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This paper studies transfers from halo orbit to halo orbit in the Jovian system. I wonder if there are any locally stable orbits connecting Earth-Moon L1 and L2 points.

EDIT: One should also consider trajectories based on low-thrust engines (ion, solar sails, magsails).

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A connection between two different periodic orbits around Lagrange / libration points is often termed a heteroclinic connection. Although the L1 and L2 points in the Earth-Moon system are themselves unstable equilibrium points, there are stable periodic (or quasi-periodic) orbits that can be found around each point. From these orbits, it is possible to find trajectories (manifolds) that connect the orbits and provide cheap transfers. These manifolds are termed stable or unstable depending on whether they are directed to or away from the periodic orbit, respectively.

Consider this scenario that could result in a cheap transfer:

  • Establish a periodic orbit about a Lagrange point
  • Perform a maneuver to approximate being on an unstable manifold from this orbit that is also a heteroclinic connection to another periodic orbit's stable manifold
  • Ride this heteroclinic connection for free until reaching the vicinity of the other periodic orbit
  • Perform a maneuver that places you on the other periodic orbit

For a specific example, consider the Genesis mission design, which employs manifold transfers between periodic orbits. Technically, there is some math that says it takes infinite time to arrive from the manifold, so if we could wait an infinite time, it would be "free"--but we don't have time for that, so maneuvers will solve that issue. Here's a paper from CalTech that looks at heteroclinic connections between perioidic orbits--but not specifically in the Earth-Moon system (but, as an aside, the great thing about the system dynamics in the circular-restricted three body problem, the only real free parameter is the system mass ratio, usually $\mu$, that you can easily change to represent the system)

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