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
- ↑ http://www.springerlink.com/content/u712vtp4122544x4/ D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98.
- ↑ http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1 D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.
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