Fine structure of the region of initial conditions for close to periodic orbits in the general three-body problem

Abstract

The region of initial conditions for close to periodic orbits is studied in the general three-body problem with components of equal mass and zero angular momentum. A method proposed earlier, based on minimization of a functional equal to the sum of the squares of the differences between the initial and current coordinates and velocities of the bodies, is used to search for such orbits. The search was conducted among orbits with periods T ≤ 2 000τ, where τ is the mean time for a component to cross the triple system. Elongated structures are found in the region of initial conditions, each of which corresponds to a certain periodic orbit. The detected structures seem to be conentrated along characteristic curves corresponding to the exact periodic orbits. A boundary zone of the initial conditions has been discovered, to the left and right of which orbits arising from the Schubart orbit and S orbit lie. Close to periodic orbits in the boundary zone possess the properties of both types of orbits. As a rule, these have periods of ~10² τ. Examples of trajectories of the bodies are presented. Dynamical and geometrical properties of the studied orbits are described.