On the figure-8 periodic solutions in the three-body problem

We study the properties of the figure-8 periodic solution in the planar three-body problem with equal masses. The three masses have a triple overlap orbit: they travel over one and the same geometric curve. This solution also has the classical isosceles symmetry property. Therefore, we study the relative periodic solutions in a rotating frame, the well-known rotating Jacobian frame. We discover that the original figure-8 is a member of a general manifold consisting basically of two branches connected at a bifurcation point that is actually extremely close to the original figure-8 in the mass space. We discover that one branch has stable orbits while the second branch has mostly unstable orbits. Finally, we emphasize the importance of the hierarchy of the three masses that is present here, because it may be relevant in cosmology, the distribution of masses in filaments, as well as theoretical studies such as the symbolic dynamics of the braids introduced by C. Moore.