Holomorphic extension of the logistic sequence

Abstract

The logistic problem is formulated in terms of the Superfunction and Abelfunction of the quadratic transfer function H(z) = uz(1 − z). The Superfunction F as holomorphic solution of equation H(F(z)) = F(z + 1) generalizes the logistic sequence to the complex values of the argument z. The efficient algorithm for the evaluation of function F and its inverse function, id est, the Abelfunction G are suggested; F(G(z)) = z. The halfiteration h(z) = F(1/2 + G(z)) is constructed; in wide range of values z, the relation h(h(z)) = H(z) holds. For the special case u = 4, the Superfunction F and the Abelfunction G are expressed in terms of elementary functions.

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Correspondence to D. Yu. Kouznetsov.

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Original Russian Text © D.Yu. Kouznetsov, 2010, published in Vestnik Moskovskogo Universiteta. Fizika, 2010, No. 2, pp. 24–31.

The article was translated by the author.

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Kouznetsov, D.Y. Holomorphic extension of the logistic sequence. Moscow Univ. Phys. 65, 91–98 (2010). https://doi.org/10.3103/S0027134910020049

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Key words

  • Logistic operator
  • Logistic sequence
  • Holomorphic extension
  • Superfunction
  • Abelfunction
  • Pomeau-Manneville scenario