The non-extensive version of the Kolmogorov-Sinai entropy at work

Abstract

We address the problem of applying the Kolmogorov-Sinai method of entropic analysis, expressed in a generalized non-extensive form, to the dynamics of the logistic map at the chaotic threshold, which is known to be characterized by a power law rather than exponential sensitivity to initial conditions. The computer treatment is made difficult, if not impossible, by the multifractal nature of the natural invariant distribution: Thus the statistical average is carried out on the power index. The resulting entropy time evolution becomes a smooth and linear function of time with the non-extensive index Q < 1 prescribed by the heuristic arguments of earlier work, thereby showing how to make the correct entropic prediction in the spirit of the single-trajectory approach of Kolmogorov.