Universality and the power spectrum at the onset of chaos

Abstract

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