Fractals & Chaos Assignment for 11/25

We did some more work collecting information for the Catalog of Behavior for the Logistic Map function Mr. Drix introduced to you last week, getting a clearer idea on the changes in behavior as a increases towards 4.0. The patterns we observed raised a few questions:

  1. Somewhere within the interval 3.0 < a < 3.1, the single fixed point attractor “bifurcates” (i.e., splits) into a 2 cycle. Where exactly does this happen?
  2. Later, the 2-cycle becomes a 4-cycle. Do all points of the cycle bifurcate simultaneously, or can one split before another? In other words, is there a 3-cycle within the interval 3.4 < a < 3.5, or does the 2-cycle split directly to a 4-cycle?
  3. Once chaos appears past a = 3.6, is that it? Or is there anything more to be found there?

Continue your explorations of this function to try and answer these questions. Use the applets posted on the left side of the page. I have a special prize for the first person who can find a 3-cycle…

 

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