Difference between revisions of "Logistic sequence"
| Line 1: | Line 1: | ||
| + | #redirect[[LogisticSequence]] |
||
| − | [[File:Elutin1a4tori.jpg|400px|right|thumb|thumb|Iterations of the logistic transfer function, $f_4^c(x)$ for $c=$0.2, 0.5, 0.8,1, 1.2, 1.5]] |
||
| − | '''Logistic sequence''' or $\mathrm{LogisticSequance}$ $F_u$ is [[Superfunction]] of the quadratic transfer function |
||
| − | : $f_u(z)=u~ z ~ (1\!-\!z)$ |
||
| − | Parameter $u$ is usually assumed to be a positive constant. For $u>1$, the logistic sequence is [[entire function]]. |
||
| − | |||
| − | The transfer function $f$ is called also [[logistic operator]]. The non-integer iterates of $f$ can be expressed through the logistic sequence and its inverse function. |
||
| − | |||
| − | For the special case $u=4$, the logistic sequence can be expressed in terms of elementary functions; for this case, the [[iteration]]s $f_4^c(x)$ are plotted versus $x$ for $c=$ 0.2, 0.5, 0.8, 1, 1.2, $1.5$ In figure at right. |
||
| − | |||
| − | ==Evaluation of the logistic sequence== |
||
| − | The non-integer iterations of the logistic operator can be constructed using the analytic continuation of the logistic sequence. The logistic sequence is function $F$ satisfying the recurrent equation |
||
| − | : $F(z\!+\!1)=f_u(F(z))$ |
||
| − | Initially, such an equation was considered for integer values of $z$, see, for example, |
||
| − | <ref name="abbott"> |
||
| − | N.B.Tufillaro, T.Abbott, J.Reilly, An Experimetal Approach To Nonlinear Dynamics and Chaos (Addison Wesley, New York, 1992). |
||
| − | </ref><ref name="strogatz"> |
||
| − | S.H.Strogatz, Nonlinear Dynamics and Chaos (Addison Wesley, Reading, MA, 1994). |
||
| − | </ref><ref name="sprott">3.J.C Sprott, Chaos and Time Series Analysis (Oxford Univ., Oxford, 2003).</ref> |
||
| − | , but then it was generalized for complex values |
||
| − | <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>. In the simplest case, the logistic sequence allow the asymptotic representation |
||
| − | : $F(z)= u^z+ a_2 u^{2z} + a_3 u^{3z}+a_4 u^{4z}+...$ |
||
| − | where $a_2$, $a_3$, .. are real coefficients. These coefficients can be found at the substitution of the asymptotic representation to the recurrent equation. In particular, |
||
| − | : $a_2=\frac{-1}{u-1}$ |
||
| − | : $a_3=\frac{2}{(u-1)(u^2-1)}$ |
||
| − | : $a_4=\frac{-5-u}{(u-1)(u^2-1)(u^3-1)}$ |
||
| − | With some [[Maple (software)|Maple]] or [[Mathematica]], one can easy calculate a dozen of such coefficients. |
||
| − | According to the [[Main_Page#Axioms|Axioms of TORI]], such representation has priority; it is considered as principal. (More complicated solutions with other asymptotic behaviors can be constructed in the similar way.) |
||
| − | |||
| − | The series above diferge, but still allow the precise evaluation of the function. |
||
| − | For value of $z$, while $u^z$ is not small, the asymptotic representation |
||
| − | : $F(z)=f_u^n(z\!-\!n)$ |
||
| − | can be used for some natural $n$, such that $u^{z-n}$ is small. |
||
| − | |||
| − | ==Inverse of the logistic sequence== |
||
| − | For the logistic transfer function, the [[Abel function]] $G$ is the inverse function of the logistic sequence $F$. The Abel function satisfies the Abel equation |
||
| − | : $G(f(z))=G(z)+1$ |
||
| − | This Abel-function can be expressed through the asimptotic representation, inverting that for the Superfunciton: |
||
| − | : $G(z)=\log_u(z+s_2 z^2+s_3 z^3+...)$ |
||
| − | The coefficient $s$ can be found substituting the representation into the Abel equation; with some |
||
| − | [[Mathematica]] of [[Maple (software)|Maple]] one can easy get a dozen of such coefficients. |
||
| − | In particular, |
||
| − | : $s_2=-a_2=\frac{1}{u-1}$ |
||
| − | : $s_3=\frac |
||
| − | {2u} |
||
| − | { (u-1)(u^2-1) }$ |
||
| − | : $s_4=\frac{(u^2-5)u}{(u-1)(u^2-1)(u^3-1)}$ |
||
| − | Again, the series is asymptotic, and if the argument is not small, the Abel function can be evaluated as |
||
| − | : $G(z)=F(f^{-n}(z))+n$ |
||
| − | for some [[natural number|natural]] $n$ such that $|f^{-n}(z)|\ll 1$. |
||
| − | |||
