Do ("non-halo") Lissajous orbits have stable/unstable manifolds?

Question

The question Did DSCOVR travel “along the stable manifold of its 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 non-halo¹ 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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¹Halo orbits are a subset of Lissajous orbits where the oscillations in the plane of the two massive bodies (e.g. Earth's orbit around the Sun) and the oscillations out of plane (perpendicular to Earth's orbit) are the same, so that in the CR3BP approximation the orbit is perfectly closed and repeating. Lissajous orbits have that constraint relaxed, so the periods could be different by a tiny amount up to tens of percent or even more.

Answer 1

It seems that they do. As @Diane noted in a comment, this paper (PDF freely available) discusses Lissajous orbits and approximation of their invariant manifolds: Europa Lander Trajectory Design Using Lissajous Staging Orbits

Abstract:

Lissajous orbits and approximation of their invariant manifolds are used to generate landing trajectories to the surface of Europa. Each lissajous is discretized into individual revolutions that each resemble a periodic orbit. The unstable man- ifolds of each individual revolution propagated forward in time generate more surface coverage than manifolds of simple libration point orbits such as halo or Lyapunov orbits. The stable manifolds propagated backwards in time from the individual lissajous revolutions provide direct connections to the last phase of a moon tour. The strategy developed produces ballistic landing trajectories with a wide surface coverage, and allows for the decoupling of the landing and moon tour phase by using the lissajous as an intermediate staging orbit. The multiple revo- lutions of the lissajous, multiple departure times along each revolution, multiple quasi periodic options at each energy, and multiple energies of the lissajous family provide many degrees of freedom in the design process.

Below are two figures as examples of the workflow to give a rough idea of what's being done, but a careful reading of the full paper is really necessary.

above: "Figure 8. The approximated unstable invariant manifold of a full lissajous orbit (Ay = 2000 km, Az = 3000 km) and potential landing trajectories. a) 200 rev lissajous, 1 rev's manifold, 100 periapsis, b) 200 rev lissajous, 100 rev's manifold, 20000 periapsis"

below: "Figure 9. Manifold trajectories of a single lissajous rev with multiple periapsis passes. a) Manifold of a single lissajous rev (Ay=1000 km, Az = 5000 km, 𝜙=171). b) latitude/longitude map of periapsis."