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English: Diagram showing the connexion between Verhulst dynamic and Mandelbrot set
Date 7 April 2008
Source Own work
Author Georg-Johann Lay
( Reusing this file)
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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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