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Parabolic saddles and Newhouse domains in Celestial Mechanics

Miguel Garrido, Pau Martín, Jaime Paradela

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 Poincaré 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.

Parabolic saddles and Newhouse domains in Celestial Mechanics

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 Poincaré map is the identity matrix), one of them, denoted here by , 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 , 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 in parameter space (the ratio of masses of the bodies) and residual subsets for which the homoclinic class of 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 -body problem such as the Sitnikov problem and the case are also considered.

Paper Structure

This paper contains 42 sections, 52 theorems, 360 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The restricted 3-body problem as an area preserving map
  3. A degenerate fixed point at infinity
  4. The set of oscillatory motions
  5. Newhouse domains in Celestial Mechanics
  6. The Sitnikov model
  7. The restricted 4-body problem
  8. Organization of the article
  9. Outline of the proof
  10. Reduction to a 3-dimensional system with a degenerate periodic orbit
  11. Local dynamics close to $O$
  12. Failure of the classical Lambda lemma
  13. The Shilnikov stable and unstable maps
  14. Secondary homoclinic tangencies to the degenerate saddle $O$
  15. Renormalization in a neighborhood of a homoclinic tangency
  16. ...and 27 more sections

Key Result

Theorem 1.1

There exists a Newhouse domain $\mathcal{N}\subset(0,1/2]$ (which contains $\{0\}$ in its closure) for the one-parameter family of diffeomorphisms $P_\mu=h_{MG}^{-1}\circ f_\mu\circ h_{MG}$. Namely, for all $\mu\in\mathcal{N}$ there exists $\Lambda_\mu$ which is a basic set for $P_\mu$ and an orbit

Figures (6)

  • Figure 1: Phase space of the $2$-body problem in McGehee coordinates. The stable and unstable manifolds of $O$ coincide along a homoclinic loop which separates the regions of bounded and unbounded motions.
  • Figure 2: Construction of the Shilnikov data.
  • Figure 3: The hyperbolic fixed point $z_{\kappa,T}$ and its local invariant manifolds.
  • Figure 4: The horizontal and vertical foliations constructed using the Shilnikov maps.
  • Figure 5: The line of tangencies is transverse to the leaves of the horizontal foliation.
  • ...and 1 more figures

Theorems & Definitions (91)

  • Remark 1
  • Theorem 1.1
  • Remark 2
  • Definition 1.2
  • Theorem 1.3
  • Remark 3
  • Theorem 1.4
  • Theorem 1.5: Chazy, 1922 Chazyclassification
  • Conjecture 1.6: Alekseev
  • Remark 4
  • ...and 81 more