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The question Did DSCOVR travel “along the stable manifold of it's future SE L1 Halo orbit” to get there? is specific to DSCOVR's trajectory from Earth to its primarily heliocentric orbit near Sun-Earth L1, which is a Lissajous oribit.

Here I would just like to ask the general question: Can Lissajous orbits have stable/unstable manifolds? For those specific orbits where the horizontal and vertical periods are rational fractions (e.g. 3/2, 5/4) would the manifold just be a "cylinder-like" extension of the closed, periodic orbit? For example, if the ratio were 2:1 would it look like a figure-eight just extruded and stretched into the 3rd dimension?

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    $\begingroup$ It is a pity that this question have not received enough attention. My intuition would say that they have (they are quite similar to Halos except for the periodicity) but it is merely opinion based $\endgroup$ – Julio Aug 26 '18 at 0:23
  • $\begingroup$ @Julio thanks for your interest. I wonder if an examination if some of the basic behaviors of these manifolds for halo orbits also occur for slightly perturbed Lissajous orbits might at least address if they have functionally or behaviorally equivalent surfaces, even if they can't rigorously be proven to be the same thing; sort of an orbital mechanical "duck typing" (1, 2). $\endgroup$ – uhoh Aug 26 '18 at 2:26
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    $\begingroup$ In short- yes they do. In fact, last Wednesday at the Astrodynamics Specialists Conference in Snowbird, Utah, Ryan Russell from the University of Texas presented "Europa Lander Trajectory Design Using Lissajous Staging Orbits" by Ricardo Restrepo, Ryan Russell (both UT) and Martin Lo (JPL), in which he discussed Lissajous manifolds. In the paper, the authors show images of the manifold tubes-- over one rev of the Liss (or quasi-halo), the orbit looks very much like a not-quite-connected halo, and the manifold tube looks very much like a halo manifold tube. $\endgroup$ – Diane Aug 27 '18 at 14:39
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    $\begingroup$ The final papers from the conference will be available in a month or so. $\endgroup$ – Diane Aug 27 '18 at 14:41
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    $\begingroup$ I just found it online searching on the title at scholar.google.com. $\endgroup$ – Diane yesterday

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