Criticality of the routes to chaos of the1−a|x|zmap

Abstract

We discuss, within a unified renormalization-group (RG) framework, the onset of chaos appearing in the $1\ensuremath{-}a{|x|}^{z}$ map for real $z\ensuremath{\ge}1$. In particular, we study with some detail the criticality associated with the $k\ensuremath{\rightarrow}\ensuremath{\infty}$ limit of the ${p}^{k}$ bifurcation sequences [$p=2, 3, \mathrm{and} 4$, which correspond, respectively, to the ${R}^{*k}$, ${(\mathrm{RL})}^{*k}$, and ${({\mathrm{RL}}^{2})}^{*k}$ Metropolis-Stein-Stein sequences]. The critical points ${a}_{p}^{*}(z)$ monotonically increase, for all values of $p$, from ${a}_{p}^{*}(1)$ [$1\ensuremath{\le}{a}_{p}^{*}(1)<2$] to 2 as $z$ increases from 1 to \ensuremath{\infty}. The $z$ dependence of ${\ensuremath{\delta}}_{p}(z)$ for $p=2$ [${\ensuremath{\delta}}_{2}(z)$ monotonically increases as $z$ increases from 1 to \ensuremath{\infty}] is different from that associated with $p>2$ [${\ensuremath{\delta}}_{p}(z)$ diverges in $z\ensuremath{\rightarrow}1$ and (possibly) $z\ensuremath{\rightarrow}\ensuremath{\infty}$ limits, presenting a minimum in the neighborhood of $z=2$]. The present RG recovers, for both ${a}_{p}$ and ${\ensuremath{\delta}}_{p}$ and all values of $p$, the exact asymptotic behaviors in the $z\ensuremath{\rightarrow}1$ limit. It suggests, in the $z\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, the (possibly exact) asymptotic behaviors ${a}_{p}^{*}\ensuremath{-}2\ensuremath{\propto}\frac{1}{z}$ and ${\ensuremath{\delta}}_{p}\ensuremath{\propto}z$.