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 = f (f (x)) can be a way of finding new cycles. Fixed points on the graph of y = f (f (x)) that are not common with the graph of y = f (x) will be the parameters of our 2-cycle. By extension, graphing y = f (f (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 (f (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.
Your homework: read pages 138-145 in Fractals: The Patterns of Chaos, about “Visualizing Chaos”
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.