| Abstract: |
| Non hyperbolic fixed points and non hyperbolic periodic orbits play and important role in the dynamics of several well known problems.
In the context of celestial mechanics, quite often objects at infinity are invariant but non hyperbolic. However, it has been
proven with several degrees of generality that these objects, although degenerate, may have invariant stable and unstable manifolds
that govern the dynamics of the system at points close to them. These type of objects also appear in problems of chemistry and economics.
In this work we consider \emph{parabolic tori}, that is, tori such that the vector field vanishes in the normal directions up to certain order, and give conditions under which these tori have invariant stable and unstable manifolds with more than one stable or unstable directions. We require the dynamics on the tori to be conjugated to a rigid rotation with Diophantine frequency vector. We find the manifolds using the parametrization method after finding suitable approximate solutions of the corresponding invariance equation.
We apply the abstract theorem to the case of the planar $(n+2)$-body problem. We prove that, for any KAM tori of the $n$-body problem, if the masses of the last two bodies are small enough, there exist solutions defined for all $t>0$ such that the first $n$ bodies tend to the prescribed KAM tori while the last two go to infinity and arrive there with zero velocity (that is, they have a parabolic motion). The set of solutions satisfying these properties is a manifold whose dimension depends on the final configuration. |
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