Dynamics of a stochastic system driven by cross-correlated sine-Wiener bounded noises

Abstract

The sine-Wiener noise, as one new type of bounded noise and a natural tool to model fluctuations in dynamical systems, has been applied to problems in a variety of areas, especially in biomolecular networks and neural models. In this paper, by virtue of the Novikov theorem, Fox’s approach, and the ansatz of Hanggi, an approximate Fokker–Planck equation is derived for an one-dimensional Langevin-type equation with cross-correlated sine-Wiener noise. Meanwhile, the dynamical characters of a bistable system driven by cross-correlated sine-Wiener noise are investigated by applying the approximate theoretical method. For the bistable system, the cross-correlation intensity \(\lambda \) can induce the reentrance-like phase transition, while the other noise intensities and the self-correlation time, except for the self-correlation time of additive bounded noise, can induce the first-order-like phase transition. The transition from the stable state to another one can be accelerated by \(\alpha \) (additive bounded noise intensity), \(\tau _1\) (the self-correlation time of the multiplicative bounded noise), and \(\tau _2\) (the self-correlation time of the additive bounded noise) and can be restrained with \(\lambda \) and \(\tau _3\) (self-correlation time of the cross-correlation bounded noise). It is interesting that the noise-enhanced stability phenomenon is observed with D (multiplicative bounded noise intensity) varying for the positive correlation (\(\lambda >0\)) and is enhanced as \(\lambda \) increases. The numerical results are in basic agreement with the theoretical predictions.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

References

  1. 1.

    Gammaitoni, L., Hänggi, P., Jung, P., Marchesoni, F.: Stochastic resonance. Rev. Mod. Phys. 70, 223 (1998)

    Article  Google Scholar 

  2. 2.

    Horsthemke, W., Lefever, R.: Noise-Induced Transitions: Theory and Applications in Physics, Chemistry and Biology. Springer, Berlin (1984)

    Google Scholar 

  3. 3.

    Sagues, F., Sancho, J.M., Ojalvo, J.G.: Spatiotemporal order out of noise. Rev. Mod. Phys. 79, 829 (2007)

    Article  Google Scholar 

  4. 4.

    Reimann, P.: Brownian motors: noisy transport far from equilibrium. Phys. Rep. 361, 57 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. 5.

    Hänggi, P., Marchesoni, F.: Artificial Brownian motors: controlling transport on the nanoscale. Rev. Mod. Phys. 81, 387 (2009)

    Article  Google Scholar 

  6. 6.

    Mantegna, R.N., Spagnolo, B.: Noise enhanced stability in an unstable system. Phys. Rev. Lett. 76, 563 (1996)

    Article  Google Scholar 

  7. 7.

    Hänggi, P.: Stochastic resonance in biology how noise can enhance detection of weak signals and help improve biological information processing. ChemPhysChem 3(3), 285–290 (2002)

    Article  Google Scholar 

  8. 8.

    Tsimring, L.S.: Noise in biology. Rep. Prog. Phys. 77(2), 026601 (2014)

    Article  Google Scholar 

  9. 9.

    Ma, J., Tang, J.: A review for dynamics in neuron and neuronal network. Nonlinear Dyn. 89, 1569 (2017)

    Article  MathSciNet  Google Scholar 

  10. 10.

    Wang, C.N., Ma, J.: A review and guidance for pattern selection in spatiotemporal system. Int. J Mod. Phys. B 32, 1830003 (2018)

    Article  MathSciNet  Google Scholar 

  11. 11.

    Wu, F., Wang, C., Jin, W., Ma, J.: Dynamical responses in a new neuron model subjected to electromagnetic induction and phase noise. Physica A 469, 81–88 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. 12.

    d’Onofrio, A.: “Fuzzy oncology”: fuzzy noise induced bifurcations and their application to anti-tumor chemotherapy. Appl. Math. Lett. 21, 662 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. 13.

    d’Onofrio, A.: Bounded-noise-induced transitions in a tumor-immune system interplay. Phys. Rev. E 81, 021923 (2010)

    Article  Google Scholar 

  14. 14.

    Bobryk, R.V., Chrzeszczyk, A.: Transition induced by bounded noise. Physica A 358, 263–272 (2005)

    Article  Google Scholar 

  15. 15.

