Logistic sequence

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 iterations $f_4^c(x)$ 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, , but then it was generalized for complex values . 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 or Mathematica, one can easy calculate a dozen of such coefficients. According to the 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 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 $n$ 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 .

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 (used as logo of the Physics Department of the MSU and as part of the logo of TORI), and also the $\sqrt{\exp}$, discussed in , and various superfunctions, including the Ackermann functions.