File:Verhulst-Mandelbrot-Bifurcation.jpg

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Description

English: Diagram showing the connexion between Verhulst dynamic and Mandelbrot set

Date

7 April 2008

Source

Own work

Author

Georg-Johann Lay

Permission
( Reusing this file)

I, the copyright holder of this work, release this work into the public domain. This applies worldwide.
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I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

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Note

The Verhulst process

$v_k : x\mapsto kx(1-x)$

is equivalent to the process

$f_c: z\mapsto z^2+c$

by means of a linear transformation $\Phi$ , i.e.

$\Phi^{-1} \circ v_k \circ \Phi = f_c$

Just let

$\Phi: x \mapsto \tfrac{1}{2}-\tfrac{1}{k}x$

and observe that the Parameters $c$ and $k$ are connected via

$1-4c \,=\, (k-1)^2$

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The following pages on Schools Wikipedia link to this image (list may be incomplete):

Mandelbrot set

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