Logistic sequence

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Iterations of the logistic transfer function, for 0.2, 0.5, 0.8,1, 1.2, 1.5

Logistic sequence is Superfunction of the quadratic transfer function

Parameter is usually assumed to be a positive constant. For , the logistic sequence is entire function; as the real part of approaches to minus infinity.

The transfer function is called also logistic operator. The non-integer iterates of can be expressed through the logistic sequence and its inverse function.

For the special case , the logistic sequence can be expressed in terms of elementary functions; for this case, the iterations are plotted versus for 0.2, 0.5, 0.8, 1, 1.2, 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 satisfying the recurrent equation

Initially, such an equation was considered for integer values of , 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

where , , .. are real coefficients. These coefficients can be found at the substitution of the asymptotic representation to the recurrent equation. In particular,

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 such that is not small, the asymptotic representation

can be used for some natural such that is small.

Inverse of the logistic sequence

For the logistic transfer function, the Abel function is the inverse function of the logistic sequence . The Abel function satisfies the Abel equation

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

The coefficients 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,

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

for some natural such that .

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

Iterations of the logistic operator

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

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 , the representation through the elementary function could be used too. Such a representation is described below.

Special case

The logistic sequence is relatively simple superfunction, and in the case , it can be expressed through the elementary function,

In this case, the Abel function

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

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

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

In such a way, for , 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.


For values , the logistic sequence appears as superfunction of the logistic operator . Together with the Abel function , 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 that .

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 [5] (used as logo of the Physics Department of the MSU and as part of the logo of TORI [6] ), and also the , discussed in [7][8][9], and various superfunctions, including the Ackermann functions.

This article had been copypasted and adopted (in particular, $...$ were replaced to <math>...</math>) from [10].


  1. Tufillaro, N.B.; T.Abbott, J.Reilly (1992). "An Experimetal Approach To Nonlinear Dynamics and Chaos". Addison Wesley, New York.
  2. Strogatz, S.H. (1994). "Nonlinear Dynamics and Chaos". Addison Wesley, Reading, MA.
  3. Sprott, J.C (2003). "Chaos and Time Series Analysis". Oxford Univ., Oxford.
  4. Kouznetsov, Dmitrii (2010). "Holomorphic extension of the logistic sequence". Moscow University Physics Bulletin: 91-98.
  5. 5.0 5.1 Kouznetsov, Dmitrii; Trappmann Henryk (2010). "Superfunctions and square root of factorial". Moscow University Physics Bulletin 65 (1): 6-12.
  6. http://tori.ils.uec.ac.jp/TORI , Tools for Outstanding Research and Investigation
  7. (1950) "Reele analytische Losungen der Gleichung und verwandter Funktionalgeichungen.". Journal fur die reine und angewandte Mathematik 187: 56–67.
  8. Kouznetsov, Dmitrii (2009). "Analytic solution of F(z+1)=exp(F(z)) in complex z-plane". Mathematics of Computation 78: 1647-1670.
  9. Kouznetsov, Dmitrii; H.Trappmann (2010). "Portrait of the four regular super-exponentials to base sqrt(2)". Mathematics of Computation 79: 1727-175. .
  10. http://tori.ils.uec.ac.jp/TORI/index.php/Logistic_sequence