    Mandelbrot, B.: The variation of certain speculative prices. In: Fractals and Scaling in Finance, pp. 371–418. Springer, New York (1997)

  16. 16.

    Gurley, K., Kareem, A.: Simulation of correlated non-Gaussian pressure fields. Meccanica 33(3), 309–317 (1998)

    Article  MATH  Google Scholar 

  17. 17.

    Stathopoulos, T.: PDF of wind pressures on low-rise buildings. J. Struct. Div. 106(5), 973–990 (1980)

    Google Scholar 

  18. 18.

    Ochi, M.K.: Non-Gaussian random processes in ocean engineering. Probab. Eng. Mech. 1(1), 28–39 (1986)

    Article  Google Scholar 

  19. 19.

    Andrews, D.F., Herzberg, A.M.: Data: A Collection of Problems from Many Fields for the Student and Research Worker. Springer, Berlin (2012)

    Google Scholar 

  20. 20.

    Iyengar, R.N., Jaiswal, O.R.: A new model for non-Gaussian random excitations. Probab. Eng. Mech. 8(3–4), 281–287 (1993)

    Article  Google Scholar 

  21. 21.

    Borland, L.: Ito-Langevin equations within generalized thermostatistics. Phys. Lett. A 245, 67 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. 22.

    Wio, H.S.: On the role of non-Gaussian noises on noise-induced phenomena. In: Nonextensive Entropy: Interdisciplinary Applications, pp. 177–193 (2003)

  23. 23.

    Wio, H.S., Toral, R.: Effect of non-Gaussian noise sources in a noise-induced transition. Physica D 193, 161 (2004)

    Article  MATH  Google Scholar 

  24. 24.

    Doering, C.R.: A stochastic partial differential equation with multiplicative noise. Phys. Lett. A 122(3–4), 133 (1987)

    Article  MathSciNet  Google Scholar 

  25. 25.

    Sargsyan, K.V., Smereka, P.: A numerical method for some stochastic differential equations with multiplicative noise. Phys. Lett. A 344(2–4), 149 (2005)

    MATH  Google Scholar 

  26. 26.

    Dubkov, A.A., Spagnolo, B., Uchaikin, V.V.: Lévy flight superdiffusion: an introduction. Int. J. Bifurc. Chaos 18(09), 2649 (2008)

    Article  MATH  Google Scholar 

  27. 27.

    Augello, G., Valenti, D., Spagnolo, B.: Non-Gaussian noise effects in the dynamics of a short overdamped Josephson junction. Eur. Phys. J. B 78(2), 225 (2010)

    Article  Google Scholar 

  28. 28.

    Zhu, W.Q., Cai, G.Q.: On Bounded stochastic processes. In: d’Onofrio, A. (ed.) Bounded Noises in Physics, Biology and Engineering. Verlag, Birkauser (2016)

    Google Scholar 

  29. 29.

    Tsallis, C.: Possible generalization of Boltzmann–Gibbs statistics. J. Stat. Phys. 52, 479 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  30. 30.

    Ning, L.J., Liu, P.: The effect of sine-Wiener noises on transition in a genotype selection model with time delays. Eur. Phys. J. B 89(9), 201 (2016)

    Article  Google Scholar 

  31. 31.

    Liu, P., Ning, L.J.: Transitions induced by cross-correlated bounded noises and time delay in a genotype selection model. Physica A 441, 32–39 (2016)

    Article  Google Scholar 

  32. 32.

    de Franciscis, S., d’Onofrio, A.: Spatiotemporal bounded noises and transitions induced by them in solutions of the real Ginzburg–Landau model. Phys. Rev. E 86(2), 021118 (2012)

    Article  Google Scholar 

  33. 33.

    de Franciscis, S., d’Onofrio, A.: Spatio-temporal sine-Wiener bounded noise and its effect on Ginzburg–Landau model. Nonlinear Dyn. 74(3), 607 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. 34.

    Bobryk, R.V., Chrzeszczyk, A.: Transitions in a Duffing oscillator excited by random noise. Nonlinear Dyn. 51, 541 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  35. 35.

    de Franciscis, S., d’Onofrio, A.: Cellular polarization: interaction between extrinsic bounded noises and the wave-pinning mechanism. Phys. Rev. E 88(3), 032709 (2013)

    Article  Google Scholar 

  36. 36.

