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File:Verhulst-Mandelbrot-Bifurcation.jpg

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)
Public domain I, the copyright holder of this work, release this work into the public domain. This applies worldwide.
In some countries this may not be legally possible; if so:
I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

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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