Chaos Lab, Flash #1

Koenigs-Lemeray Construction for the Logistic Family

Buttons

  • KL Step: does one step of KL process (vertigo-horrid)
  • Clear: removes red cobweb, retains current value of X
  • Reset: removes red cobweb, resets X to slider value X0

Sliders

  • R may be changed by dragging the blue disk up and down. When you change R, you:
    • adjust the control parameter [ R in the logistic function, R*X*(1-X) ]
    • redraw the yellow graph of the logistic function, X between 0 and 1,
    • the red cobweb is remembered, and
    • the current value of X is remembered.
  • X0 may be changed by sliding the green square along the red diagonal. When you change X0, you:
    • change the value of X0 for the next KL click
    • clear the red cobweb, and
    • reset the value of X to X0.

Suggestions

Experiment with values of R everywhere, but especially:
  • R in [0.5, 2.8]: an attractive fixed point
  • R around 3.2: an attractive 2-periodic cycle
  • R around 3.5: an attractive 4-periodic cycle
  • R in [3.6, 4.0]: lots of chaotic behavior
  • R around 3.83: an attractive 3-periodic cycle
Experiment with a few different values of X0, eg:
  • 0.5
  • in interval (0.0, 0.5)
  • in interval (0.5, 1.0)

Revised 20 September 2005 by Ralph Abraham