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Author Topic: Hidden Structures of the Mandelbrot and Julia Sets  (Read 118 times)


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Hidden Structures of the Mandelbrot and Julia Sets
« on: August 11, 2018, 01:13:06 PM »

An exploration of the correlations between the Mandelbrot Set and the Julia Sets for z^2 + c. Followed by some flybys of the full Mandelbrot/Julia structures. Finally, an attempt to show the complete 4D Mandelbrot Set..with partial success.


Mandelbrot set - from order to chaos:

This video illustrates three different ways of thinking about the Mandelbrot set. I recommend you watch this video fullscreen in high definition.

In the top-right corner is the normal view of the Mandelbrot set.

On the left is a graph known as a cobweb plot, which shows what's happening underneath the little pink dot on the green line. The blue curve represents the equation for the Mandelbrot set, z = zē + c. As the pink dot slowly moves to the right, along the horizontal axis of the Mandelbrot set, the blue curve slowly moves downwards. The faint yellow line in the plot shows the iterations of the Mandelbrot equation, and at the moment, the iterations are spiralling inwards towards a single point, known as an attractor, marked by the red dot. The plot is looking rather boring at the moment, but things will get progressively more interesting!

When the pink dot crosses over from the main cardioid to the big circle, the yellow line will stop spiralling towards a single point, and will start alternating between two points. This change of behaviour is known as a bifurcation, and the new attractor is called a limit cycle, because the iterations cycle back and forth between the two points. You can see the yellow spiral is slowly becoming more and more densely packed, and... there! The red dot has split into two, and we see the new limit cycle attractor.

In the bottom-right corner is a graph known as a bifurcation diagram, which gives you an overview of what's happening in the cobweb plot, and the green line indicates the current state. As you can see, in the bifurcation diagram, the single curve has now split into two, and these will soon split again into four, when the pink dot crosses over from the big circle to the little circle. In the cobweb plot, the yellow line will suddenly start visiting four red points instead of two. And there it goes!

And now, something very surprising will happen! The orderly behaviour we've been seeing will suddenly disappear, and the cobweb plot will become a mass of chaotic lines.

Occasionally, the plot will become orderly again for a moment, whenever the pink dot passes over a baby Mandelbrot set, but it'll quickly return to the chaotic behaviour. Here, we're approaching a baby Mandelbrot set, and the cobweb plot will become orderly for a moment... and then return to chaos. We're now approaching a bigger baby Mandelbrot, and the plot will stay orderly for longer... and then return to chaos again.

The amazing thing is that, although this chaotic behaviour looks very different from the orderly behaviour, it's created by exactly the same equation, with just a small change to the input value.
« Last Edit: September 29, 2018, 02:26:15 PM by Unbeliever »
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