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MattB's excellent and well-sourced answer to Where do the butterflies land on this bifurcation plot? (Earth-Moon three-body butterfly orbits) begins:

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.

Question: What are "transcritical, pitchfork, period-doubling, torus, and subharmonic bifurcations" in the context of three-body orbits?

Is it possible (and it might not be) to just write short if incomplete description of what each of these kinds of bifurcations are specifically within the context of three body orbit design for spaceflight?

The International Journal of Bifurcation and Chaos is the name of the mathematics journal in which the linked paper is published so the study of bifurcations is a big field and there will be many scholarly works that go into detail, but is it possible to write short blurbs for each from which halo orbit fans might at least glean some basic insight?


Also related:

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  • $\begingroup$ I think there's a video game or movie plot somewhere in this chaos... $\endgroup$
    – uhoh
    Commented Feb 25, 2022 at 1:26

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I can attempt an answer by summarizing some of my favorite dissertations. Forgive some of the word vomit below, this is certainly not the most elegant or succinct explanation. I'll give a bit of a preamble and then get right into quoting some other authors in an attempt to directly answer your questions. I will assume you already understand that an orbits stability can be evaluated by the eigenvalues of its monodromy matrix and that there are strict rules governing the eigenvalues of that monodromy matrix (more on that at the end if you'd like). I also put a TL;DR near the end.

A bifurcation is a change in the qualitative nature of a family of solutions. That’s pretty general, but there are basically two things that a bifurcation can signify:

  • A branching of a family into a new family with distinctly different motion (such as the period-doubling bifurcation from the halo family to the butterfly family).
  • A change in the order of instability within an orbit family (that is, a change in the eigenstructure or number of stable/unstable eigenvalues).

Sometimes both of these occur at a bifurcation, and sometimes only one occurs.

Transcritical, pitchfork, period-doubling, torus, and subharmonic bifurcations are different taxonomizations of the above behavior. There are other types of bifurcations too: secondary Hopf bifurcations, modified secondary Hopf bifurcations, period-multiplying (doubling, tripling, etc.).

Some of these bifurcations are grouped together. I don’t know if there is an official breakdown of these groups, but I’ll use Dr. Emily Zimovan Spreen’s organization from her dissertation. In fact, let me quote a huge block from her PhD, as I can't say it any better:

  1. Tangent Bifurcation: As a general rule, a transition in stability characteristics along an orbit family denotes a bifurcation (as noted by changes in the value of the stability index). In the case of a change in family stability occurring simultaneously with two nontrivial eigenvalues of the monodromy matrix going to unity, λj = 1/λj = +1, a tangent bifurcation has occurred. There is a change in the order of instability along a family as a tangent bifurcation occurs [55], [57]. A tangent bifurcation is further delineated as either a cyclic fold, pitchfork bifurcation, or transcritical bifurcation:
  • Cyclic Fold Bifurcation: The orbits along a single periodic orbit family change order of instability but do not intersect with any other family at a cyclic fold [55]. A cyclic fold occurs at an extremum in Jacobi constant value [57].
  • Pitchfork Bifurcation: At a change in stability characteristics along a family, two new families are formed that both possess the same stability as members of the original family prior to the bifurcation.
  • Transcritical Bifurcation: At a transcritical bifurcation, a stable and unstable orbit family intersect; at this intersection, the stability characteristics of the families are swapped.
  1. Period-Doubling Bifurcations: At a period-doubling bifurcation, two nontrivial monodromy matrix eigenvalues collide at −1 and depart from the unit circle to the negative real line or vice versa [57]. The stability characteristics (i.e., order of instability) of an orbit family changes at a period-doubling bifurcation.
  2. Period-Multiplying Bifurcations: A period-multiplying bifurcation (of multiplying factor m where m is an integer greater than 2) occurs when two nontrivial monodromy matrix eigenvalues evolve such that λj, 1/λj = 1^(1/m) = cos(2π/m) ± i sin(2π/m) [55]. Note, these are the first and (m − 1)th complex roots of unity. As an example, when two nontrivial eigenvalues of the monodromy matrix, λj, 1/λj = 1^(1/3) = −0.5±0.8660i, a period-tripling bifurcation occurs. Period-multiplying bifurcations do not require a collision of eigenvalues on the unit circle and thushere is not necessarily a corresponding change in orbital stability along the family.
  3. Secondary Hopf and Modified Secondary Hopf Bifurcations: A less typical type of bifurcation is described as a secondary Hopf bifurcation; this type occurs when two eigenvalues collide on the unit circle and depart into the complex plane at a location other than ±1 along the real axis (therefore, a change in stability also occurs). A modified secondary Hopf bifurcation is triggered when two eigenvalues collide on the real line and depart into the complex plane (other than at ±1), however, this scenario does not occur with a change in stability. In the special case of a secondary Hopf bifurcation (modified or regular), after departure from the unit circle or real line, eigenvalues are complex but with magnitude greater than unity, indicating the existence of spiral manifolds (oscillatory and departing/approaching flow) [57]. In some special cases, periodic solutions are produced from secondary Hopf bifurcations, however, in general, invariant tori surrounding the periodic orbit are formed [57]–[59].

