The rectilinear three-body problem using symbol sequence II: role of the periodic orbits

Abstract

We study the change of phase space structure of the rectilinear three-body problem when the mass combination is changed. Generally, periodic orbits bifurcate from the stable Schubart periodic orbit and move radially outward. Among these periodic orbits there are dominant periodic orbits having rotation number (n − 2)/n with n ≥ 3. We find that the number of dominant periodic orbits is two when n is odd and four when n is even. Dominant periodic orbits have large stable regions in and out of the stability region of the Schubart orbit (Schubart region), and so they determine the size of the Schubart region and influence the structure of the Poincare section out of the Schubart region. Indeed, with the movement of the dominant periodic orbits, part of complicated structure of the Poincare section follows these orbits. We find stable periodic orbits which do not bifurcate from the Schubart orbit.