    Guo, W., Du, L.C., Mei, D.C.: Transitions induced by time delays and cross-correlated sine-Wiener noises in a tumor-immune system interplay. Physica A 391, 1270 (2012)

    Article  Google Scholar 

  37. 37.

    d’Onofrio, A., Gandolfi, A.: Resistance to antitumor chemotherapy due to bounded noise induced transitions. Phys. Rev. E 82(6), 061901 (2010)

    Article  MathSciNet  Google Scholar 

  38. 38.

    Guo, W., Mei, D.C.: Stochastic resonance in a tumor-immune system subject to bounded noises and time delay. Physica A 416, 90 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. 39.

    Long, F., Guo, W., Mei, D.C.: Stochastic resonance induced by bounded noise and periodic signal in an asymmetric bistable system. Physica A 391(22), 5305–5310 (2012)

    Article  Google Scholar 

  40. 40.

    de Franciscis, S., Caravagna, G., d’Onofrio, A.: Bounded noises as a natural tool to model extrinsic fluctuations in biomolecular networks. Nat. Comput. 13, 297 (2014)

    Article  MathSciNet  Google Scholar 

  41. 41.

    d’Onofrio, A., Caravagna, G., de Franciscis, S.: Bounded noise induced first-order phase transitions in a baseline non-spatial model of gene transcription. Physica A 492, 2056–2068 (2018)

    Article  MathSciNet  Google Scholar 

  42. 42.

    Caravagna, G., Mauri, G., d’Onofrio, A.: The interplay of intrinsic and extrinsic bounded noises in biomolecular networks. PLoS ONE 8(2), e51174 (2013)

    Article  Google Scholar 

  43. 43.

    de Franciscis, S., Caravagna, G., d’Onofrio, A.: Gene switching rate determines response to extrinsic perturbations in the self-activation transcriptional network motif. Sci. Rep. 6, 26980 (2016)

    Article  Google Scholar 

  44. 44.

    Fuliński, A., Telejko, T.: On the effect of interference of additive and multiplicative noises. Phys. Lett. A 152(1), 11 (1991)

    Article  Google Scholar 

  45. 45.

    Madureira, A.J., Hänggi, P., Wio, H.S.: Giant suppression of the activation rate in the presence of correlated white noise sources. Phys. Lett. A 217(4), 248 (1996)

    Article  Google Scholar 

  46. 46.

    Berdichevsky, V., Gitterman, M.: Stochastic resonance in linear systems subject to multiplicative and additive noise. Phys. Rev. E 60(2), 1494 (1999)

    Article  Google Scholar 

  47. 47.

    Zeng, C., Zeng, J., Liu, F., Wang, H.: Impact of correlated noise in an energy depot model. Sci. Rep. 6, 19591 (2016)

    Article  Google Scholar 

  48. 48.

    Wu, D.J., Cao, L., Ke, S.Z.: Bistable kinetic model driven by correlated noises: steady-state analysis. Phys. Rev. E 50, 2496 (1994)

    Article  Google Scholar 

  49. 49.

    Bemmo, D.T., Siewe, M.S., Tchawoua, C.: Combined effects of correlated bounded noises and weak periodic signal input in the modified FitzHugh–Nagumo neural model. Commun. Nonlinear Sci. Numer. Simul. 18(5), 1275–1287 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  50. 50.

    Yao, Y., Deng, H., Yi, M., Ma, J.: Impact of bounded noise on the formation and instability of spiral wave in a 2D Lattice of neurons. Sci. Rep. 7, 43151 (2017)

    Article  Google Scholar 

  51. 51.

    Yao, Y., Deng, H., Ma, C.Z., Yi, M., Ma, J.: Impact of bounded noise and rewiring on the formation and instability of spiral waves in a small-world network of Hodgkin–Huxley neurons. PLoS ONE 12(1), e0171273 (2017)

    Article  Google Scholar 

  52. 52.

    Yao, Y., Ma, C.Z., Wang, C.J., Yi, M., Gui, R.: Detection of sub-threshold periodic signal by multiplicative and additive cross-correlated sine-Wiener noises in the FitzHugh–Nagumo neuron. Physica A 492, 1247–1256 (2018)

    Article  MathSciNet  Google Scholar 

  53. 53.

