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Title:
Invariant stable manifolds associated to parabolic objects with applications to Celestial Mechanics
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Abstract:
In this talk we present some results on the existence and regularity of invariant manifolds (or whiskers) for either equilibrium points or invariant tori (including periodic orbits) whose normal directions are neutral in the sense that the corresponding eigenvalues are one. We call these objects parabolic. The sufficient conditions for their existence come from higher order terms in the Taylor expansion of the normal dynamics. We consider both the case of maps and the case of differential equations with special attention for the maps case. Some of the analytical results are presented in \textit{a posteriori} format, meaning that if we have a sufficiently good approximation of the manifolds together with some non-degeneracy conditions, then there is a true manifold nearby. A methodology to find this approximated manifold is also provided by using the celebrated parametrization method. Some examples showing the differences between the \textit{non degenerate} (hyperbolic) case and our case are also provided along this talk. When the stable invariant manifold associated to the eigenvalue one, is one dimensional, we prove the Gevrey character of this manifolds. We apply our result to prove that the set of parabolic orbits in the Sitnikov problem and in the planar restricted three body problem is an invariant manifold which is $1/3$-Gevrey. The ``whiskers'' (stable and unstable manifolds) of Diophantine parabolic tori are present in several problems of celestial mechanics for open sets of the mass parameters. More concretely, we will show that one can find them in the restricted planar $n$-body problem and in the full planar $n$-body problem, for any $n\ge 1$, at least for values of the masses corresponding to the planetary problem (that is, where one mass is much larger than the rest). The whiskers of these parabolic tori have an important role in the dynamics of the $n$-body problem and may be used to obtain oscillatory orbits (solutions where one of the masses goes closer and closer to infinity but always returning to a fixed neighborhood of the rest) or diffusion orbits (solutions where the semiaxis of the orbits change along time). This is a joint work with E. Fontich and P.Martin.
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