Ensemble averages and nonextensivity at the edge of chaos of one-dimensional maps

Abstract

Ensemble averages of the sensitivity to initial conditions $\xi (t)$ and the entropy production per unit time of a {\it new} family of one-dimensional dissipative maps, $x_{t+1}=1-a{e}^{-1/|x_t{|}^z}(z>0)$, and of the known logistic-like maps, $x_{t+1}=1-a|x_t{|}^z(z>1)$, are numerically studied, both for {\it strong} (Lyapunov exponent ${\lambda }_1>0$) and {\it weak} (chaos threshold, i.e., ${\lambda }_1=0$) chaotic cases. In all cases we verify that (i) both $<{\ln }_q\xi >[{\ln }_{q}x\equiv (x^{1-q}-1)/(1-q);{\ln }_{1}x=\ln x]$ and $<S_q>[S_q\equiv (1-{\sum }_i{p}_i^q)/(q-1);S_1=-{\sum }_i{p}_i\ln p_i]$ {\it linearly} increase with time for (and only for) a special value of $q$, $q_{sen}^{av}$, and (ii) the {\it slope} of $<{\ln }_q\xi >$ and that of $<S_q>$ {\it coincide}, thus interestingly extending the well known Pesin theorem. For strong chaos, $q_{sen}^{av}=1$, whereas at the edge of chaos, $q_{sen}^{av}(z)<1$.

5 pages, 5 figures

References in corpus (1)

• Nonextensive Pesin identity. Exact renormalization group analytical results for the dynamics at the edge of chaos of the logistic map

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