Periodically driven linear system with multiplicative colored noise

Abstract

A periodically driven linear system subject to multiplicative correlated noise is considered. It has been argued recently by several authors that such a simple system exhibits stochastic resonance. By introducing a general type of composite stochastic process, bridging two previously considered limiting cases of dichotomous and Gaussian noise, it is proved that, indeed, the amplitude of the average of the driven linear process at long times shows a pronounced maximum both as a function of the noise strength and as a function of the autocorrelation time. However, this kind of stochastic resonant behavior can be experimentally observable only in a special case where the initial phase of the external forcing is somehow fixed. Additional averaging over the uniform distribution of the initial random phase, inherent in most physical systems, leads to that the periodic output vanishes identically at long times. Moreover, the system response is typically defined in terms of the power spectrum rather than the amplitude of the average. The output signal given by the spectral density corresponding to the frequency of the external forcing is calculated via the long-time phase-averaged correlation function. It appears that the output signal simply diverges upon approaching the second moment instability point with increasing noise strength. No stochastic resonance is observed for any parameter settings. Interestingly, the resonancelike behavior of the system response as a function of the autocorrelation time is retained.