Periodic Solutions of the Planetary 5:2 Resonance Three-Body Problem

Abstract

An important information about a dynamical system is provided by the study of its periodic solutions and their stability. This is due to the fact that the periodic trajectories already contain a great deal of information about the dynamics of the system, and they are substantially easier to work with: “What renders these periodic solutions so precious to us is that they are, so to speak, the only breach through which we might try to penetrate into a stronghold hitherto reputed unassailable.” (Poincaré, 1892). It is well known (Arnold, 1978) that periodic orbits occur in one-parameter families, each of which defines a continuous curve in the phase space of studied problem. This set of curves is characteristic of the Hamiltonian and the topology of the phase space is determined by its branchings.