Chaos Lab #1a

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

  • When you move the R slider, you:
    • adjust the control parameter [ R in the logistic function, R*X*(1-X) ]
    • redraw the green graph of the logistic function, X between 0 and 1, and
    • clear the screen of the red KL web
    • BUT, the current value of X is remembered.
  • When you drag the X0 slider, you:
    • Change the value of X0 for the next Reset click
    • Leave the red cobweb and current value of X in place

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 (sorry for typo!!)
  • in interval (0.0, 0.5)
  • in interval (0.5, 1.0)

Revised 09 September 2002 by Ralph Abraham