# Fractals & Chaos Recap for 12/9

We wrapped up our conversation about numbers of cycles findable in the Feigenbaum Plot, noting again that for every attracting cycle we find, there is an accompanying “evil twin” repelling cycle. These “evil twin” repelling cycles serve a purpose. As the members of the attracting cycle bifurcate, their patterns branch out further and further. When this pattern of branching reaches one of the values of the repelling cycle – indeed it will reach all members of the repelling cycle simultaneously – the branching stops and the system dissolves into chaos once more. This explains why these spontaneously generating cycles appear in “windows” in the Feigenbaum Plot, with an abrupt end to chaos on the left, and an abrupt resuming of chaos on the right. We wrapped up today with a recap of the work we’ve done so far, occupying our time exclusively within the Real numbers. The last portion of the course will now branch into the Complex numbers, effectively repeating the analyses we’ve done with iterating functions into the complex plane. Our first stop will be to understand how to represent geometric transformation with operations on complex numbers. We modeled this in class by showing how the repetition of a series of geometric transformations can result in the same “fixed point attracting” behavior as we saw with the real numbers.

# Fractals & Chaos Recap for 12/6

We have observed that cycles are born in the Feigenbaum Plot in one of two ways: bifurcations of lower cycles and spontaneously out of chaos. Yesterday we understood how these cycles are spontaneously born and how this phenomenon coupled with the self-similarity of the Feigenbaum Plot suggests an order to cycles. What it also gives us is a way of counting how many cycles of each type there are.

We know there are two fixed points on the original f (x) = ax(1 – x). One is attracting for a < 3, the other (at zero) is always repelling.

For 2-cycles, we look at f (f (x)). We see as many as four fixed points on this graph. But two are already members of 1-cycles and must be eliminated. This leaves only two points eligible to be members of 2-cycles, meaning we only have one 2-cycle, which we see bifurcating at a = 3.

To count 3-cycles, we observe that f (f (f (x))) has as many as eight fixed points. A 3-cycle pattern n, ___, ____, n could be explained two ways: as a 1-cycle (n, n, n, n) or as a real 3-cycle (n, o, p, n). So again, we remove the two 1-cycle points and are left with six points eligible to be members of 3-cycles, suggesting two 3-cycles. We see one in the Feigenbaum plot at a ≈ 3.83, but that’s an attracting fixed point. Where’s the other one? It turns out that for every cycle born out of chaos, there is an “evil twin” repelling cycle born as well. As a result, there are actually two 3-cycles born at a ≈ 3.83: one attracting and one repelling.

To count 4-cycles, we observe that f(f(f(f(x)))) has as many as 16 fixed points. But if we see a pattern of n, ___, ____, ____, n, we could explain this by:

• A 1-cycle: n, n, n, n, n
• A 2-cycle: n, o, n, o, n
• A real 4-cycle: n, o, p, q, n

So we eliminate the two 1-cycle points and the two 2-cycle points. This leaves 12 points eligible to be members of 4-cycles, suggesting three 4-cycles. One is the bifurcating cycle we clearly see in the Feigenbaum plot, the second and third are an attracting/repelling 4-cycle pair found at a ≈ 3.96.

To count 5-cycles, we start with the 32 fixed points, eliminate the two that are members of 1-cycles, and observe that the 30 remaining points must create six 5-cycles. One attracting/repelling pair we see at a ≈ 3.74, but the other two are harder to find (one is at a ≈ 3.906, the other pair at a ≈ 3.99028).

Your homework for this weekend: continue this train of thought and identify how many 6-, 7-, and 8-cycles there are. If you’re feeling ambitious, try to locate all of them!

# Fractals & Chaos Recap for 12/5

We observed yesterday that cycles appear to be “born” in one of two ways: bifurcations of “lower” cycles, and spontaneously arising from chaos. We’ve already shown why cycles bifurcate, so we started today with an explanation about how cycles spontaneously emerge from chaos.

