  We demonstrate the remarkable effectiveness of boundary value formulations
coupled to numerical continuation for the computation of stable and unstable
manifolds in systems of ordinary differential equations. Specifically, we
consider the Circular Restricted Three-Body Problem (CR3BP), which models the
motion of a satellite in an Earth- Moon-like system. The CR3BP has many
well-known families of periodic orbits, such as the planar Lyapunov orbits and
the non-planar Vertical and Halo orbits. We compute the unstable manifolds of
selected Vertical and Halo orbits, which in several cases leads to the
detection of heteroclinic connections from such a periodic orbit to invariant
tori. Subsequent continuation of these connecting orbits with a suitable end
point condition and allowing the energy level to vary, leads to the further
detection of apparent homoclinic connections from the base periodic orbit to
itself, or the detection of heteroclinic connections from the base periodic
orbit to other periodic orbits. Some of these connecting orbits could be of
potential interest in space-mission design.
