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 Sq ≡ [1 - Σ_i=1^W p_i^q]/[q - 1] (with S₁ = - Σ_i=1^W P_i 1n p_i) 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→∞ lim_W→∞ lim_N→∞ 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 f(α) function).