Chaos edges of $z$-logistic maps: Connection between the relaxation and sensitivity entropic indices

Details Chaos edges of $z$-logistic maps: Connection between the relaxation and sensitivity entropic indices Chaos edges of $z$-logistic maps: Connection between the relaxation and sensitivity entropic indices

2005-02-25

arxiv.org/abs/cond-mat/0502607v1

Phys. Rev. E 73 (2006) 037201

Physics

Condensed Matter

Statistical Mechanics

5 pages, 5 figures

Scientific paper

10.1103/PhysRevE.73.037201

Chaos thresholds of the $z$-logistic maps $x_{t+1}=1-a|x_t|^z$ $(z>1; t=0,1,2,...)$ are numerically analysed at accumulation points of cycles 2, 3 and 5. We verify that the nonextensive $q$-generalization of a Pesin-like identity is preserved through averaging over the entire phase space. More precisely, we computationally verify $\lim_{t \to\infty}< S_{q_{sen}^{av}} >(t)/t= \lim_{t \to\infty}< \ln_{q_{sen}^{av}} \xi >(t)/t \equiv \lambda_{q_{sen}^{av}}^{av}$, where the entropy $S_{q} \equiv (1- \sum_i p_i^q)/ (q-1)$ ($S_1=-\sum_ip_i \ln p_i$), the sensitivity to the initial conditions $\xi \equiv \lim_{\Delta x(0) \to 0} \Delta x(t)/\Delta x(0)$, and $\ln_q x \equiv (x^{1-q}-1)/ (1-q) $ ($\ln_1 x=\ln x$). The entropic index $q_{sen}^{av}<1$, and the coefficient $\lambda_{q_{sen}^{av}}^{av}>0$ depend on both $z$ and the cycle. We also study the relaxation that occurs if we start with an ensemble of initial conditions homogeneously occupying the entire phase space. The associated Lebesgue measure asymptotically decreases as $1/t^{1/(q_{rel}-1)}$ ($q_{rel}>1$). These results led to (i) the first illustration of the connection (conjectured by one of us) between sensitivity and relaxation entropic indices, namely $q_{rel}-1 \simeq A (1-q_{sen}^{av})^\alpha$, where the positive numbers $(A,\alpha)$ depend on the cycle; (ii) an unexpected and new scaling, namely $q_{sen}^{av}(cycle n)=2.5 q_{sen}^{av}(cycle 2)+ \epsilon $ ($\epsilon=-0.03$ for $n=3$, and $\epsilon = 0.03$ for $n=5$).

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