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That our mapping ${\displaystyle f(z)}$ from the plane into itself is differentiable as a complex function, means that it is differentiable as a real function - that is, that its two components ${\displaystyle f_{x}(x,y)}$ and ${\displaystyle f_{y}(x,y)}$ are differentiable - and that these two components satisfy the Cauchy-Riemann differential equations:

${\displaystyle {\frac {\partial }{\partial {x}}}f_{x}={\frac {\partial }{\partial {y}}}f_{y}}$ and ${\displaystyle {\frac {\partial }{\partial {y}}}f_{x}=-{\frac {\partial }{\partial {x}}}f_{y}}$

if so, these two numbers are the real and imaginary part of ${\displaystyle f'(z)}$, respectively.

It is this condition that causes the characteristic features of the Mandelbrot and Julia sets for complex iteration. The usual family of iterations ${\displaystyle z^{2}+c}$ can (in coordinate form) be written ${\displaystyle (x,y)}$ → ${\displaystyle (x^{2}-y^{2},2xy)+(u,v)}$ (if c = u + iv), and if we here replace the y-coordinate of the function, that is 2xy, by 1.95×xy, the shapes in the Sea Horse Valley become distorted.

This thread-like and tattered look is typical for the real - or non-complex - fractals. For a function which is not, as in this case, the result of a mild interference in a complex function, the picture is often very chaotic, and the colouring can be impossible at most places, because our method of colouring presupposes that the sequences of iteration converge to a finite cycle, and for a non-complex iteration the terminus need not be a finite set. The terminal set is now called an attractor, and attractors can have very surprising shapes. Because of this, such an attractor is known as a strange attractor.

• 2010-04-21 09:44 Gertbuschmann 800×600× (431790 bytes) {{Information |Description = Non-complex Mandelbrot set |Source = I (~~~) created this work entirely by myself. |Date = ~~~~~ |Author = Gert Buschmann |other_versions = }}
• 2010-04-15 00:27 Gertbuschmann 800×600× (413599 bytes) {{Information |Description = Real fractal |Source = I (~~~) created this work entirely by myself. |Date = 16/4-10 |Author = Gert Buschmann |other_versions = }}