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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: Do 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
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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 Apr 22 '19 at 12:48
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    $\begingroup$ Unstable, definitely yes. Stable, nope not for Earth-Sun Lagrange points. There are quasi-stable manifolds, but not truly stable ones. Here "stable" would have limits for any orbit but if the orbit decay on the order of months, I don't really consider that stable when Earth orbiting spacecraft have stability for many years or even much longer. $\endgroup$ – honeste_vivere Feb 24 at 16:22

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