Deviations from spectral Dirac comb due to semiperiodic pulses

Abstract: In the frequency power spectral density, periodic oscillations appear as a Dirac comb at integer multiples of the frequency of the period. In weakly nonlinear systems or systems close to the primary instability threshold, the periodicity may be perturbed, resulting in deviations from the Dirac comb. We present a stochastic model of such semiperiodic fluctuations, describing them as a superposition of pulses with a fixed shape. Closed form expressions are derived for the frequency power spectral density in the case of periodic pulse arrivals and a random distribution of pulse amplitudes and durations. In general, the spectrum is a Dirac comb located at multiples of the inverse periodicity time and modulated by the pulse spectrum. Deviations from strict periodicity in the arrivals are considered in two ways: either as a random offset to each periodic arrival (jitter) or as independently distributed waiting times between arrivals (renewal). Both ways remove the Dirac comb with remarkable efficiency, leaving mainly the spectrum of the pulse function. Where the jitter process modulates the mass of the higher harmonics, the renewal process leads to spectral broadening. We demonstrate the applicability of normally distributed waiting times to modelling. Contrary to the previous literature, we argue that negative waiting times do not pose problems for the theory, broadening the applicability of the normal approximation. Randomness in the pulse arrival times is investigated by numerical realizations of the process, and the model is used to describe time series of kinetic energy of fluctuating motions in two-dimensional thermal convection.