Parabolic saddles and Newhouse domains in Celestial Mechanics

TL;DR

This paper investigates the complex dynamics near parabolic orbits in the restricted 3-body problem, demonstrating the existence of Newhouse domains with rich bifurcation structures and dense elliptic islands, extending classical homoclinic bifurcation results.

Contribution

It extends classical homoclinic bifurcation theory to degenerate parabolic settings in celestial mechanics, establishing the existence of Newhouse domains with maximal Hausdorff dimension homoclinic classes.

Findings

Existence of Newhouse domains in the parameter space of the 3-body problem.

Homoclinic classes with maximal Hausdorff dimension are dense in these domains.

Elliptic islands form unbounded subsets of the phase space and accumulate hyperbolic sets.

Abstract

In the 70s McGehee introduced a compactification of the phase space of the restricted 3-body problem by gluing a manifold of periodic orbits "at infinity". Although from the dynamical point of view these periodic orbits are parabolic (the linearization of the Poincar\'{e} map is the identity matrix), one of them, denoted here by $O$, possesses stable and unstable manifolds which, moreover, separate the regions of bounded and unbounded motion. This observation prompted the investigation of the homoclinic picture associated to $O$, starting with the work of Alekseev and Moser. We continue this research and extend, to this degenerate setting, some classical results in the theory of homoclinic bifurcations. More concretely, we prove that there exist Newhouse domains $\mathcal N$ in parameter space (the ratio of masses of the bodies) and residual subsets $\mathcal R\subset \mathcal N$ for which the homoclinic class of $O$ has maximal Hausdorff dimension and is accumulated by generic elliptic periodic orbits. One of the main consequences of our work is the fact that, for a (locally) topologically large set of parameters of the restricted 3-body problem the union of its elliptic islands forms an unbounded subset of the phase space and, moreover, the closure of the set of generic elliptic periodic orbits contains hyperbolic sets with Hausdorff dimension arbitrarily close to maximal. Other instances of the restricted $n$-body problem such as the Sitnikov problem and the case $n=4$ are also considered.