, and a large set of Kepler motions in which the paths of the asteroid and the smaller body do not intersect persist as quasi-periodic motions in the perturbed system. However, these quasi-periodic motions with incommensurable frequencies are not possible for Kepler motions in which the paths intersect due to the potential for close interactions between the asteroid and the smaller body. The existence of hyperbolic sets in which the asteroid repeatedly comes close to a collision was proven independently by Bolotin and MacKay and by Font, Nunes, and Simó. My result, currently in preparation, is that there exists stable motions of the asteroid near resonant Kepler orbits in which the asteroid repeatedly undergoes close interactions with the smaller body.