Generalization to nonextensive systems of the rate of entropy increase: The case of the logistic map

The asymptotic rate of increase $\kappa_1$ (expected to coincide with the Kolmogorov-Sinai entropy rate) of $S_1(t) \equiv - \sum_i p_i(t) \ln p_i(t)$ associated with a given dynamical system is generically a well--defined quantity (positive in case of strong chaos, i.e., {\it exponential} sensitivity to initial conditions, and zero otherwise). Typically $\kappa_1$ vanishes at the chaos threshold (weak chaos, i.e., typically {\it power-law} sensitivity to initial conditions), hence a better characterization is desirable there. On the simple example of the logistic map, we show that the recently introduced generalization $\kappa_q \equiv \lim_{t \to \infty} \frac{S_q(t)}{t}$ with $S_q(t) \equiv \frac{1-\sum_i[p_i(t)]^q}{q-1}$ provides a very convenient such characterization. There exists at the edge of chaos a unique value $q^*$ such that $0<\kappa_{q^*}< \infty$, whereas $\kappa_q$ vanishes (diverges) for $q>q^*$ ($q<q^*$). Moreover, the entropic index $q^*$ precisely coincides with the one which emerges from the analysis of the sensitivity to initial conditions, which coincides in turn with that calculated from the multifractal $f(\alpha)$ function.