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

Oct, 2001
9 pages
Published in:
  • Phys.Lett.A 289 (2001) 51-58
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Abstract: (arXiv)
We numerically calculate, at the edge of chaos, the time evolution of the nonextensive entropic form Sq≡[1−∑i=1Wpiq]/[q−1]S_q \equiv [1-\sum_{i=1}^W p_i^q]/[q-1] (with S1=−∑i=1Wpiln⁡piS_1=-\sum_{i=1}^Wp_i \ln p_i) for two families of one-dimensional dissipative maps, namely a logistic- and a periodic-like with arbitrary inflexion zz at their maximum. At t=0t=0 we choose NN initial conditions inside one of the WW 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∗<1q^*<1 exists such that the lim⁡t→∞lim⁡W→∞lim⁡N→∞Sq(t)/t\lim_{t\to\infty} \lim_{W\to\infty} \lim_{N\to\infty} S_q(t)/t is {\it finite}, {\it thus generalizing the (ensemble version of) Kolmogorov-Sinai entropy} (which corresponds to q∗=1q^*=1 in the present formalism). This special, zz-dependent, value q∗q^* numerically coincides, {\it for both families of maps and all zz}, with the one previously found through two other independent procedures (sensitivity to the initial conditions and multifractal f(α)f(\alpha) function).
Note:
  • 6 pages and 6 figs