    Yao, Y., Ma, J.: Weak periodic signal detection by sine-Wiener-noise-induced resonance in the FitzHugh–Nagumo neuron. Cogn. Neurodyn. 12(3), 343–349 (2018)

    Article  MathSciNet  Google Scholar 

  54. 54.

    d’Onofrio, A.: A Bounded Noises in Physics, Biology, and Engineering. Modeling and Simulation in Science, Engineering and Technology. Springer, New York (2013)

    Google Scholar 

  55. 55.

    Zhu, S.Q.: Steady-state analysis of a single-mode laser with correlations between additive and multiplicative noise. Phys. Rev. A 47, 2405–2408 (1993)

    Article  Google Scholar 

  56. 56.

    Van Kampen, N.G.: Stochastic differential equation. Phys. Rep. 24, 171–228 (1976)

    Article  MathSciNet  Google Scholar 

  57. 57.

    Liang, G.Y., Cao, L., Wu, D.J.: Approximate Fokker–Planck equation of system driven by multiplicative colored noises with colored cross-correlation. Physica A 335, 371 (2004)

    Article  Google Scholar 

  58. 58.

    Sancho, J.M., Miguel, S.M., Katz, S.L., Gunton, J.D.: Analytical and numerical studies of multiplicative noise. Phys. Rev. A 26, 1589 (1989)

    Article  Google Scholar 

  59. 59.

    Novikov, E.A.: Functionals and the random-force method in turbulence theory. Sov. Phys. JETP 20, 1290 (1965)

    MathSciNet  Google Scholar 

  60. 60.

    Owchar, M.: Wiener integrals of multiple variations. Proc. Am. Math. Soc. 3, 459 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  61. 61.

    Furutsu, K.: On the statistical theory of electromagnetic waves in a fluctuating medium (I). J. Res. NBS D 67, 303 (1963)

    MATH  Google Scholar 

  62. 62.

    Hänggi, P., Mroczkowski, T.T., Moss, F., McClintock, P.V.E.: Bistability driven by colored noise: theory and experiment. Phys. Rev. A 32, 695 (1985)

    Article  Google Scholar 

  63. 63.

    Castro, F., Sanchez, A.D., Wio, H.S.: Reentrance phenomena in noise induced transitions. Phys. Rev. Lett. 75, 1691 (1995)

    Article  Google Scholar 

  64. 64.

    Liu, Q., Jia, Y.: Fluctuations-induced switch in the gene transcriptional regulatory system. Phys. Rev. E 70, 041907 (2004)

    Article  Google Scholar 

  65. 65.

    Jia, Y., Li, J.R.: Reentrance phenomena in a bistable kinetic model driven by correlated noise. Phys. Rev. Lett. 78, 994 (1997)

    Article  Google Scholar 

  66. 66.

    Mei, D.C., Xie, C.W., Zhang, L.: Effects of cross correlation on the relaxation time of a bistable system driven by cross-correlated noise. Phys. Rev. E 68, 051102 (2003)

    Article  Google Scholar 

  67. 67.

    Masoliver, J., West, B.J., Lindenbergerg, K.: Bistability driven by Gaussian colored noise: first-passage times. Phys. Rev. A 35, 3086 (1987)

    Article  Google Scholar 

  68. 68.

    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(6), 5764 (1996)

    Article  Google Scholar 

  69. 69.

    Wang, C.J., Yang, K.L., Du, C.Y.: Multiple cross-correlation noise induced transition in a stochastic bistable system. Physica A 470, 261–274 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China ( Grant No. 11205006), the Natural Science New Star of Science and Technologies Research Plan in Shaanxi Province of China (Grant No. 2014KJXX-77), the Science Foundation of the Education Bureau of Shaanxi Province, China (Grant No. 15JK1045), and the Natural Science Foundation of Shaanxi Province, China (Grant No. 2018JM1034).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Can-Jun Wang.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, C., Lin, Q., Yao, Y. et al. Dynamics of a stochastic system driven by cross-correlated sine-Wiener bounded noises. Nonlinear Dyn 95, 1941–1956 (2019). https://doi.org/10.1007/s11071-018-4669-0

Download citation

Keywords

  • Cross-correlated sine-Wiener bounded noises
  • Fokker–Planck equation
  • Bistable dynamical system