Lissajous and Halo orbits in the restricted three-body problem

We study the dynamics near the collinear Lagrangian points of the spatial, circular, restricted three-body problem. Following a standard procedure, we reduce the system to the center manifold and we analyze the Lissajous orbits as well as the halo orbits, the latter ones arising from bifurcations of the planar Lyapunov family of periodic orbits. To obtain the Lissajous orbits, we perform a classical perturbation theory and we provide a formal approximate solution under suitable non-degeneracy and non-resonance conditions. As for the halo orbits, we construct a normal form adapted to the synchronous resonance: introducing a detuning, measuring the displacement from the resonance, and expanding the energy in series of the detuning, we are able to evaluate the energy level at which the bifurcation takes place. Except for a particular case, the analytical values obtained after a second order resonant perturbation theory are in very good agreement (in some cases up to the fourth decimal digit) with the numerical values found in the literature.