Generalization of the Kolmogorov-Sinai entropy: Logistic-like and generalized cosine maps at the chaos threshold

Ugur Tirnakli, Garin Fedor Janampa Añaños, Constantino Tsallis

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Abstract

We numerically calculate, at the edge of chaos, the time evolution of the nonextensive entropic form Sq ≡ [1 - Σi=1W piq]/[q - 1] (with S1 = - Σi=1W Pi 1n pi) for two families of one-dimensional dissipative maps, namely a logistic-like and a generalized cosine with arbitrary inflexion z at their maximum. At t = 0 we choose N initial conditions inside one of the W small windows in which the accessible phase space is partitioned; to neutralize large fluctuations we conveniently average over a large amount of initial windows. We verify that one and only one value q* < 1 exists such that the limt→∞ limW→∞ limN→∞ Sq (t)/t is finite, thus generalizing the (ensemble version of the) Kolmogorov-Sinai entropy (which corresponds to q* = 1 in the present formalism). This special, z-dependent, value q* numerically coincides, for both families of maps and all z, with the one previously found through two other independent procedures (sensitivity to the initial conditions and multifractal f(α) function).

Original languageEnglish
Pages (from-to)51-58
Number of pages8
JournalPhysics Letters, Section A: General, Atomic and Solid State Physics
Volume289
Issue number1-2
DOIs
StatePublished - 8 Oct 2001
Externally publishedYes

Bibliographical note

Funding Information:
One of us (U.T.) acknowledges the partial support of BAYG-C program by TUBITAK (Turkish agency) and CNPq (Brazilian agency) as well as by the Ege University Research Fund under the project no. 2000FEN049. This work has also been partially supported by PRONEX/FINEP, CNPq and FAPERJ (Brazilian agencies).

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