The complexity of the logistic map at the chaos threshold

We apply a generalized version of the Kolmogorov-Sinai entropy, based on a non-extensive form, to analyzing the dynamics of the logistic map at the chaotic threshold, the paradigm of power-law sensitivity to initial conditions. We make the statistical averages on the distribution of the power indexes beta, and we show that the resulting entropy time evolution becomes a linear function of time if we assign to the non-extensive index q the value Q < 1 prescribed by the heuristic arguments of earlier work. We also show that the emerging entropy index Q is determined by the asymptotic mean value of the index beta, and that this same mean value determines the strength of the logarithmic time increase of entropy, stemming from the adoption of the ordinary Shannon form. (C) 2001 Elsevier Science B.V. All rights reserved.