Resonant Phenomenon in a Stochastic Delayed Bistable Chemical System

Abstract

In this paper, the resonant phenomenon for a bistable chemical system in the presence of noises and delayed feedback is investigated. The signal-to-noise ratio (SNR) is calculated when periodic signal is introduced addi- tively (or multiplicatively). The impacts of the parameter μ of the reaction, time delay τ , strength K of the feedback loop, multiplicative (D) and additive (Q) noise strengths and cross-correlation strength λ between two noises on the SNR are discussed. When the periodic signal is introduced additively, our results show (i) the SNR as a function of the parameter μ exhibits a maximum, the existence of the max- imum is a characteristic of the parametric resonance (PR) phenomenon; (ii) the SNR as a function of D exhibits only a maximum, however, for the case of SNR as a function of Q exhibits not only a maximum, but also a minimum. The existence of the maximum and minimum in the SNR is the identifying characteristics of the stochastic resonance (SR) and reverse-resonance (RR); and (iii) the increases of τ , K and λ enhance the SR and weaken the RR. Finally, we compare the resonant phenomenon for the additive peri- odic signal with that for multiplicative one in the chemical system.