A perturbation calculation shows that the power spectrum of strange attractor near the accumulation parameter of a period-doubling bifurcation sequence consists of peaks broadened by a phase modulation, with broad skirts created by an amplitude modulation. Moving towards the accumulation parameter, at each bifurcation the total noise power decreases by a factor of 10.48, the average peak width decreases by a factor of 20.96, and the spectral bandwidth of the skirts decreased by a factor of 2.
Farmer, J.D. (1981). 'Spectral Broadening of Period-Doubling Bifurcation Sequences'. Physical Review Letters, 47(3), pp.179-182.