Resonance transitions associated to weak capture in the restricted three-body problem

An interesting dynamics is studied in the restricted three-body problem where a particle abruptly transitions between resonance states, called a resonance hop. It occurs in a region about the secondary mass point which supports weak capture. This region, called a weak stability boundary, was recently proven to give rise to chaotic dynamics. Although it was numerically known that the resonance hop was associated with this boundary, this process was not well understood. In addition, the dynamical structure of the weak stability boundary has not been well understood. In this paper, we give a way to reveal the global structure of the weak stability boundary associated to resonance motions. This structure is shown to be surprisingly rich in resonant periodic motions interconnected by invariant manifolds. In this case, nearly all the motions are approximately resonant in nature where resonance hops can occur. The correlation dimension of orbits undergoing resonant motions, associated to the weak stability boundary, is also examined. The dynamics analyzed in the present paper is related to that studied by J. Marsden et al. under the perspective of Lyapunov orbits and the associated invariant manifolds. Applications are discussed.