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

Abstract

We numerically calculate, at the edge of chaos, the time evolution of the nonextensive entropic form $S_q\equiv [1-{\sum }_{i=1}^W{p}_i^q]/[q-1]$ (with $S_1=-{\sum }_{i=1}^W{p}_i\ln p_i$) for two families of one-dimensional dissipative maps, namely a logistic- and a periodic-like 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 $q^{\ast }<1$ exists such that the ${\lim }_{t\to \infty }{\lim }_{W\to \infty }{\lim }_{N\to \infty }{S}_q(t)/t$ is {\it finite}, {\it thus generalizing the (ensemble version of) Kolmogorov-Sinai entropy} (which corresponds to $q^{\ast }=1$ in the present formalism). This special, $z$-dependent, value $q^{\ast }$ numerically coincides, {\it 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 $f(\alpha )$ function).

6 pages and 6 figs

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• Entropy production in the cyclic lattice Lotka-Volterra model
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