Feigenbaum

You can find a program exploring the Feigenbaum set at this location: https://sourceforge.net/projects/heptagram-problem/files/Feigenbaum/

The program starts out like this

This shows the standard function f(a,x) = a*x*(1-x), where the paramater a = 2 can be manipulated using up and down arrow keys and the parameter x0 = 0.5 by using the left and right arrow keys. The starting value (a=2, x0= 0.5) do not produce any interesting results, but increasing the value of a gives a little more:

Now you see that the iterates of x0 seem to tend to some limit value, also illustrated by this image, where the iteration number is on the horizontal axis, and the iteration value on the vertical axis:

The next image shows the Feigenbaum tree, here the value a is along the horizontal axis, and along the vertical axis I have iterates x0 = 0.5 1000 times without showing anything, and then showing the values of the following 1000 iterates. The vertical line shows a=2.96 (the two values a=0.90… and y=0.334… show me where I had the mouse when I activated the print screen). You can see, that a=2.96 is close to a bifurcation point, which is why it took some time for the iterates to settle on the limiting value.

Next I change the value of a using the up or down arrow key and get this image:

On returning to the first view showing the function f(a,x) = a*x*(1-x) with the new value of a = 3.244… I get this:

It shows that there is a cycle in the values of the iterates. We can now provoke things by moving x0 to a point close to the intersection between the graf of f and the line y=x; that intersection ought to be a fixed point for f, and it is, but an unstable one:

This can also be shown using the iteration screen (iteration number n horizontal, and iteration value x_n vertical):

Initially the values of x_n do not move that much, but eventually they tend to the two values of the stable 2-cycle. There are more functions stored in the file start_pictures, as shown here:

I have chosen to show number two f(a,x) = (a*x*(1-x))^2. We see that iterates that get “dangerously” close to zero or to one, will end up in zero after a number of iterates. Therefore numbers close to 0 or 1 end up close to 0; for a = 4 this happens for x < 0.0727 approximately, and for x>0.0272; so the Feigenbaum tree will show no x-values (vertical) in those two intervals. Final two pictures: