Fractals & Chaos Assignment for 11/27

We explored the behavior of the Logistic Population formula some more and formulated a few questions:

  1. The end behavior of iterations is a single fixed point up to a = 3.0, and at a = 3.1, the population has bifurcated into a two-cycle. Question 1: At what value of a does this bifurcation actually happen?
  2. There is a two-cycle up until a = 3.4, with a four-cycle at a = 3.5. Question 2: Do both points of the two-cycle bifurcate simultaneously, or does one split before the other? In other words, can you find a three-cycle between a = 3.4 and a = 3.5?
  3. After a = 3.6, we observe bands of chaos. Question 3: Once chaos appears, is that it? Or is there something to be found between bands of chaos?

Questions? Comments?

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