I have been rather creative and explored an interesting Julia set. I will say that I picked it for the way it looks, rather than anything scientific. I make no claims that this Julia set is of any real mathematical interest, nor that it is related to any interesting dynamical system or anything like that.

Here is part of the Julia set for \(F_{c}=(1 + \sin(z) ) \log(|z|)\) and with \(c = – 0.5 i + 2 \). I have included grid lines to help us navigate.

So, let us have a closer look.

The self-similarity in this Julia set is quite striking. the generic features here are also quite generic; the branching off and swirls.

Above is my attempt at creating the famous Barnsley fern [1]. It is one of the better known examples of a self-similar set, the other key example is the Sierpinski gasket.

The Barnsley fern resembles real species of Asplenium ferns, see below.

References

[1] Michael Barnsley, Fractals Everywhere, Boston, MA: Academic Press, 1993

Links

Wikipedia entry on the Barnsley Fern

Just for fun I though I would have a look at some Julia sets with random noise. So I decided to have a look at the Julia set for \(F_{c}= \exp(\frac{z^{2}}{2})\) and \(c = 2- 0.5I\). This was chosen for no particular reason.

To this I modified the algorithm to include some noise in the form of a random complex number. The random number is of the form \(R_{\#} = \frac{z_{R}}{\#}\)

where \(|z_{R}| \leq \sqrt{2}\) and \(\#\) is a real number that scales the random number.

Basically, as the random numbers become larger then the fractal pattern gets “dissolved” in random noise. I won’t claim there is any real scientific value in this experiment. Enjoy the pictures.

Here we have no noise. As the pictures go down the noise increases.