3. MANDELBROT SETS WITH MATLAB

The instruction pcolor of MATLAB generates color density plots, i.e. plane representations of real valued functions of two variables where the same value of the function corresponds to the same colour on the plot.
Since it is always possible to represent a COMPLEX variable as a couple of real variables in Gauss' plane, MATLAB proves extremely efficient in calculating and painting the approximate domains of non convergence of the series of complex functions needed in the study of MANDELBROT sets.

A variant of the classical escape method is applied here: instead of evaluating the number of cycles required in order that the modulus of the partial sum exceeds a fixed bound, we evaluate the value of that modulus for a fixed number of cycles.

That procedure:

allows to show the interior structure of the sets;

eliminates conditional instructions fastening the program execution.

COLORMAPS

The use of the different system MATLAB colormaps provides different effects; in particular colormap copper produces interesting metal coloured shapes.

Here are some examples.

f(z)=z²+c MATLAB Script M.1

QUESTION

What does it happen if instead of power two, different positive integer powers are introduced into the series?

ANSWER

A Mandelbrot like set is generated with as many heads as the exponent of the power minus one.

Look at the following images!

f(z)=z+c MATLAB Script M.2

f(z)=z³+c MATLAB Script M.3

f(z)=z⁴+c MATLAB Script M.4

f(z)=z⁵+c MATLAB Script M.5

f(z)=z⁵⁰+c MATLAB Script M.6

INDEX

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