Referring to our previous proof, graphing the function y = ((x)) can be a way of finding new cycles. Fixed points on the graph of y((x)) that are not common with the graph of y = (x) will be the parameters of our 2-cycle. By extension, graphing y = f (((x))), or any number of nested iterations, will give us a tool of finding new cycles. More crucially, this also shows us why cycles spontaneously appear from chaos. Explore the graph here. For a = 3.84, the “wiggliness” of the graph of y = f (((x))) is enough for the fingers of the graph to touch the line y = x. But for a < 3.84, it isn’t. The moment that a becomes large enough for those fingers to touch the line y = x is the moment that a 3-cycle is born.

The graph above shows that it is not possible to get a 3-cycle before a = 3.84, meaning the 3-cycle “window” we see in the Feigenbaum Plot is the first time we get a 3-cycle (addressing one of the other questions we asked yesterday). It also gives a clue to the order of cycles. We’ve already noticed that the Feigenbaum Plot exhibits fractal-like self-similar behavior, and the 6-cycle we observed at a = 3.63 could almost be viewed as two groups of three. If we consider that the 3-cycle at a = 3.84 is “born” from the original fixed point trend we observed for a < 3.0, then we could argue that the 6-cycle is actually two conjoined 3-cycles, each born from the first bifurcation at a = 3.0. This would suggest that there is a 12-cycle for an even lower value of a, born from the second bifurcations 4-cycle (and indeed there is, at a = 3.5821).

The 5-cycle we see at a = 3.74 then is mirrored with a 10-cycle at a = 3.6053, and a 20-cycle at a = 3.5775. This pattern could continue forever, to find any cycle, of any length.

This argument forms the basis for the Sharkovskii order we saw in yesterday’s article. The 3 cycle is the very last cycle to be born out of the chaos of this trend. The 5-cycle is the second-to-last, and the 7-cycle and every other odd-numbered cycle comes before those. But before we get to any odd-numbered cycle, we first would find the 6-cycle (2 x 3). Before that, the 10 cycle (2 x 5); before that, the 14 -cycle (2 x 5), and so on. But before any of those, we find the 12-cycle (4 x 3); before that the 20-cycle (4 x 5); before that the 28-cycle (4 x 7). And so on, reading the Feigenbaum plot right-to-left, until we find our “un-bifurcating” powers of two cycles, stitching back together to 16, to 8, to 4, to 2, and then finally back to 1.

If you’d like to learn a bit more about the Universality of the Feigenbaum Plot, check out this long article from a class similar to ours at Georgia Tech.

# Fractals & Chaos Recap for 12/4

We spent the first portion of class today exploring the Feigenbaum Plot today by asking a few supplemental questions to question 3 (is there anything to be found after chaos):

• Which comes first, a 3 cycle (so far found at a = 3.84) or the 5 cycle (found at a = 3.74 but also a = 3.906)?
• Why do some cycles happen more than once (see the 5-cycle above, or the two 6-cycles at a = 3.63 and a = 3.845 or two 4-cycles at a = 3.5 and a = 3.961)?
• In general:
• Where do cycles come from? How are they “born”?
• Is there some order to cycles?

Instructions on using Paul Fischer-York’s Bifurcation Diagram

• Use the Darkness slider to make the image darker and easier to see.
• Left-click and drag a rectangle around any portion of the diagram to zoom in on that portion.
• Right-click anywhere in the diagram to run a series of iterations at that value of a along the x-axis. The software will identify the magnitude of the cycle you’re looking at (though sometimes it makes a mistake, so look at the list as well!)

After some time to explore, we made a few observations about the general questions. The first is that cycles are born in two ways:

1. Bifurcations of “lower” cycles, or
2. Spontaneously arising from chaos

This suggests a certain ordering of cycles: the 2-cycle comes first, then the 4-cycle, the 8-cycle, the 16-cycle, and so on for powers of 2. Eventually, these cycles become so large that the system becomes chaotic. But this only explains cycles that are powers of 2. What about any other? For this, please read this article about the Feigenbaum plot, its constant, and the ordering of cycles.