Fractals & Chaos Lesson Recap for 12/3

We finally found our first 3-cycle, at a value of a = 3.83. Oddly, this was after chaos emerged, suggesting that sometimes, inexplicably, chaos can turn into order. This of course raised a slew of new questions:

  1. We also found a 5-cycle at a = 3.74 and a = 3.906. Is there a 3-cycle before a = 3.74? Or does the 3-cycle necessarily come after the 5-cycle?
  2. For that matter, why do some cycles happen more than once? A 6-cycle can be found at a = 3.63 and 3.845. A 4-cycle can be found at a = 3.5 and 3.96. Where are these coming from?
  3. What even is the order of cycles, anyway? It seems that powers of 2 come first, born out of bifurcations from our original pattern of fixed points.
  4. Finally, where do the cycles that emerge from chaos come from? Why do they emerge? How are they “born”?

To make exploring these equations easier, I’ve created a Logistic Orbit Iterator on Google Sheets. Make a copy of it and save it to your drive, and you can edit the value of a and the seeds to get a picture of the destinations of the orbit (or scroll down to the bottom and see how it behaves after 500 iterations).

Tomorrow, we’ll finally see the full picture of what we’ve been discussing.

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