In the case of tangent bifurcations (other than the cyclic fold type), period-doubling, and period-multiplying bifurcations, a new family of periodic orbits intersects with the current family...

I had to alter some of the equation formatting. So I think the above should answer your questions in a literal sense. But I'll try to expand with some examples. Here is a figure from Dr. Eric Campbell’s dissertation that can elucidate some of the above bifurcations.

  • A tangent bifurcation occurs in part (a). This is showing one pair of eigenvalues, both before and after the tangent bifurcation. Note that the directions of the arrows could be reversed.
  • A period doubling bifurcation is shown in part (b). Again, this is showing one pair of eigenvalues and the arrows could be reversed.
  • A secondary Hopf bifurcation is shown in part (c). In contrarary to parts (a) and (b), here we are looking at four eigenvalues (two pairs), and we see each pair before and after the bifurcation.

Types of bifurcations from Dr. Eric Campbell's PhD

The eigenstructure of secondary Hopf bifurcations is particularly confusing in my opinion. In parts (a) and (b) it is clear which eigenvalue is paired with its reciprocal, but that information is harder to identify when you depart the real line or the unit circle. Knowing the rules of eigenvalues of the monodromy matrix is critical to understanding these bifurcations. Again, I'll put in a paragraph at the end just in case someone wants it, but the important thing to remember about the eigenstructure is this: for a periodic orbit, real eigenvalues occur in reciprocal pairs and complex eigenvalues occur in conjugate pairs (or, equivalently, the eigenstructure of the monodromy matrix is symmetric across the real axis and unit circle). To help show this I've included an example figure of this eigenstructure below. Note the subscript number of each lambda. The gray circle shows the unit circle and the gray horizontal line is the real axis.

enter image description here

TL;DR? Some Practical Examples of Bifurcations and their Context in the CRTBP

  • Period-doubling bifurcation: The butterfly family arises out of a period-doubling bifurcation from the halos. You can see these especially clearly when you look at the early members of the butterfly family; they look like a double NRHO.
  • Period-multiplying bifurcation: Similar to period-doubling. These are new families of solutions that exhibit similar but distinct behavior to the underlying family. In Dr. Zimovan Spreen's dissertation there are some wonderful examples of this in Chapters 5 and 6.
  • Secondary Hopf Bifurcation: I encountered this type of bifurcation when examining the butterfly family of orbits in my masters research. I never looked into the existence of new families, but this bifurcation did indicate the existence of spiral manifolds. In practice these aren't much different than the good old "stable" or "unstable" manifolds most of us have heard of. However, spiral manifolds introduce an extra dimension, which is realized as a "spiraling" behavior around what would otherwise be simple asymptotic approach or departure.
  • General Order of Instability Changes: This is critically important when examining the stability evolution of a family of orbits. As an example, consider quasi-periodic orbits in the CRTBP. A quasi-periodic solution can only exist if the underlying periodic orbit possesses a center subspace (i.e., it has monodromy eigenvalues on the unit circle). If all of your eigenvalues are on the real line, you can't generate a QPO around this orbit. However, if you detect a bifurcation where one or two pairs of eigenvalues are now on the unit circle, it is possible to generate a QPO.

