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

TL;DR

This paper investigates the chaos thresholds of $z$-logistic maps, verifying a generalized entropy identity, exploring the connection between sensitivity and relaxation indices, and discovering new scaling relations across different cycles.

Contribution

It provides the first numerical verification of a generalized Pesin-like identity for $z$-logistic maps and uncovers novel scaling laws linking sensitivity and relaxation indices.

Findings

Verified nonextensive entropy identity at chaos thresholds.

Discovered a quantitative relation between sensitivity and relaxation indices.

Identified new cycle-dependent scaling behavior of entropic indices.

Abstract

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 $n=5$).