Logistic sequence

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Logistic sequence ${\displaystyle F_u}$ is Superfunction of the quadratic transfer function

${\displaystyle f_u(z)=uz(1-z)}$

Parameter ${\displaystyle u}$ is usually assumed to be a positive constant. For ${\displaystyle u>1}$, the logistic sequence is entire function; ${\displaystyle F_u(z)\approx u^z+{𝒪}(u^{2z})}$ as the real part of ${\displaystyle z}$ approaches to minus infinity.

The transfer function ${\displaystyle f}$ is called also logistic operator. The non-integer iterates of ${\displaystyle f}$ can be expressed through the logistic sequence and its inverse function.

For the special case ${\displaystyle u=4}$, the logistic sequence can be expressed in terms of elementary functions; for this case, the iterations ${\displaystyle f_4^c(x)}$ are plotted versus ${\displaystyle x}$ for ${\displaystyle c=}$ 0.2, 0.5, 0.8, 1, 1.2, ${\displaystyle 1.5}$ In figure at right.

Evaluation of the logistic sequence[edit]

The non-integer iterations of the logistic operator can be constructed using the analytic continuation of the logistic sequence. The logistic sequence is function ${\displaystyle F}$ satisfying the recurrent equation

${\displaystyle F(z+1)=f_u(F(z))}$

Initially, such an equation was considered for integer values of ${\displaystyle z}$, see, for example, ^[1][2][3] , but then it was generalized for complex values ^[4]. In the simplest case, the logistic sequence allow the asymptotic representation

${\displaystyle F(z)=u^z+a_2{u}^{2z}+a_3{u}^{3z}+a_4{u}^{4z}+...}$

where ${\displaystyle a_2}$, ${\displaystyle a_3}$, .. are real coefficients. These coefficients can be found at the substitution of the asymptotic representation to the recurrent equation. In particular,

${\displaystyle a_2=\frac{-1}{u-1}}$

${\displaystyle a_3=\frac{2}{(u-1)(u^2-1)}}$

${\displaystyle a_4=\frac{-5-u}{(u-1)(u^2-1)(u^3-1)}}$

With some Maple or Mathematica, one can easy calculate a dozen of such coefficients. More complicated solutions with other asymptotic behaviors can be constructed in the similar way.

The series above diferge, but still allow the fast and precise evaluation. For value of ${\displaystyle z}$ such that ${\displaystyle u^z}$ is not small, the asymptotic representation

${\displaystyle F(z)=f_u^n(z-n)}$

can be used for some natural ${\displaystyle n}$ such that ${\displaystyle u^{z-n}}$ is small.

Inverse of the logistic sequence[edit]

For the logistic transfer function, the Abel function ${\displaystyle G}$ is the inverse function of the logistic sequence ${\displaystyle F}$. The Abel function satisfies the Abel equation

${\displaystyle G(f(z))=G(z)+1}$

This Abel-function can be expressed through the asimptotic representation, inverting that for the Superfunciton:

${\displaystyle G(z)={\log }_u(z+s_2{z}^2+s_3{z}^3+...)}$

The coefficients ${\displaystyle s}$ can be found substituting the representation into the Abel equation; with some Mathematica of Maple one can easy get a dozen of such coefficients. In particular,

${\displaystyle s_2=-a_2=\frac{1}{u-1}}$

${\displaystyle s_3=\frac{2u}{(u-1)(u^2-1)}}$

${\displaystyle s_4=\frac{(u^2-5)u}{(u-1)(u^2-1)(u^3-1)}}$

Again, the series is asymptotic, and if the argument is not small, the Abel function can be evaluated as

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

for some natural ${\displaystyle n}$ such that ${\displaystyle |f^{-n}(z)|\ll 1}$.

The inverse function for the logistic operator can be expressed as follows:

${\displaystyle f^{-1}(z)=\frac{1}{2}-\sqrt{\frac{1}{4}-\frac{z}{u}}}$

Iterations of the logistic operator[edit]

As usually, the combination of the superfunction (which is logistic sequence ${\displaystyle F}$) and the Abel function (which is ${\displaystyle G}$, the inverse of the logistic sequence) allows to evaluate the arbitrary (in particular, fractional and even complex) iterations of the logistic transfer function:

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

This representation is used to plot the non-integer iterates of the logistic operator, shown in the upper right corner of this article. Namely for the case ${\displaystyle u=4}$, the representation through the elementary function could be used too. Such a representation is described below.

Special case ${\displaystyle u=4}$[edit]

The logistic sequence is relatively simple superfunction, and in the case ${\displaystyle u=4}$, it can be expressed through the elementary function,

${\displaystyle F(z)=\frac{1}{2}(1-\cos (2^z))}$

In this case, the Abel function

${\displaystyle G(z)={\log }_2(\arccos (1-2z))}$

Such a representation follows also from the table of superfunctions ^[5].

The combination gives the expression for the iteration of the transfer function:

${\displaystyle f^c(z)=\frac{1}{2}(1-\cos ({\exp }_2(c+{\log }_2(\arccos (1-2z))))}$

Such a representation can be simplified, this leads to the expression

${\displaystyle f^c(z)=\frac{1}{2}(1-\cos (2^c\arccos (1-2z)\left)\right)}$

In such a way, for ${\displaystyle u=1}$, 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[edit]

For values ${\displaystyle u>1}$, the logistic sequence ${\displaystyle F_u}$ appears as superfunction of the logistic operator ${\displaystyle f_u}$. Together with the Abel function ${\displaystyle G_u}$, this allows to evaluate various iterates of the logistic operator. In particular, the square root of the logistic operator can be evaluated, id est, such function ${\displaystyle h}$ that ${\displaystyle h(h(z))=f(z)}$.

In the similar way, the superfunctions and the Abel functions can be evaluated for various transfer functions. One may evaluate the square root of factorial ^[6] (used as logo of the Physics Department of the MSU and as part of the logo of TORI ^[7] ), and also the ${\displaystyle \sqrt{\exp }}$, discussed in ^[8][9][10], and various superfunctions, including the Ackermann functions.

This article had been copypasted and adopted (in particular, $...$ were replaced to <math>...</math>) from ^[11].

References[edit]