I'm sure there are other examples I can't think of. And I know this post does not describe every type of bifurcation you asked about. Perhaps they are different names for bifurcations I discussed, or something else entirely. I hope this was helpful! I highly recommend reading about Broucke's Stability Diagram as well, which you can find in both of those dissertations. It's another bifurcation detection tool.

Post-hoc edit: I remembered an occasion where the occurrence of a secondary Hopf bifurcation really did matter. You may have heard of "stability indices." This is a way of combining the 4 eigenvalues into two stability metrics, v1 and v2, which are calculated like this: vi = .5 * (Li + 1/Li) where Li is one of the non-trivial eigenvalues. This equation works well when your eigenvalues are real or are on the unit circle, as the resulting stability indices will have only a real part and no imaginary part: the inverse and the conjugate of a complex number are the same thing if that complex number has unit magnitude (i.e., is on the unit circle), so the imaginary part cancels out when you do Li + 1/Li. But this is not true if your complex number is off the unit circle. In such cases where the eigenvalues are in the complex plane and not on the unit circle, your stability indices then have real and imaginary parts. Many papers that present the stability of families use this stability index metric, as their great benefit is that the x-axis can be a parameter like period or periapse radius while the y-axis can simply show the stability indices. If your stability index has both a real and imaginary part, it's annoying to plot. Also, you can get caught unawares by this behavior if you're not thinking about these complex, non-unity eigenvalues. Most plotting packages require you to plot real-values scalars. So, when you the author are generating your stability index plot, you call something like real(stab_indices) to turn your complex stability index array into something plot-able. This is fine when your eigenvalues are real or on the unit circle because, again, the complex part cancels out. But if you write that function without thinking about complex, non-unit eigenvalues, you are dropping the complex part without realizing it. I distinctly remember sitting in on research meetings where Dr. Zimovan Spreen showed this and came up with an alternative formulation for the stability index. I can thank her for saving me from making such mistakes in my research. Obviously you don't need an in-depth understanding of bifurcations to notice that you have non-unit complex eigenvalues, but still, it's helpful.

Here is a post script about the eigenstructure of the monodromy matrix. We identify bifurcations (and orbit stability) by examining the eigenvalues of the monodromy matrix (the STM after one full rev). There are 6 complex eigenvalues. For a periodic orbit, real eigenvalues occur in reciprocal pairs and complex eigenvalues occur in conjugate pairs (or, equivalently, the eigenstructure of the monodromy matrix is symmetric across the real axis and unit circle). Further, one pair of eigenvalues is always equal to unity, signifying that the solution is periodic and that it belongs to a family of motion. These two eigenvalues are called the trivial pair, and as they exist for any periodic solution we tend to ignore them. We are thus left with 4 eigenvalues, subject to the reciprocal and conjugate rules. As you examine the eigenstructure of periodic orbits in a family, you will see that it evolves. That is, each pair of non-trivial eigenvalues is constantly changing as you evolve a parameter in the family.

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    $\begingroup$ +n! for an incredibly informative (and thoroughly entertaining) answer to my question, which is "certainly not the most elegant or succinct" either! :-) I will now start down the road of taking this all in. Though I'm extremely chaotic in nature, chaos proper is a challenging subject for me. So it's going to take some time, and your "P.S. Monodromy matrix." is exactly where I'll begin. $\endgroup$
    – uhoh
    Commented Feb 24, 2023 at 0:09

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