Physics 5  Lecture Notes - Mandelbrot and Julia Sets

To get a more detailed understanding of how the Mandelbrot program works, you can write code to plot each iteration of the formula  zas they develop.  To do this, I modified the Complex class to include getReal(), getImag(), setReal() and setImag() member functions for getting and setting values of the _real and _imag parts.

Then this fairly simple DarkGDK() function:

A key part of this loop is the scaling formula used to convert from the Complex number to a point on the dbScreen:
                a = (int)(629*(x.getReal()+2)/4);

      b = (int)(479*(x.getImag()+2)/4);

Which makes the dbBox color lighter and lighter as they progress.  Here is one output of the program (it has been inverted so the boxes get darker) with the initial point -0.7 + 0.3i  point just barely outside the Mandelbrot set, so the iteration diverge outward:

This prompted a few changes in the code  making the dbBox a bit smaller and tweaking the colors a bit:

Which lead to this output:

This led to another variation,

Again the picture is inverted to black on white instead of white on black.

Another rendition of the loop:

With the initial value -0.688+0.28i, the first 10000 iterates are plotted below.

This point is near the boundary of the Mandelbrot set and on the interior, so it takes many iterations to converge to a point somewhere near the center of the white spot in the middle.  Going a tad further downwards towards the border produces a divergent orbit:

That appears to be right near the boundary…is it repeating points exactly, or only approximately?

Mandelbrot with his hair on fire.

So what’s going on here?  For those, like me, who are interested  here’s a bit of theory.

As H.-O.Peitgen and H.R.Richter wrote in their monograph of 1986, The Beauty of Fractals, The Mandelbrot set embodies a transition from order to chaos. [..] Think of attractors as points in the plane competing for influence on other points in the plane: if an initial point x₀ leads to convergence of the Mandelbrot iterates  x_n+1 = x²_n  + c  to a point inside the plane, then x₀ is said to be in the basin of attraction for the point c.  This basin must also include all the inverse iterates (preimages) of x₀.  The boundary such a basin of attraction is an intricate curve or sequence of curves that may or may not be connected.  The orbit of a point on the boundary of a a Julia set will include other points on the boundary, none of which are in one basin of attraction or another and so flitter about with seemingly random abandon.  Further, Peitgen and Richter put it, 

	It can happen that the boundary falls to dust, and this decay is one of the most important scenarios.

 

Mandelbrot’s iteration formula is very simple:

Consider the very simple case where c = 0.  Then there are three cases:

·         If |x₀| < 1 then x_n → 0 as n →∞

·         If |x₀| > 1 then x_n → ∞ as n →∞

·         If |x₀| = 1 then |x_n| = 1  as n →∞ and the sequence stays on…but may hop about on the boundary.

Consider, for instance, the value c = -0.12375 + 0.56508i and the iterates  for each x₀ in the complex plane.  If the iterates for a particular x₀  diverge to infinity then x₀ is not in the basin of attraction for this c, whereas if the iterates don’t diverge (which can mean either (1) converging to a fixed point or (2) cycling through a finite set of points or wandering aimlessly within a fixed region) then the boundary of the basin of attraction is on or more closed curves with an intricate shape: a fractal.  Boundaries of this kind are known as Julia sets after the French mathematician Gaston Julia.  The Julia set contains the unstable fixed point of the mapping  together with all of its preimages; it contains an infinite number of unstable periodic sequences also with their preimages, and above all it contains chaotic sequence of points that never approach any kind of regularity.

As an example, try x₀ = c = -0.12+0.74i.  This leads to the first iterate c² + c = 0.0995+0.005265i and (c² + c)² + c = -0.653+0.563i and ((c² + c)² + c)²+c = -0.1199+0.7399i brings us back (nearly exactly) to where we started: this is a three-cycle.  The basin of attraction of this three-cycle is not a single closed curve (as in the previous case) but infinitely many intricate closed curves.  Here’s the Julia fractal followed by the code I used to produce it: