Instability and Diffusion in the Elliptic Restricted Three-Body Problem

Xinhao Liao*, Donald G. Saari, Zhihong Xia

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


The importance of the stability characteristics of the planar elliptic restricted three-body problem is that they offer insight about the general dynamical mechanisms causing instability in celestial mechanics. To analyze these concerns, elliptic-elliptic and hyperbolic-elliptic resonance orbits (periodic solutions with lower period) are numerically discovered by use of Newton's differential correction method. We find indications of stability for the elliptic-elliptic resonance orbits because slightly perturbed orbits define a corresponding two-dimensional invariant manifold on the Poincaré surface-section. For the resonance orbit of the hyperbolic-elliptic type, we show numerically that its stable and unstable manifolds intersect transversally in phase-space to induce instability. Then, we find indications that there are orbits which jump from one resonance zone to the next before escaping to infinity. This phenomenon is related to the so-called Arnold diffusion.

Original languageEnglish (US)
Pages (from-to)23-39
Number of pages17
JournalCelestial Mechanics and Dynamical Astronomy
Issue number1
StatePublished - Jan 1 1998


  • Diffusion
  • Instability
  • Resonant jump
  • Resonant orbit
  • Transversal intersection

ASJC Scopus subject areas

  • Astronomy and Astrophysics
  • Space and Planetary Science


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