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Reliability of quasi integrable and non-resonant Hamiltonian systems under fractional Gaussian noise excitation

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Abstract

The reliability of quasi integrable and non-resonant Hamiltonian system under fractional Gaussian noise (fGn) excitation is studied. Noting rather flat fGn power spectral density (PSD) in most part of frequency band, the fGn is innovatively regarded as a wide-band process. Then, the stochastic averaging method for quasi integrable Hamiltonian systems under wide-band noise excitation is applied to reduce 2n-dimensional original system into n-dimensional averaged Itô stochastic differential equations (SDEs). Reliability function and mean first passage time are obtained by solving the associated backward Kolmogorov equation and Pontryagin equation. The validity of the proposed procedure is tested by applying it to an example and comparing the numerical results with those from Monte Carlo simulation.

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References

  1. Zhou, W.N., Zhou, X.H., Yang, J., et al.: Stability analysis and application for delayed neural networks driven by fractional Brownian noise. IEEE Trans. Neural Netw. Learn. Syst. 29(5), 1491–1502 (2017)

    Article  MathSciNet  Google Scholar 

  2. Kumar, M., Kumar, S., Das, M.K., et al.: Securing images with a diffusion mechanism based on fractional Brownian motion. J. Inf. Secur. Appl. 40, 134–144 (2018)

    Google Scholar 

  3. Fouque, J., Hu, R.M.: Optimal portfolio under fractional stochastic environment. Math. Finan. 29(3), 697–734 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  4. Zhang, X., Xiao, W.: Arbitrage with fractional Gaussian processes. Phys. A 471, 620–628 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Calif, R., Schmitt, F.G.: Modeling of atmospheric wind speed sequence using a lognormal continuous stochastic equation. J. Wind Eng. Ind. Aerodyn. 109, 1–8 (2012)

    Article  Google Scholar 

  6. Guo, W.J., Wang, Y.X., Xie, M.X., et al.: Modeling oil spill trajectory in coastal waters based on fractional Brownian motion. Mar. Pollut. Bull. 58(9), 1339–1346 (2009)

    Article  Google Scholar 

  7. Mei, R.X., Xu, Y., Kurths, J.: Transport and escape in a deformable channel driven by fractional Gaussian noise. Phys. Rev. E 100(2), 022114 (2019)

    Article  Google Scholar 

  8. Vojta, T., Skinner, S., Metzler, R.: Probability density of the fractional Langevin equation with reflecting walls. Phys. Rev. E 100(4), 042142 (2019)

    Article  MathSciNet  Google Scholar 

  9. Biagini, F., Hu, Y.Z., Ksendal, B., et al.: Stochastic Calculus for Fractional Brownian Motion and Applications. Springer Science & Business Media, Berlin (2008)

    Book  Google Scholar 

  10. Mishura, Y.S.: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Springer Science & Business Media, Berlin (2008)

    Book  MATH  Google Scholar 

  11. Pei, B., Xu, Y., Wu, J.L.: Stochastic averaging for stochastic differential equations driven by fractional Brownian motion and standard Brownian motion. Appl. Math. Lett. 100, 106006 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. Xu, Y., Pei, B., Guo, R.: Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete Contin. Dyn. Syst. Ser. B 20(7), 2257–2267 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Xu, Y., Pei, B., Wu, J.: Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion. Stochast. Dyn. 17(2), 1750013 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Deng, M.L., Lü, Q.F., Zhu, W.Q.: Stochastic averaging of quasi integrable and non-resonant Hamiltonian systems excited by fractional Gaussian noise with Hurst index H∈(1/2, 1). Int. J. Nonlinear Mech. 98, 43–50 (2018)

    Article  Google Scholar 

  15. Crandall, S.H., Chandiramani, K.L., Cook, R.G.: Some first-passage problems in random vibration. J. Appl. Mech. 88, 532–538 (1966)

    Article  Google Scholar 

  16. Roberts, J.B.: First passage probability for nonlinear oscillators. J. Am. Soc. Civil Eng. 102, 851–866 (1976)

    Google Scholar 

  17. Roberts, J.B.: First-passage time for oscillators with nonlinear damping. J. Appl. Mech. 45, 175–180 (1978)

    Article  MATH  Google Scholar 

  18. Zhu, W.Q., Deng, M.L., Huang, Z.L.: First-passage failure of quasi-integrable Hamiltonian systems. J. Appl. Mech. 69(3), 274–282 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhu, W.Q., Wu, Y.J.: First-passage time of Duffing oscillator under combined harmonic and white-noise excitations. Nonlinear Dyn. 32(3), 291–305 (2003)

    Article  MATH  Google Scholar 

  20. Wu, Y.J., Fang, W.: Stochastic averaging method for estimating first-passage statistics of stochastically excited Duffing–Rayleigh–Mathieu system. Acta. Mech. Sin. 24(5), 575–582 (2008)

    Article  MATH  Google Scholar 

  21. Deng, M.L., Zhu, W.Q.: Stochastic averaging of MDOF quasi integrable Hamiltonian systems under wide-band random excitation. J. Sound Vib. 305(4–5), 783–794 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mandelbrot, B.B., van Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10(4), 422–437 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rangarajan, G., Ding, M.: Processes with Long-range Correlations: Theory and Applications. Springer Science & Business Media, Berlin (2003)

    Book  Google Scholar 

  24. Papoulis, A.: Probability, Random Variables, and Stochastic Processes, pp. 345–348. McGraw-Hill, New York (1991)

    Google Scholar 

  25. Zhu, W.Q.: Nonlinear stochastic dynamics and control in Hamiltonian formulation. Appl. Mech. Rev. 59(4), 230–248 (2006)

    Article  Google Scholar 

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Acknowledgements

The work reported in this paper was supported by National Key R&D Program of China (Grant No. 2018YFC0809400), Zhejiang Provincial Natural Science Foundation of China (Grant No. LY16A020001) and National Natural Science Foundation of China (Grant No. 11802267).

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Authors and Affiliations

  1. Department of Mechanics, State-Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou, 310027, China

    Q. F. Lü, W. Q. Zhu & M. L. Deng

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  1. Q. F. Lü
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  2. W. Q. Zhu
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  3. M. L. Deng
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Correspondence to M. L. Deng.

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Lü, Q.F., Zhu, W.Q. & Deng, M.L. Reliability of quasi integrable and non-resonant Hamiltonian systems under fractional Gaussian noise excitation. Acta Mech. Sin. 36, 902–909 (2020). https://doi.org/10.1007/s10409-020-00962-3

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  • DOI: https://doi.org/10.1007/s10409-020-00962-3

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