TY - JOUR

T1 - Generalization of the Kolmogorov-Sinai entropy: Logistic-like and generalized cosine maps at the chaos threshold

AU - Tirnakli, Ugur

AU - Ananos, Garin F.J.

AU - Tsallis, Constantino

PY - 2001/10/8

Y1 - 2001/10/8

N2 - 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). © 2001 Elsevier Science B.V. All rights reserved.

AB - 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). © 2001 Elsevier Science B.V. All rights reserved.

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U2 - 10.1016/S0375-9601(01)00570-9

DO - 10.1016/S0375-9601(01)00570-9

M3 - Article

SP - 51

EP - 58

JO - Physics Letters, Section A: General, Atomic and Solid State Physics

JF - Physics Letters, Section A: General, Atomic and Solid State Physics

SN - 0375-9601

ER -