Bridges between the Generalized Sitnikov Family and the Lyapunov Family of Periodic Orbits

The linearization of the spatial restricted three–body problem at the collinear equilibrium point 2has two pairs of pure imaginary eigenvalues and one pair of real eigenvalues so the center manifold is four dimensional. By the classical Lyapunov center theorem there are two families of periodic solutions emanating from this equilibrium point. Using normal form techniques we investigate the existence of bridges of periodic solutions connecting these two Lyapunov families. A bridge is a third family of periodic solutions which bifurcates from both the Lyapunov families. We show that for the mass ratio parameterμnear 1/2 and near 0 there are many bridges of periodic solutions.