File:Elutin1a4tori.jpg

${\displaystyle f(x)=4x(1\!-\!x)}$

${\displaystyle y=f^{c}(x)}$

${\displaystyle c=}$

${\displaystyle f}$

${\displaystyle c}$

${\displaystyle c}$

${\displaystyle f_{u}(x)=u~x~(1\!-\!x)}$

${\displaystyle u\!=\!4}$

F[c_, z_] = 1/2 (1 - Cos[2^c ArcCos[1 - 2 z]]) 
Plot[{F[1.5, x], F[1, x], F[.8, x], F[.5, x], F[.2, x]}, {x, 0, 1}] 

In order to keep the code short, the colors are not adjusted. The code above can be obtained from the representation of the superfunction ${\displaystyle F}$ and the Abel function ${\displaystyle G}$

${\displaystyle f^{c}(z)=F(c+G(z))}$ 

at

Failed to parse (syntax error): {\displaystyle F(z)= \frac{1}{2}(1−\cos(2z))}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(z)=F^{-1}(z)=\log_2(\arccos(1\!−\!2z))}

^[2]. Expressions for the more general case are suggested in the article Logistic sequence.

Copyleft 2011 by Dmitrii Kouznetsov. Previously, this image appeared at http://tori.ils.uec.ac.jp/TORI/index.php/File:Elutin1a4tori.jpg ; the free use is allowed.

References

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