Invariant manifolds of a non-autonomous quasi-bicircular problem computed via the parameterization method

The parameterization method (pm) has been used to compute high-order parameterizations of invariant manifolds of vector fields at fixed points. This paper extends such approach to invariant manifolds of periodically-perturbed vector fields about a periodic orbit with the same frequency, with a direct application on the libration points of the Sun–Earth–Moon system. The Sun–Earth–Moon environment is modeled by the so-called quasi-bicircular model (qbcp), which is a coherent restricted four-body model that describes the motion of a spacecraft under the simultaneous gravitational influences of the Earth, the Moon, and the Sun. The pm is adapted to account for the explicit time-dependency of the corresponding vector field. This new procedure yields high-order periodic semi-analytical approximations of the center manifolds about the libration points $L_{1,2}$ of the periodically-perturbed Sun-(Earth  +  Moon) and Earth–Moon systems. These approximations are then used to initialize the computation of Poincaré maps, which allow to get a qualitative description of the non-autonomous dynamics near the equilibrium points. It is shown that, with this new approach, the semi-analytical description of the center manifolds in a coherent four-body environment is valid in a neighborhood significant enough to be used in practice. In particular, the well-known Halo orbit bifurcation is recovered in all cases. (paper)