File:Elutin1a4tori.jpg

Iterations of the logistic transfer function $f(x)=4x(1\!-\!x)$ (shown qith thick black line) $y=f^c(x)$ for $c=$ 0.2, 0.5, 0.8, 1, 1,5 .

Function $f$ is iterated $c$ times; however, the number $c$ of iterations has no need to be integer.

This pic was generated with the "universal" algorithm that evaluates the iterations of more general function $f_u(x)=u~x~ (1\!-\!x)$; see ^[1]. Namely for $u\!=\!4$, the iterates can be expressed through the elementary function, and such a plot can be generated, for example, in Mathematica with very simple code:

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 representation above can be obtained from the representation of the superfunction $F$ and the Abel function $G$:

$f^c(z)=F(c+G(z))$

at

$F(z)= \frac{1}{2}(1−\cos(2z))$

$G(z)=F^{-1}(z)=\log_2(\arccos(1\!−\!2z))$

As an exercise, one may check the property $f^{c+d}(z)=f^c(f^d(z))$.

More superfunctions represented through elementary functions can be found in ^[2].

Copyleft 2011 by Dmitrii Kouznetsov. The free use is allowed.

References