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== Summary ==
== Summary ==
{{Image_Details|user-pd
{{Image_Details|user-pd
|description  = Iterations of the [[logistic transfer function]] <math>f(x)=4x(1\!-\!x)</math> (shown with thick black line) <math>y=f^c(x)</math> for <math>c=</math> 0.2, 0.5, 0.8, 1, 1,5 .  Function <math>f</math> is iterated <math>c</math> times; however, the number <math>c</math> of iterations has no need to be [[integer]].  This pic was generated with the "universal" algorithm that evaluates the iterations of more general function <math>f_u(x)=u~x~ (1\!-\!x)</math>; see <ref name="logistic"> http://www.springerlink.com/content/u712vtp4122544x4/ D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98. </ref>. Namely for <math>u\!=\!4</math>, the iterates can be expressed through the elementary function, and such a plot can be generated, for example, in [[Mathematica]] with very simple code:
|description  = Iterations of the [[logistic transfer function]] <math>f(x)=4x(1\!-\!x)</math> (shown with thick black line): <math>y=f^c(x)</math> for <math>c=</math> 0.2, 0.5, 0.8, 1, 1,5 .  Function <math>f</math> is iterated <math>c</math> times; however, the number <math>c</math> of iterations has no need to be [[integer]].  This pic was generated with the "universal" algorithm that evaluates the iterations of more general function <math>f_u(x)=u~x~ (1\!-\!x)</math>; see <ref name="logistic"> http://www.springerlink.com/content/u712vtp4122544x4/ D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98. </ref>. Namely for <math>u\!=\!4</math>, 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]])  
  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}]  
  Plot[{F[1.5, x], F[1, x], F[.8, x], F[.5, x], F[.2, x]}, {x, 0, 1}]  

Revision as of 01:43, 18 May 2011

Summary

Title / Description


Iterations of the logistic transfer function (shown with thick black line): for 0.2, 0.5, 0.8, 1, 1,5 . Function is iterated times; however, the number of iterations has no need to be integer. This pic was generated with the "universal" algorithm that evaluates the iterations of more general function ; see [1]. Namely for , 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 code above can be obtained from the representation of the superfunction and the Abel function

 

at

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

Failed to parse (syntax error): {\displaystyle G(z)=F^{-1}(z)=\log_2(\arccos(1\!−\!2z))}

Citizendium author


Dmitrii Kouznetsov
Date created


March 2011
Country of first publication


Japan
Notes


Expressions for the more general case are suggested in the article Logistic sequence.

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

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

  1. http://www.springerlink.com/content/u712vtp4122544x4/ D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98.
  2. 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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