Fractals & Chaos Assignment for 5/3

We’ve spent some time exploring the logistic function analyzers linked at right, and have found some patterns. For example, when a = 3.0, we have a fixed point attractor, but at a = 3.1, there’s a 2 cycle. There’s a 4 cycle at a = 3.5, and while there appear to be separate bands of chaos at a = 3.6, they’ve merged by the time we get to a = 3.7. Please investigate the logistic function some more, with the following questions in mind:

  1. Where between 3.0 << 3.1 does the first bifurcation occur? When is the 2-cycle born?
  2. Can a 3 cycle be found in the interval 3.4 << 3.5? If so, this would suggest that one part of a cycle can bifurcate before the other. If not, perhaps bifurcation is an all or nothing thing.
  3. Is there only chaos to be found for a > 3.6? Or is there something else hidden in there?
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