  When the planar circular restricted 3-body problem (PCRTBP) is periodically
perturbed, as occurs in many useful astrodynamics models, most unstable
periodic orbits persist as whiskered tori. Intersections between stable and
unstable manifolds of such tori provide natural heteroclinic pathways enabling
spacecraft to greatly modify their orbits without using propellant. However,
the 2D Poincar\'e sections used in PCRTBP studies no longer work to find these
intersections. Thus, in this study, we develop new fast methods to search for
and compute such heteroclinics. First, the dynamics are used to restrict the
intersection search to only certain manifold subsets, greatly reducing the
required computational effort. Next, we present a massively parallel procedure
for carrying out this search by representing the manifolds as discrete meshes
and adapting methods from computer graphics collision detection algorithms.
Implementing the method in Julia and OpenCL, we obtain a 5-7x speedup by
leveraging GPUs versus CPU-only execution. Finally, we show how to use manifold
parameterizations to refine the approximate intersections found in the mesh
search to very high accuracy, as well as to numerically continue the
connections through families of tori; the families' Whitney differentiability
enables interpolation of needed parameterizations. The ability to very rapidly
find a heteroclinic intersection between tori of fixed frequencies thus allows
the systematic exploration of intersections for tori of nearby frequencies as
well, yielding a variety of potential zero-fuel spacecraft trajectories. We
demonstrate the tools on the Jupiter-Europa planar elliptic RTBP.