| − | The inverse function for the logistic operator can be expressed as follows: |
||
| − | :$f^{-1}(z)= \frac{1}{2} - \sqrt{\frac{1}{4}-\frac{z}{u}}$ |
||
| − | |||
| − | ==Iterations of the logistic operator== |
||
| − | As usually, the combination of the Superfunction (which is logistic sequence $F$) and the Abel function |
||
| − | (which is $G$, the inverse of the logistic sequence) allows to evaluate the arbitrary (in particular, fractional and even complex) iterations of the logistic transfer function: |
||
| − | : $f^c(z)=F(c+G(z))$ |
||
| − | This representation is used to plot the non-integer iterates of the logistic operator, shown in the upper right corner of this article. |
||
| − | However, namely for the case $u=4$, the representation through the elementary function could be used too. Such a representation is suggested below. |
||
| − | |||
| − | ==Special case $u=4$== |
||
| − | The logistic sequence is relatively simple [[superfunction]], and in the case $u\!=\!4$, it can be expressed through the elementary function, |
||
| − | : $F(z)=\frac{1}{2}(1-\cos(2^z))$ |
||
| − | In this case, the Abel function |
||
| − | : $G(z)=\log_2(\arccos(1-2z))$ |
||
| − | Such a representation follows also from the table of superfunctions |
||
| − | <ref name="factorial"> |
||
| − | 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. (Russian version: p.8-14) |
||
| − | </ref>. |
||
| − | |||
| − | The combination gives the expression for the iteration of the transfer function: |
||
| − | : $ f^c(z)=\frac{1}{2} \Big(1-\cos\Big(\exp_2(c+\log_2(\arccos(1-2z)) \Big)\Big) $ |
||
| − | Such a representation can be simplified, this leads to the expression |
||
| − | : $ f^c(z)=\frac{1}{2} \Big(1-\cos\Big(2^c~\arccos(1-2z) \Big)\Big) $ |
||
| − | In such a way, for $u\!=\!4$, the iterations of the logistic operator, as well as its [[Superfunction]] and the [[Abelfunction]] can be expressed through the elementary functions. The last expression could be obtained also using the [[Schroeder function]] |
||
| − | of the logistic operator. |
||
| − | |||
| − | ==Conclusion== |
||
| − | For values $u>1$, the logistic sequence $F_u$ appears as [[superfunction]] of the logistic operator $f_u$. |
||
| − | Together with the Abel function $G_u$, this allows to evaluate various iterates of the logistic operator. |
||
| − | In particular, the square root of the logistic operator (its half-iteration) can be evaluated, id est, such function $h$ that |
||
| − | $h(h(z))=f(z)$. |
||
| − | |||
| − | In the similar way, the superfunctions and the Abel function can be evaluated for various transfer functions. |
||
| − | One may evaluate the [[square root of factorial]] |
||
| − | <ref name="factorial"> |
||
| − | 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. (Russian version: p.8-14) |
||
| − | </ref> (used as logo of the [[Physics Department of the MSU]] and as part of the logo of [[Main Page|TORI]]), |
||
| − | and also the $\sqrt{\exp}$, discussed in |
||
| − | <ref name="kneser"> |
||
| − | http://www.ils.uec.ac.jp/~dima/Relle.pdf |
||
| − | Reele analytische L\"osungen der Gleichung $\varphi(\varphi(x))=e^x$ und |
||
| − | verwandter Funktionalgeichungen. |
||
| − | Journal fur die reine und angewandte Mathematik. {\bf 187}, 56–67 (1950) |
||
| − | </ref><ref name="moce"> |
||
| − | http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html |
||
| − | D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78, 1647-1670 (2009), |
||
| − | </ref><ref name="sqrt2"> |
||
| − | http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html |
||
| − | D.Kouznetsov, H.Trappmann. |
||
| − | Portrait of the four regular super-exponentials to base sqrt(2). |
||
| − | Mathematics of Computation, 2010, v.79, p.1727-1756. |
||
| − | </ref>, and various [[superfunction]]s, including the [[Ackermann function]]s. |
||
| − | |||
| − | ==References== |
||
| − | <references/> |
||
| − | |||
| − | [[Category:Mathematical functions]] |
||
| − | [[Category:Superfunctions]] |
||
| − | [[Category:Logistic sequence]] |
||
| − | [[Category:Articles in English]] |
||
Latest revision as of 07:02, 1 December 2018
Redirect to:
