The rate of entropy increase at the edge of chaos

Under certain conditions, the rate of increase of the statistical entropy of a simple, fully chaotic, conservative system is known to be given by a single number, characteristic of this system, the Kolmogorov-Sinai entropy rate. This connection is here generalized to a simple dissipative system, the logistic map, and especially to the chaos threshold of the latter, the edge of chaos. It is found that, in the edge-of-chaos case, the usual Boltzmann-Gibbs-Shannon entropy is not appropriate. Instead, the non-extensive entropy S-q = (1 - Sigma(i)(w) = (1) p(i)(q)) / (q - 1), must be used. The latter contains a parameter q, the entropic index which must be given a special value q* not equal 1 (for q = 1 one recovers the usual entropy) characteristic of the edge-of-chaos under consideration. The same q* enters also in the description of the sensitivity to initial conditions, as well as in that of the multifractal spectrum of the attractor. (C) 2000 Published by Elsevier Science B.V.