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Computational information for the logistic map at the chaos threshold

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  • 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.
    Mathematics Subject Classification: 28D20, 37E05, 37M25, 60J10.

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