1
$\begingroup$

Below is a $\mu = 0.01215$ bifurcation plot from E. J. Doedel, E.J. et al's Elemental periodic orbits associated with the libration points in the circular restricted 3-body problem International Journal of Bifurcation and Chaos, Vol. 17, No. 08, pp. 2625-2677 (2007) (paywalled, but also readable in ResearchGate)

Doedel et al (2007) Fig. 3. Bifurcation diagram for the Earth–Moon system (μ = 0.01215), showing families of periodic orbits that emanate from the libration points and from subsequent branch points

Fig. 3. Bifurcation diagram for the Earth–Moon system (μ = 0.01215), showing families of periodic orbits that emanate from the libration points and from subsequent branch points. The red cubes are the libration points. Small white spheres denote branch points, and small dark-red spheres denote collision orbits. The planar families C1, C2, and D1, are only partially represented; in particular, the fact that D1 arises from C1 via a period-doubling bifurcation is not indicated in the diagram. A glossary of the notation used is given in Table 1.

Due to recent insomnia I've written this answer to Where is Artemis on this Earth-Moon three-body bifurcation plot? Where's the near-rectilinear halo orbit for example?

Now I'm looking for a similar answer for butterfly orbits as discussed in @MattB's excellent answer to How does a butterfly orbit move in 3D? Way to generate or visualize?

Question: Where do the butterflies land on this bifurcation plot? (Earth-Moon three-body butterfly orbits)

$\endgroup$

1 Answer 1

1
$\begingroup$

I'm fairly certain that the butterfly family is not present in this diagram, sadly. The butterfly family originates from a period-doubling bifurcation in the halo family, and in that paper, they say "In this paper we use the term branch point to denote transcritical and pitchfork bifurcations, thereby excluding period-doubling, torus, and subharmonic bifurcations." Emphasis mine.

I can say that the Halo orbits themselves are the dark blue H1, H2, and H3 branches that bifurcate from the red lines. The butterflies, at least the ones I've worked with, bifurcate out of the L2 Halo family (I'm honestly not sure if butterflies exist for the L1 family), and we can see that there are no bifurcations in the given diagram.

However, if you'd like some more info on bifurcations, I present a pretty basic discussion of it in my thesis. Specifically I talk about how the nature of the halo-butterfly bifurcation. But to be honest, Emily Zimovan-Spreen gives it a much better treatment in her PhD dissertation. She talks about bifurcations in general in section 4.4, and some Halo-specific bifurcations in section 5.2. Section 5.2 shows the butterflies (as well as some other families that are, honestly, much cooler-looking). When I was doing my research, I relied heavily on Eric Campbell's PhD, "Bifurcations from Families of Periodic Solutions in the Circular Restricted Problem with Application to Trajectory Design," for my own learning.

$\endgroup$
3

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.