Abstract
We investigate the escape problem of a Brownian particle over a potential barrier driven by bounded noise, which is different form the common unbounded noise. Some novel stochastic phenomena and results are founded. For weak frequency of bounded noise, the mean first passage time (MFPT) shows a non-monotonic dependence on the intensity of white noise. One remarkable behavior is the phenomenon of co-occurrence of noise-enhanced stability and resonant activation. Another novel finding is that the position of minimum of MFPT about frequency shows monotone non-decreasing or non-increasing function with the variation of amplitude of bounded noise, while the minimum of MFPT monotonously depends on the amplitude of bounded noise. More important, we uncover the relationship of the parameters of bounded noise which induce the occurrence of the phenomenon of resonant activation. Besides, resonant activation can also be induced by simplified bounded noise, called sine-Wiener noise. The behaviors presented in our work driven by bounded noise show distinctive features compared with the ones inspired by the common noise.
Similar content being viewed by others
References
Kramers, H.A.: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284–304 (1940)
Hänggi, P., Talkner, P., Borkovec, M.: Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys. 62, 251–341 (1990)
Benzi, R., Sutera, A., Vulpiani, A.: The mechanism of stochastic resonance. J. Phys. A 14, L453–L457 (1981)
Gammaitoni, L., Hänggi, P., Jung, P., Marchesoni, F.: Stochastic resonance. Rev. Mod. Phys. 70, 223–288 (1998)
Mantegna, R.N., Spagnolo, B.: Noise enhanced stability in an unstable system. Phys. Rev. Lett. 76, 563–566 (1996)
Mielke, A.: Noise induced stability in fluctuating, bistable potentials. Phys. Rev. Lett. 84, 818–821 (2000)
Fiasconaro Spagnolo, B., Boccaletti, S.: Signatures of noise-enhanced stability in metastable states. Phys. Rev. E 72, 061110 (2005)
Doering, C.R., Gadoua, J.C.: Resonant activation over a fluctuating barrier. Phys. Rev. Lett. 69, 2318–2321 (1992)
Xu, Y., Wang, X., Zhang, H.Q., Xu, W.: Stochastic stability for nonlinear systems driven by Lévy noise. Nonlinear Dyn. 68, 7–15 (2012)
Xu, Y., Gu, R.C., Zhang, H.Q., Xu, W., Duan, J.Q.: Stochastic bifurcations in a bistable Duffing–van der Pol oscillator with colored noise. Phys. Rev. E 83, 056215 (2011)
Xu, Y., Ma, S., Zhang, H.Q.: Hopf bifurcation control for stochastic dynamical system with nonlinear random feedback method. Nonlinear Dyn. 65, 77–84 (2011)
Marchi, M., Marchesoni, F., Gammaitoni, L., Menichella-Saetta, E., Santucci, S.: Resonant activation in a bistable system. Phys. Rev. E 54, 3479–3487 (1996)
Flomenbom, O., Klafter, J.: Resonant activation in discrete systems. Phys. Rev. E 69, 051109 (2004)
Pankratova, E.V., Polovinkin, A.V., Mosekilde, E.: Resonant activation in a stochastic Hodgkin-Huxley model: interplay between noise and suprathreshold driving effects. Eur. Phys. J. B 45, 391–397 (2005)
Maddox, J.: Surmounting fluctuating barriers. Nature 359, 771 (1992)
Beece, D., Eisenstein, L., Frauenfelder, H., Good, D., Marden, M.C., Reinisch, L., Reynolds, A.H., Sorensen, L.B., Yue, K.T.: Solvent viscosity and protein dynamics. Biochemistry 19, 5147–5157 (1980)
Leibler, S.: Moving forward noisily. Nature 370, 412–413 (1994)
Zurcher, U., Doering, C.R.: Thermally activated escape over fluctuating barriers. Phys. Rev. E 47, 3862–3869 (1993)
Bier, M., Astumian, R.D.: Matching a diffusive and a kinetic approach for escape over a fluctuating barrier. Phys. Rev. Lett. 71, 1649–1652 (1993)
