iconEuler Examples

Feigenbaum Iteration

by R. Grothmann

>load "feigenbaum"

Try the following notebook, which produces the famous chaotic behaviour, which was investigated by Feigenbaum.

>type f
function f (x, l)
return l*x*(1-x);

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.

>l=2.9; fwebplot("f(x,l)",0,1,0.5,20):

Feigenbaum Iteration

However, for l>3 it switches between two points.

>l=3.4; fwebplot("f(x,l)",0,1,0.5,20):

Feigenbaum Iteration

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.

>l=3.4; fwebplot("f(f(x,l),l)",0,1,0.5,20):

Feigenbaum Iteration

For larger l, we the iteration switches again.

>l=3.47; fwebplot("f(f(x,l),l)",0,1,0.5,20):

Feigenbaum Iteration

We can investigate f(f(f(f(x)))) now.

>l=3.47; fwebplot("f(f(f(f(x,l),l),l),l)",0,1,0.5,20):

Feigenbaum Iteration

The orginal iteration exhibits 4 limit points now.

>l=3.47; fwebplot("f(x,l)",0,1,0.5,20):

Feigenbaum Iteration

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.

>h=niterate("f",x',1000;l'); h=h[:,500:1000];

xmark is a bit slow for that many points.


Feigenbaum Iteration