by R. Grothmann
Try the following notebook, which produces the famous chaotic behaviour, which was investigated by Feigenbaum.
function f (x, l) return l*x*(1-x); endfunction
We use a so-called web plot to visualize the iteration. Follow the red line to see the iteration values.
The iteration converges well for l<3.
However, for l>3 it switches between two points.
With the following animation, we can experience the changes in convergence, when the parameter l changes from 0 to 4. The initial value is 2.5.
Use the cursor up or down key to change l.
To see, what happens, we iterate f times f.
For larger l, we the iteration switches again.
We can investigate f(f(f(f(x)))) now.
The orginal iteration exhibits 4 limit points now.
We now try to display all limit points for various l.
The function f is iterated, starting from 0.5, 1000 times, and we take the last 500 iterations.
Note: The niterate function is given an additional argument l.
xmark is a bit slow for that many points.