Generalization of the Kolmogorov–Sinai entropy: logistic-like and generalized cosine 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 $S_1=-\sum_{i=1}^Wp_i \ln p_i$) 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 $q^*<1$ 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 $q^*=1$ in the present formalism). This special, $z$-dependent, value $q^*$ 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).