August  2002, 2(3): 415-431. doi: 10.3934/dcdsb.2002.2.415

Computational information for the logistic map at the chaos threshold

1. 

Department of Mathematics, University of Pisa, via Buonarroti, 2/a, 56127 Pisa, Italy

2. 

Centro Interdisciplinare per lo Studio dei Sistemi Complessi, University of Pisa, via Bonanno, 25/b, 56126 Pisa, Italy

Received  August 2001 Revised  November 2001 Published  May 2002

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.
Citation: C. Bonanno, G. Menconi. Computational information for the logistic map at the chaos threshold. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 415-431. doi: 10.3934/dcdsb.2002.2.415
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