Generalization of the Kolmogorov-Sinai entropy: logistic-like and generalized cosine maps at the chaos threshold

We numerically calculate, at the edge of chaos, the time evolution of the nonextensive entropic form S-q equivalent to [1 - Sigma (W)(i=1) P-i(q)]/[q-1] (with S-1 equivalent to -Sigma (W)(i=1) pi ln pi) for two families of one-dimensional dissipative maps, namely a logistic-like and a generalized cosine with arbitrary inflexion z at their maximum. At t = 0 we choose N initial conditions inside one of the W small windows in which the accessible phase space is partitioned; to neutralize large fluctuations we conveniently average over a large amount of initial windows. We verify that one and only one value q* < 1 exists such that the lim(t --> infinity)limW(--> infinity) limN(--> infinity) S-q (t)/t is finite, thus generalizing the (ensemble version of the) Kolmogorov-Sinai entropy (which corresponds to q* = 1 in the present formalism). This special, z-dependent, value q* numerically coincides, for both families of maps and all z, with the one previously found through two other independent procedures (sensitivity to the initial conditions and multifractal 0 f (alpha) function). (C) 2001 Elsevier Science B.V. All rights reserved.