Where Fractals meet Dynamic Systems: Theory and Computer Experiments.

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You might remember my swift overview I gave about some interesting results concerning chaotic behavior of the dynamics produced from the logistic map. In this article I want to proceed with this but by looking at it from a different angle, i.e. discovering a connection to the Cantor set that is a fractal.

The dynamical system we are going to look at, has the form

Or by defining (logistic map):

We will focus on values of a that are greater than 4. Let us make some observations first.

If we restrict the starting points of the above iteration to values within the interval I := [0, 1], we can ask if there are points that will leave this interval after a certain number of iterations.

Since the derivative of f is given by

we find a local maximum of f at x = 1/2. At this point we have

f(1/2) = a/4 > 1

On the other hand, obviously we have f(x) >/= 0 on I. This shows, f can leave the interval I…