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

The asymptotic rate of increase $\kappa_1$ of $S_1(t) \equiv - \sum_i p_i(t) \ln p_i(t)$ is a well-defined positive quantity in case of strong chaos and vanishes at the chaos threshold. On the simple example of the logistic map, we show that the 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 convenient way to treat the chaos threshold. In fact there exists a unique value $q^*$ such that $0<\kappa_{q^*}< \infty$, whereas $\kappa_q$ vanishes (diverges) for $q>q^*$ ($q<q^*$). The entropic index $q^*$ coincides with the values obtained from the study of the sensitivity to initial conditions, and from the multifractal $f(\alpha)$ function.