Broadening of spectral peaks at the merging of chaotic bands in period-doubling systems

For systems that undergo period-doubling cascades there also exists an ‘‘inverse cascade’’ of chaotic band mergings. The frequency spectrum associated with a chaotic orbit of $2^k$ constituent bands has $2^k$ δ-function spectral components superposed on a broadband continuous component. At merging, pairs of the $2^k$ bands join to produce $2^{k\mathrm{-}1}$ bands. Associated with this, the number of δ-function spectral peaks halves. This happens via the acquisition of a finite spectral broadened width by every other δ function (hence making them no longer δ functions). This paper investigates this transition in detail with emphasis on its scaling properties.