Stochastic resonance on a circle without excitation: physical investigation and peak frequency formula

In this article the existence of stochastic resonance (SR) without external force in a simplified circular system for different values of the control parameter b is considered. The average power spectra are calculated as well as the signal-to-noise ratio as a measure for stochastic resonance. It is shown that in the monostable and semistable (b<1 and b=1) cases coherent oscillations occur and SR exists. For the case b>1, the system is oscillatory and noise plays only a destructive role; therefore no SR occurs. The rotation number of the system is calculated and compared to the peak frequency of the power spectrum. Although the coincidence in the noisy case is not as good as that in the deterministic case, we can derive an empirical formula between the peak frequency of the power spectrum and the rotation number of the system, which is in good agreement with results of numerical simulations.