File:Verhulst-Mandelbrot-Bifurcation.jpg

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DescriptionVerhulst-Mandelbrot-Bifurcation.jpg	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 domainPublic domainfalsefalse 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.
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Note[edit]

The Verhulst process

${\displaystyle v_{k}:x\mapsto kx(1-x)}$

is equivalent to the process

${\displaystyle f_{c}:z\mapsto z^{2}+c}$

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

${\displaystyle \Phi ^{-1}\circ v_{k}\circ \Phi =f_{c}}$

Just let

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

and observe that the Parameters ${\displaystyle c}$ and ${\displaystyle k}$ are connected via

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