Stochastic resonance in bistable systems: The effect of simultaneous additive and multiplicative correlated noises

We analyze the effect of the simultaneous presence of correlated additive and multiplicative noises on the stochastic resonance response of a modulated bistable system. We find that when the correlation parameter is also modulated, the system’s response, measured through the output signal-to-noise ratio, becomes largely independent of the additive noise intensity. ∗Electronic address: [email protected], Fellow of CNEA †Electronic address: [email protected], Member of CONICET, Argentina and Regular Associate ICTP. http://www.cab.cnea.edu.ar/Cab/invbasica/FisEstad/estadis.htm 1 The phenomenon of stochastic resonance (SR) has attracted considerable interest in the last decade due, among other aspects, to its potential technological applications for optimizing the output signal-to-noise ratio (SNR) in nonlinear dynamical systems. The phenomenon shows the counterintuitive role played by noise in nonlinear systems as it contributes to enhance the response of a system subject to a weak external signal. There is a wealth of papers, conference proceedings and reviews on this subject, Ref. [1] being the most recent one, showing the large number of applications in science and technology, ranging from paleoclimatology, to electronic circuits, lasers, and noise-induced information flow in sensory neurons in living systems. A tendency shown in recent papers, and determined by the possible technological applications, points towards achieving an enhancement of the system response (that is obtaining a larger output SNR) by means of the coupling of several SR units [2–6] in what conforms an “extended medium” [7]. Another aspect that has also attracted interest is finding a system where the SNR becomes largely independent of external parameters such as noise intensity [4,8]. In this work we present some results related to the latter aspect. In order to reach such a goal we study a bistable system subject to both an additive and a multiplicative noise but, at variance with the work in Ref. [9], we consider that both noises are correlated. In addition to the modulation of the bistable potential by a weak external signal, we also consider the effect of a modulation of the correlation between both noises. We show that the result of this contribution is to widen the maximum of the SNR as a function of the additive noise intensity, making the detection of the signal less sensitive to the actual value of that noise. It is worth remarking here that on one hand, the additive (external) noise can be (naturally) assumed as white, while on the other hand, the multiplicative (internal) one, that generally involves characteristic time scales of the system, would not necessarilly be white. However, as indicated in [10], as a first step in this study, we can approximate the multiplicative colored noise by a white one. Here we exploit the results of Madureira et al. [10] where it was shown that the activation rate in a bistable system subject to correlated 2 additive and multiplicative noises is dramatically suppressed or enhanced according to the sign of the correlation. The model equation we consider corresponds to a onedimensional bistable system described by the Langevin equation ẋ = ε(t) + x− x + xξ1(t) + ξ2(t), (1) where ε(t) = ε0 cos(Ωt), ε0 and Ω are the intensity and frequency of the potential modulation respectively. The additive and multiplicative Gaussian white noises, indicated by ξ1(t) and ξ2(t), satisfy 〈ξi(t)〉 = 0, i = 1, 2 〈ξ1(t)ξ1(t)〉 = 2δ(t− t)Q 〈ξ2(t)ξ2(t)〉 = 2δ(t− t)D 〈ξ1(t)ξ2(t)〉 = 2ρ(t) δ(t− t) √ DQ. (2) The correlation intensity between both noises is indicated by the parameter ρ(t), that fulfills the condition |ρ| ≤ 1. The associated the Fokker-Planck equation (Stratonovich prescription) ∂tP (x, t) = − ∂x [(