Bifurcation Diagrams

First, create an example of a bifurcation diagram by clicking the "Bifurcation" button (to the left). Once the applet is loaded, click on the "Plot" button to make a bifurcation diagram for .

This will take a short while to calculate and display because the algorithm which produces the diagram requires many iterations of .

Bifurcation Diagram Algorithm

  1. Start with = 0, x = 0.5.
  2. Calculate the first 30 iterates of .
  3. Plot the value of for iterates 31 through 120.
  4. Increment a little, reset x = 0.5.
  5. If <= 4, go to step 2, else exit.

The idea is that after the first 30 iterates the values of are either fixed points or points on a periodic orbit. Note: this is not true when convergence to a fixed point or a periodic point is slow, such as for = 1 and = 3 -- observe the "smearing" around these values of .

Remark: it is good to use the critical point x = 0.5 as the iteration seed since it is always attracted to an attracting periodic point. However, you get qualitatively the same diagram by starting with any point in (0, 1).

Next: Follow the "As Increases" link.