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The paper Earth-Moon Near Rectilinear Halo and Butterfly Orbits for Lunar Surface Exploration (AAS 18-406) says

Periodic Orbits in the Earth-Moon System

The current investigation focuses on three types of periodic orbits in the vicinity of the Moon: Near Rectilinear Halo Orbits (NRHOs), which are periodic in the Earth-Moon rotating frame in the CR3BP; butterfly orbits, a periodic family that bifurcates off the NRHOs; and circular Low Lunar Orbits (LLOs), which are periodic in a Keplerian sense.

and shows the drawing below. There are several questions and answers here about near-rectilinear halo orbits and the ever-so-popular YouTube video Near Rectilinear Halo Orbit Explained and Visualized Okay, I like it at least.

But I can't understand nor visualize what these butterfly orbits look like in 3D or how objects move in them. Is it possible to generate one and show it here with a few views, or to find a resource where this is done, or even to just provide an algorithm and a python script that makes them? Even a set of initial state vectors would be enough to get started.

Figure 1: L2 Family of NRHOs and Butterflies

Figure 1: L2 Family of NRHOs and Butterflies

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Sorry that I'm so late to this, but I worked with the butterfly family quite a lot when I was a student in Professor Howell's group so I feel compelled to answer! It looks like you found some good resources already, but I can expand a bit and show some more of the family :)

I have quite a few visualizations of the L2 southern butterfly family in the Earth-Moon CRTBP. I've uploaded a bunch of videos to a playlist on YouTube. I also have a bunch of 3D figures in Plotly which are quite neat but I will hold off on sharing those for the moment. The videos in that playlist detail a few things, including the evolution of the family and what a few of the orbits would actually look like with a spacecraft in them. The family is extensive and becomes quite complex, so the examples I've shown in the videos only cover a small subset of the available motion. Hopefully, though, the examples I did provide will allow you to visualize the rest of the family in your head.

I'll highlight some videos:

I also detailed the evolution of the family in my thesis. In particular, figures 5.8—5.13 detail the evolution of the family. I tried my best to highlight the motion and how it changes, but it's difficult in 2D.

I also recommend Dan Grebow's thesis, which gives more example orbits and provides initial conditions like you were interested in.

There is so much to talk about. I could go on and on! But hopefully this answers your question.

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    $\begingroup$ +n! Jackpot! Oh this is fantastic :-) will go get those theses and spend the weekend poring over them. In the mean time, if you click the three-body and halo-orbit tags under the question you will find more unanswered or partially answered questions, some of which may pique your interest as well. Thanks! $\endgroup$
    – uhoh
    Commented Nov 27, 2020 at 0:06
  • $\begingroup$ It's really wonderful that you've uploaded those videos to YouTube as well.On behalf of all the readers here Thank you! $\endgroup$
    – uhoh
    Commented Nov 27, 2020 at 0:10
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    $\begingroup$ I'm glad I could help! If I can clarify anything, feel free to reach out. $\endgroup$
    – Matt B
    Commented Nov 27, 2020 at 20:55

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