The Method of Spreading Cumulative Twist and Its Application to the Restricted Circular Planar Three Body Problem

The purpose of this paper is twofold. First we show that the dynamics of a Sun-Jupiter-Comet system and under some simplifying assumptions has a semi-infinite region of instability. This is done by reducing the dynamics to the study of a certain exact area preserving (EAP) map and showing applicability of Aubry–Mather theory. Second, we give a sufficient “cumulative twist” condition to verify that an EAP map is an EAP twist (EAPT) map in a certain coordinate system. We construct such a coordinate system by “spreading the cumulative twist” which arises from the long term dynamics of system. These results along with the prequel paper [GK1] show that the aforementioned EAPT map admits an application of Aubry–Mather theory and has no invariant curves in a certain semi-infinite region. This, in particular, leads to existence of orbits of the Comet with initial eccentricity 0.66 and “visible in the Solar system” (see figure1) which get ejected to infinity. Alternatively certain orbits of the Comet can come from infinity and be captured so that they approach orbits of eccentricity 0.66 in the future. More generally, Aubry–Mather theory implies that in the instability region above eccentricity 0.66 there are all possible Chazy instabilities (see Section 1.1 for details).