The Rectilinar Three-body Problem using Symbol Sequence II. Role of the periodic orbits

TL;DR

This paper investigates how the phase space structure of the rectilinear three-body problem changes with mass variations, focusing on the bifurcation and influence of dominant periodic orbits on the system's stability and Poincare section structure.

Contribution

It identifies the role of dominant periodic orbits with specific rotation numbers and their impact on the phase space and stability regions, including orbits not bifurcating from the Schubart orbit.

Findings

Number of dominant periodic orbits depends on parity of n

Dominant orbits influence the size of the Schubart region

Existence of stable periodic orbits not bifurcating from Schubart orbit

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 follow these orbits. We find stable periodic orbits which do not bifurcate from the Schubart orbit.