Abstract
The effects of time delay on stochastic resonance (SR) in a bistable system with time delay, correlated noises and periodic signal are studied by using the theory of signal-to-noise ratio (SNR). The expression of the SNR is derived under the adiabatic limit and the small delay time approximation. It is found that: (i) For the case of no correlations between multiplicative and additive noise, the delay time τ can enhance the SNR as a function of the multiplicative noise intensity α and it can restrain the SNR as a function of the additive noise intensity D; (ii) For the case of correlations between multiplicative and additive noise, τ can induce a minimum and maximum in curve of the SNR as a function of α, and can intensively restrain the SNR as a function of the D and there is a critical value of delay tim τ c =0.1 in the height of the SNR peak with change of τ, i.e., when τ takes value blow τ c , the τ boosts up the SNR as a function of the strength λ of correlations between multiplicative and additive noise, however, when τ takes value above τ c , the τ restrains that.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benzi, R., Sutera, A., Vulpiani, A.: The mechanism of stochastic resonance. J. Phys., A, Math. Gen. 14, L453 (1981)
Benzi, R., Sutera, A., Vulpiani, A.: Stochastic resonance in climatic change. Tellus 34, 10 (1982)
Nicolis, C.: Solar variability and stochastic effects on climate. Sol. Phys. 74, 473 (1981)
Nicolis, C.: Stochastic aspects of climatic transitions-response to a periodic forcing. Tellus 34, 1 (1982)
Fauve, S., Heslot, F.: Stochastic resonance in a bistable system. Phys. Lett. A 97, 5 (1983)
McNamara, B., Wiesenfeld, K., Roy, R.: Observation of stochastic resonance in a ring laser. Phys. Rev. Lett. 60, 2626 (1988)
McNamara, B., Wiesenfeld, K.: Theory of stochastic resonance. Phys. Rev. A 39, 4854 (1989)
Dykman, M.I., Mannella, R., McClintock, P.V.E., Stocks, N.G.: Comment on “Stochastic resonance in bistable systems”. Phys. Rev. Lett. 65, 2606 (1990)
Hu, G., Nicolis, G., Nicolis, C.: Periodically forced Fokker-Planck equation and stochastic resonance. Phys. Rev. A 42, 2030 (1990)
Zhou, T., Moss, F., Jung, P.: Escape-time distributions of a periodically modulated bistable system with noise. Phys. Rev. A 42, 3161 (1990)
Park, S.H., Kim, S., Pyo, H.B., Lee, S.: Effects of time-delayed interactions on dynamic patterns in a coupled phase oscillator system. Phys. Rev. E 60, 4962 (1999)
Kim, S., Park, S.H., Pyo, H.B.: Stochastic resonance in coupled oscillator systems with time delay. Phys. Rev. Lett. 82, 1620 (1999)
Kim, S., Park, S.H., Ryu, C.S.: Multistability in coupled oscillator systems with time delay. Phys. Rev. Lett. 79, 2911 (1997)
Ramana Reddy, D.V., Sen, A., Johnston, G.L.: Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators. Phys. Rev. Lett. 85, 3381 (2000)
Herrero, R., Figueras, M., Rius, J., Pi, F., Orriols, G.: Experimental observation of the amplitude death effect in two coupled nonlinear oscillators. Phys. Rev. Lett. 84, 5312 (2000)
Stephen Yeung, M.K., Strogatz, S.H.: Time delay in the Kuramoto model of coupled oscillators. Phys. Rev. Lett. 82, 648 (1999)
Schuster, H.G., Wagner, P.: Mutual entrainment of two limit cycle oscillators with time delayed coupling. Prog. Theor. Phys. 81, 939 (1989)
Masoller, C., Cavalcante, de Hugo, L.D.S., Rios Leite, J.R.: Delayed coupling of logistic maps. Phys. Rev. E 64, 037202 (2001)
Kori, H., Kuramoto, Y.: Slow switching in globally coupled oscillators: robustness and occurrence through delayed coupling. Phys. Rev. E 63, 046214 (2001)
Nie, L.R., Mei, D.C.: Noise and time delay: Suppressed population explosion of the mutualism system. Europhys. Lett. 79, 20005 (2007)
Jia, Y., Yu, S.N., Li, J.R.: Stochastic resonance in a bistable system subject to multiplicative and additive noise. Phys. Rev. E 62, 1869 (2000)
Jia, Y., Zheng, X.P., Hu, X.M., Li, J.R.: Effects of colored noise on stochastic resonance in a bistable system subject to multiplicative and additive noise. Phys. Rev. E 63, 031107-8 (2001)
Luo, X.Q., Zhu, S.Q.: Stochastic resonance driven by two different kinds of colored noise in a bistable system. Phys. Rev. E 67, 021104 (2003)
Jin, Y.F., Hu, H.Y.: Coherence and stochastic resonance in a delayed bistable system. Physica A 382, 423 (2007)
Wu, D., Zhu, S.Q.: Stochastic resonance in a bistable system with time-delayed feedback and non-Gaussian noise. Phys. Lett. A 363, 202 (2007)
Nie, L.R., Mei, D.C.: Effects of time delay on the bistable system subjected to correlated noises. Chin. Phys. Lett. 24, 3074 (2007)
Frank, T.D.: Delay Fokker-Planck equations, Novikov’s theorem, and Boltzmann distributions as small delay. Phys. Rev. E 72, 011112 (2005)
Jia, Y., Li, J.R.: Transient properties of a bistable kinetic model with correlations between additive and multiplicative noises: Mean first-passage time. Phys. Rev. E 53, 5764 (1996)
Fox, R.F.: Functional-calculus approach to stochastic differential equations. Phys. Rev. A 33, 467 (1986)
Hänggi, P., Marchesoni, F., Grigolini, P.: Bistable flow driven by coloured gaussian noise: A critical study. Z. Phys. B 56, 333 (1984)
Madureira, A.J.R., Hänggi, P., Wio, H.S.: Giant suppression of the activation rate in the presence of correlated white noise sources. Phys. Lett. A 217, 248 (1996)
Zhu, S.Q.: Steady-state analysis of a single-mode laser with correlations between additive and multiplicative noise. Phys. Rev. A 47, 2405 (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mei, D.C., Du, L.C. & Wang, C.J. The Effects of Time Delay on Stochastic Resonance in a Bistable System with Correlated Noises. J Stat Phys 137, 625–638 (2009). https://doi.org/10.1007/s10955-009-9864-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-009-9864-4