Computational information for the logistic map at the chaos threshold

Abstract

We study the logistic map $f(x)=\lambda x(1-x)$ on the unit square at the chaos threshold. By using the methods of symbolic dynamics, the information content of an orbit of a dynamical system is defined as the Algorithmic Information Content (AIC) of a symbolic sequence. We give results for the behaviour of the AIC for the logistic map. Since the AIC is not a computable function we use, as approximation of the AIC, a notion of information content given by the length of the string after it has been compressed by a compression algorithm, and in particular we introduce a new compression algorithm called CASToRe. The information content is then used to characterise the chaotic behaviour.