Fractals & Chaos Recap for 11/18

We discussed the last of the exercises from #3 of the Iterated Functions sheet, then debriefed all of the observations of Ax+B again, deciding once and for all that Ax+B is solved. A fixed point can be found by the expression x = B/(1-A), but only the value of A impacts the behavior of the fixed point:

  • If |A| < 1, the fixed point is attracting
  • If |A| > 1, the fixed point is repelling
  • If A > 0, the pattern is direct
  • If A < 0, the pattern is alternating
  • If A = 1, there is no fixed point
  • If A = -1, the fixed point is neutral
  • If A = 0, the fixed point is “super-attracting”

We also observed that as |A| gets nearer to 1, the attracting/repelling behavior becomes slower.

We looked at another hypothetical non-linear system, then set to work on exploring more of #4 from the IF sheet, using the Iterated Functions Supplement as a guide. Your homework is to finish up through #4h (by tomorrow), and read Chaos and Fractals in Human Physiology, from the February, 1990 edition of Scientific American

 

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