Approach to chaos: Universal quantitative properties of one-dimensional maps

Using the renormalization-group method, numerical results, and analytical arguments, we obtain the universal metric properties of one-dimensional iterated maps which exhibit period doublings. Such maps can be classified according to their behaviors around the extremum. For maps with the extremum (at $x=0\mathrm{}$) of the symmetric form $f\left(x\right)\mathrm{}\sim {\left|x\right|}^{z}b\left(x\right)\mathrm{}$, with $z>\mathrm{}1\mathrm{}$, where the function $b\left(x\right)\mathrm{}$ modifies ${\left|x\right|}^z$ by less than any power of $\left|x\right|$, the quantitative universal properties of the perioddoubling approach to chaos is described for asymptotically large $n$, by ${\delta }_n=\frac{({\lambda }_n-{\lambda }_{n-1\mathrm{}})}{({\lambda }_{n+1\mathrm{}}-{\lambda }_n)}=\delta [1+\epsilon \mathrm{}(n\left)\mathrm{}\tau \ln \alpha \right]$ and ${\alpha }_n=\frac{-d_n}{d_{n+1\mathrm{}}}=\alpha [1+\epsilon \mathrm{}(n\left)\mathrm{}\kappa 1\mathrm{n}\alpha \right]$, where ${\lambda }_n$ is the value of the parameter at the $n$th period-doubling bifurcation, $d_n$ is the typical distance from the extremum at the $n$th bifurcation point, and $\epsilon \mathrm{}\left(n\right)={\left[\frac{\partial \ln b\left(x\right)\mathrm{}}{\partial \ln \left|x\right|}\right]}_{\mathrm{}\left|x\right|\mathrm{}\sim {\alpha }^{-n}}$, where terms of order $n^{-2\mathrm{}}$ are neglected. Here $\delta$, $\alpha$, $\kappa$, and $\tau$ are all functions of $z$ only; they obey the relations $\frac{d\ln \alpha }{\mathrm{dz}}=\kappa \ln \alpha$ and $\frac{d\ln \delta }{\mathrm{dz}}=\tau \ln \alpha$, where $\alpha$ and $\delta$ are the Feigenbaum constants. In particular, for $f\left(x\right)\mathrm{}\approx {\left|x\right|}^z{\left|\ln \right(x^2\left)\right|}^p {\left|\ln \right|\ln \left(x^2\right)\left| \right|}^q\cdots$, where $z>\mathrm{}1\mathrm{}$, then $d_n\approx {\alpha }^{-n}{n}^{\kappa p}{\left(\ln n\right)}^{\kappa q}\cdots$ and ${\lambda }_{\infty }-{\lambda }_n\approx {\delta }^{-n}{n}^{\tau p}{\left(\ln n\right)}^{\tau q}\cdots$, where ${\lambda }_{\infty }$ is the critical value of the parameter beyond which is chaos. For $z=2\mathrm{}$, $\alpha =2.502\mathrm{}91$, $\delta =4.669\mathrm{}20$, $\kappa =-0.444\mathrm{}53$, and $\tau =0.369\mathrm{}54$. For the sake of completeness, an analysis of maps with asymmetric extrema is also presented. In this case, $\alpha$ and $\delta$ are functions of $z$ as well as the size of the asymmetry.