3.2 Chaotic Fractals

Another category of fractals represents the so-called chaotic, non-linear fractals. This fractal type is connected with the theory of chaos, and its elements are obtained by a simple mathematical equation[01] .For visualizing them, each point on the paper or screen is related to a certain number - e.g. in the case of the “Mandelbrot set” this is a complex number. This number is then iterated, that means it is used in a formula and the new number resulting from that is then again used in the same formula, which leads to the next iteration. This sequence of operations is "similar" to the work of the "copy-machine" of linear fractals - with regard to insertion.

The insertion is repeated until the numerical values approach infinity, converge or fluctuate between several numbers. Depending on the result, the original point may be colored differently.

3.2.1 The Mandelbrot Set

The Mandelbrot-fractal itself is the picture of the Mandelbrot set - the Mandelbrot set being the numerical set of the complex numbers for that is valid if, being repeatedly put into the formula , the absolute value remains a finite number[02] . The plane in which it is drawn is called the complex plane where each point represents a complex number of the form C=a+bi - with and . The system of co-ordinates of the plane or screen is defined with the x-axes representing the value of “a” respectively “X” and the y-axes the value of “bi” respectively “Yi” with “i” being the root of -1. Numbers multiplied by “i” are called imaginary numbers “b”, those which are not are called real numbers “a”.

For visualizing the set, each pixel of the screen, representing a certain complex number, is iterated in the formula . For each pixel, point, , mostly fixed in the zero point , remains the same but the value of “C=a+bi” is chosen differently. After a certain number of iterations there are two possibilities, first the iteration for a specific “C” diverges to infinity or second it does not diverge but approximates a certain number or fluctuates between several numbers, so-called fix-points. If the iteration remains limited, the analyzed pixel for a specific “C”, is an element of the Mandelbrot set and colored black. If it diverges to infinity mostly the steps are counted and, depending on how fast it diverges, the pixel is given a special color. Then the procedure is repeated for another value of “C” and so on[03] . What is fractal about that are the infinitely small boundaries between being a finite value or jumping between certain numbers and diverging towards infinity - see picture 14.

picture 14: Mandelbrot set and Julia sets

On the left side the sequence of , named “orbit”, is shown for three different starting points “C”: the 1st example converges to a fix-point and the 3rd jumps between two points which means that both starting points “C” belong to the Mandelbrot-set and are colored black. The 2nd example grows very fast from which follows that there is a high probability that reaches infinity and therefore the starting point is colored.
For each point “C” there also exists a corresponding Julia set that can be found on the right side. In the case of the Julia set the constant “C” remains the same for different starting points - the sequences can be found on the next page "Mandel.pdf".

3.2.2 The Julia Sets

The Julia sets use the same formula as for the Mandelbrot set, whereas this time for one certain Julia set, the value of “C” keeps the same and is changing. The behavior of each starting point is examined by iterating it in the formula , whereby the same complex plane as for the Mandelbrot set is applied. Finally the examined point belongs to the Julia set if keeps a finite value. The procedure is repeated for each pixel, containing different starting values , of the screen[04] .

For each value of “C” in the Mandelbrot set there exists one specific Julia set, which shows us that there are unlimited Julia sets as there are infinite points of the Mandelbrot set[05] . Julia sets for points outside the Mandelbrot set look like dust or a disconnected cloud of points, they consist of infinite loose points. Whereas Julia sets for points being elements of the Mandelbrot set itself are connected, which means that each point of the real Julia set, mostly colored black, has a neighboring-point. Those for values of “C” in the heart-shaped main-body of the Mandelbrot set look like rugged, deformed circles - see picture 14.

How can it be proved easily that a certain set is not interrupted? Gaston Julia found out that one only has to analyze the behavior of the critical zero-point, . The resulting picture of “C” is continuous if the set of iterations of these points does not grow too extremely.

Footnotes

[01] Jürgens Hartmut, Peitgen Heinz-Otto, Saupe Dietmar: Fraktale - eine neue Sprache für komplexe Strukturen, Spektrum der Wissenschaft (9/1989), p.59.
[02] http://nb.in-berlin.de/scut/essay/frak/mandel.html (07.06.1996);
[03] Programs which compute such a Mandelbrot set assign each pixel of the viewing screen to a complex number. If each of these points were computed with infinite iterations, it would take infinite time. Therefore the number of iterations is for example reduced to 500 or 1000 steps. Then it is checked if the absolute number of “Zn“ grows to infinity on high possibility, that is if Zn>2. In this case it can be said with high probability that “Zn“ is not an element of the Mandelbrot set.
[04] For finding the Julia set it is also possible to reverse this procedure. This transformation has the effect that an outer or inner point aims at the Julia set. What does the new transformation then look like? As the cubic root of real but also of complex numbers can be positive or negative, there are two possible formulas: f1(u)=+Ö(u-C) and f2(u)=-Ö(u-C). These non-linear transformations are then used in the copy-machine, which applies certain rules, transformations, to a starting image - circle, square, or any other picture. The difference to linear fractals lies in the circumstance that in general straight lines are not duplicated on straight lines any more but on curves. Jürgens Hartmut, Peitgen Heinz-Otto, Saupe Dietmar: Fraktale - eine neue Sprache für komplexe Strukturen, Spektrum der Wissenschaft (9/1989), p.60.
[05] Any Julia set belongs to a certain value of “C” which in turn represents a specific point of the Mandelbrot set.