Universality and the power spectrum at the onset of chaos

Abstract

Two one-dimensional maps are iterated to evaluate the average height $\ensuremath{\varphi}(k)$ of the peaks in the power spectrum corresponding to frequencies ${\ensuremath{\omega}}_{k,l}=\frac{(2l\ensuremath{-}1)\ensuremath{\pi}}{{2}^{k}}$, where $l=1,2,\dots{},{2}^{k\ensuremath{-}1}$ and $k=1,2,\dots{}$ at the onset of chaos. It is shown that the ratio $\frac{\ensuremath{\varphi}(k)}{\ensuremath{\varphi}(k+1)}$ is nearly constant and for large $k$ approaches a universal limit $2{\ensuremath{\beta}}^{(2)}=20.963\dots{}$.