Dirac comb and exponential frequency spectra in nonlinear dynamics

An exponential frequency power spectral density is a well known property ofmany continuous time chaotic systems and has been attributed to the presence of Lorentzian-shaped pulses in the time series of the dynamical variables. Here a stochastic modelling of such fluctuations are presented, describing these as a super-position of pulses with fixed shape and constant duration. Closed form expressions are derived for the lowest order moments, auto-correlation function and frequency power spectral density in the case of periodic pulse arrivals and a random distribution of pulse amplitudes. In general, the spectrum is a Dirac comb located at multiple sof the periodicity time and modulated by the pulse spectrum. Of the effects considered, only deviations from periodicity remove the Dirac Comb, and do so rapidly.Randomness in the pulse arrival times is investigated by numerical realizations of the process and the results are discussed in the context of the Lorenz system.