Reimann, P.: Thermally driven escape with fluctuating potentials: a new type of resonant activation. Phys. Rev. Lett. 74, 4576–4579 (1995)
Hänggi, P.: Escape over fluctuating barriers driven by colored noise. Chem. Phys. 180, 157–166 (1994)
Bartussek, R., Madureira, A., Hänggi, P.: Surmounting a fluctuating double well: a numerical study. Phys. Rev. E 52, R2149–R2152 (1995)
Iwaniszewski, J.: Escape over a fluctuating barrier: limits of small and large correlation times. Phys. Rev. E 54, 3173–3184 (1996)
Marchi, M., Marchesoni, F., Gammaitoni, L., Menichella-Saetta, E., Santucci, S.: Resonant activation in a bistable system. Phys. Rev. E 54, 3479–3487 (1996)
Reimann, P., Bartussek, R., Hänggi, P.: Reaction rates when barriers fluctuate: a singular perturbation approach. Chem. Phys. 235, 11–26 (1998)
Marchesoni, F., Gammaitoni, L., Menichella-Saetta, E., Santucci, S.: Thermally activated escape controlled by colored multiplicative noise. Phys. Lett. A 201, 275–280 (1995)
Majee, P., Goswami, G., Bag, B.C.: Colored non-Gaussian noise induced resonant activation. Chem. Phys. Lett. 416, 256–260 (2005)
Dybiec, B., Gudowska-Nowak, E.: Levy stable noise-induced transitions: Stochastic resonance, resonant activation and dynamic hysteresis. J. Stat. Mech. P05004 (2009)
d’Onofrio, A.: Bounded-noise-induced transitions in a tumor-immune system interplay. Phys. Rev. E 81, 021923 (2010)
d’Onofrio, A., Gandolfi, A.: Resistance to antitumor chemotherapy due to bounded noise induced transitions. Phys. Rev. E 82, 061901 (2010)
Wedig, W.V.: Invariant measures and Lyapunov exponents for generalized parameter fluctuations. Struct. Saf. 8, 13–25 (1990)
Wedig, W.V.: Iterative schemes for stability problems with non-singular Fokker-Planck equations. Int. J. Non-Linear Mech. 31, 707–715 (1996)
Dimentberg, M.F.: A stochastic model of parametric excitation of a straight pipe due to slug flow of a two-phase fluid. In: Proceedings of the Fifth International Conference on Flow Induced Vibrations, Brighton, pp. 207–215 (1991)
Dimentberg, M.F.: Stability and sub-critical dynamics of structures with spatially disordered parametric excitation. Probab. Eng. Mech. 7, 131–134 (1992)
Li, Q.C., Lin, Y.K.: New stochastic theory for bridge stability in turbulent flow. J. Eng. Mech. 121, 102–116 (1995)
Bobryk, R.V., Chrzeszczyk, A.: Transitions induced by bounded noise. Physica A 358, 263–272 (2005)
Bobryk, R.V., Chrzeszczyk, A.: Transitions in a Duffing oscillator excited by random noise. Nonlinear Dyn. 51, 541–550 (2008)
Liu, W.Y., Zhu, W.Q., Huang, Z.L.: Effect of bounded noise on chaotic motion of duffing oscillator under parametric excitation. Chaos Solitons Fractals 12, 527–537 (2001)
Jin, Y.F., Hu, H.Y.: Principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation. Nonlinear Dyn. 50, 213–227 (2007)
Gradiner, C.W.: Handbook of Stochastic Methods. Springer, Berlin (1985)
Fiasconaro, A., Spagnolo, B., Ochab-Marcinek, A., Gudowska-Nowak, E.: Co-occurrence of resonant activation and noise-enhanced stability in a model of cancer growth in the presence of immune response. Phys. Rev. E 74, 041904 (2006)
Acknowledgements
This work was supported by the National Natural Science Foundation of China Grant Nos. 11172233, 10972181, 11102132, and 11202160.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, D., Xu, W., Yue, X. et al. Bounded noise enhanced stability and resonant activation. Nonlinear Dyn 70, 2237–2245 (2012). https://doi.org/10.1007/s11071-012-0614-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-012-0614-9