Invariant Manifolds of Degenerate Tori and Double Parabolic Orbits to Infinity in the \((n+2)\)-Body Problem

There are many interesting dynamical systems in which degenerate invariant tori appear. We give conditions under which these degenerate tori have stable and unstable invariant manifolds, with stable and unstable directions having arbitrary finite dimension. The setting in which the dimension is larger than one was not previously considered and is technically more involved because in such case the invariant manifolds do not have, in general, polynomial approximations. As an example, we apply our theorem to prove that there are motions in the \((n+2)\) $(n+2)$ -body problem in which the distances among the first n bodies remain bounded for all time, while the relative distances between the first n-bodies and the last two and the distances between the last bodies tend to infinity, when time goes to infinity. Moreover, we prove that the final motion of the first n bodies corresponds to a KAM torus of the n-body problem.