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.

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