Abstract
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.
Similar content being viewed by others
References
Arnold, V. I.: 1964, "Instability of Dynamical Systems with Several Degrees of Freedom', Dokl. Akad. Nauk. SSSR 156, 9.
Belbruno, E. and Marsden, B.: 1995, "Resonance Hopping in Comets', Preprint.
Holmes, P. and Marsden, J.: 1982, "Melnikov Method and Arnold Diffusion for the Perturbation of Integrable Hamiltonian Systems', J. Math. Phys. 23(4), 669.
Melnikov, V. K.: 1963, "On the Stability of the Center for Time Periodic Perturbations', Trans. Moscow Math. Soc. 12, 1.
Weinstein, A.: 1973, "Lagrangian Submanifolds and Hamiltonian Systems', Ann. Math. 98, 377.
Xia, Z.: 1993, "Arnold Diffusion in the Elliptic-Restricted Three-Body Problem', J. Dynam. Diff. Equations 5, 621.
Xia, Z.: 1994, "Arnold Diffusion and Oscillatory Solutions in the Planar Three-Body Problem', J. Diff. Equations 110, 289.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Liao, X., Saari, D.G. Instability and Diffusion in the Elliptic Restricted Three-body Problem. Celestial Mechanics and Dynamical Astronomy 70, 23–39 (1998). https://doi.org/10.1023/A:1008200806138
Issue Date:
DOI: https://doi.org/10.1023/A:1008200806138