Skip to main content
SpringerLink
Log in

First-passage problem for nonlinear systems under Lévy white noise through path integral method

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, the first-passage problem for nonlinear systems driven by \(\alpha \)-stable Lévy white noises is considered. The path integral solution (PIS) is adopted for determining the reliability function and first-passage time probability density function of nonlinear oscillators. Specifically, based on the properties of \(\alpha \)-stable random variables and processes, PIS is extended to deal with Lévy white noises with any value of the stability index \(\alpha \). Application to linear and nonlinear systems considering different values of \(\alpha \) is reported. Comparisons with pertinent Monte Carlo simulation data demonstrate the accuracy of the results.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

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

Similar content being viewed by others

References

  1. Schueller, G.I.: A state-of-the-art report on computational stochastic mechanics. Prob. Eng. Mech. 12, 197–313 (1997)

    Article  Google Scholar 

  2. Dimentberg, M.F.: Statistical Dynamics of Nonlinear and Time-Varying Systems. Research Studies, Taunton (1988)

    MATH  Google Scholar 

  3. Lin, Y.K., Cai, G.Q.: Probabilistic Structural Dynamics. McGraw-Hill, New York (1995)

    Google Scholar 

  4. Roberts, J.B., Spanos, P.D.: Random vibration and statistical linearization. Wiley, New York (1990)

    MATH  Google Scholar 

  5. Schueller, G.I., Pradlwarter, H.J., Koutsourelakis, P.S.: A critical appraisal of reliability estimation procedures for high dimensions. Prob. Eng. Mech. 19, 463–474 (2004)

    Article  Google Scholar 

  6. Spanos, P.D.: Survival probability of non linear oscillators subjected to broad band random disturbances. Int. J. Non-Linear Mech. 17, 303–317 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kougioumtzoglou, I.A., Spanos, P.D.: Response and first-passage statistics of nonlinear oscillators via a numerical path integral approach. J. Eng. Mech. 139, 1207–1217 (2013)

    Article  Google Scholar 

  8. Köylüoglu, H.U., Nielsen, S.R.K., Iwankiewicz, R.: Reliability of non-linear oscillators subjected to Poisson driven impulses. J. Sound Vib. 176, 19–33 (1994)

    Article  MATH  Google Scholar 

  9. Chen, J.B., Li, J.: Dynamic response and reliability analysis of nonlinear stochastic structures. Prob. Eng. Mech. 20, 33–44 (2005)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Bergman, L., Spencer, B.: First passage time for linear systems with stochastic coefficients. Prob. Eng. Mech. 2, 46–53 (1987)

    Article  Google Scholar 

  12. Fujimura, K., Kiureghian, A.: Tail-equivalent linearization method for nonlinear random vibration. Prob. Eng. Mech. 22, 63–76 (2007)

    Article  Google Scholar 

  13. Bucher, C.: Asymptotic sampling for high-dimensional reliability analysis. Prob. Eng. Mech. 24, 505–510 (2009)

    Article  Google Scholar 

  14. Bucher, C., Sichani, M.T., Nielsen, S.: Efficient estimation of first passage probability of high-dimensional nonlinear systems. Prob. Eng. Mech. 26, 539–549 (2011)

    Article  Google Scholar 

  15. Bucher, C., Sichani, M.T., Nielsen, S.: Applications of asymptotic sampling on high dimensional structural dynamic problems. Struct. Saf. 33, 305–316 (2011)

    Article  Google Scholar 

  16. Naess, A., Johansen, M.: Response statistics of nonlinear, compliant offshore structures by path integral solution method. Prob. Eng. Mech. 8, 91–106 (1993)

    Article  Google Scholar 

  17. Naess, A., Moe, V.: Efficient path integration method for nonlinear dynamic systems. Prob. Eng. Mech. 15, 221–231 (2000)

    Article  Google Scholar 

  18. Iourtchenko, D.V., Mo, E., Naess, A.: Response probability density functions of strongly non-linear systems by the path integration method. Int. J. Non-Linear Mech. 41, 693–705 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Di Matteo, A., Di Paola, M., Pirrotta, A.: Path integral solution for nonlinear systems under parametric Poissonian white noise input. Prob. Eng. Mech. (2015). doi:10.1016/j.probengmech.2015.09.020

  20. Di Paola, M., Santoro, R.: Path integral solution for non-linear system enforced by Poison White Noise. Prob. Eng. Mech. 23, 164–169 (2008)

    Article  Google Scholar 

  21. Pirrotta, A., Santoro, R.: Probabilistic response of nonlinear systems under combined normal and Poisson white noise via path integral method. Prob. Eng. Mech. 26, 26–32 (2011)

    Article  Google Scholar 

  22. Naess, A., Iourtchenko, D.V., Batsevych, O.: Reliability of systems with randomly varying parameters via a path integration method. Prob. Eng. Mech. 26, 5–9 (2011)

    Article  MATH  Google Scholar 

  23. Iourtchenko, D.V., Mo, E., Naess, A.: Reliability of strongly nonlinear single degree of freedom dynamic systems by the path integration method. J. Appl. Mech. 75, (2008). doi:10.115/1.2967896

  24. Kougioumtzoglou, I.A., Spanos, P.D.: Stochastic response analysis of softening Duffing oscillator and ship capsizing probability determination via numerical path integral approach. Prob. Eng. Mech. 35, 67–74 (2014)

    Article  Google Scholar 

  25. Samorodnitsky, G., Taqqu, S.M.: Stable non-gaussian random processes. Stochastic models with infinite variance. Chapman and HAll/CRC, New York (1994)

    MATH  Google Scholar 

  26. Chandrasekhar, S.: Stochastic problems in physics and astronomy. Rev. Modern Phys. 15, 1–89 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mandelbrot, B.: The Pareto-Lévy law and distribution of income. Int. Econ. Rev. 1, 79–106 (1960)

    Article  MATH  Google Scholar 

  28. Stedinger, J.: Design events with specific flood risk. Water Resour. Res. 19, 511–522 (1983)

    Article  Google Scholar 

  29. Grigoriu, M.: Structural response to uncertain seismic excitations. J. Struct. Eng. 112, 1355–1365 (1986)

    Article  Google Scholar 

  30. Frendal, M., Rychlick, I.: Rainfall analysis. Markov method. Technical report, Department of Mathematical Statistics, University of Lund, Sweden. No. 6 (1992)

  31. Grigoriu, M.: Linear systems subjected to non-Gaussian \(\alpha \)-stable processes. Prob. Eng. Mech. 10, 23–34 (1995)

    Article  Google Scholar 

  32. Di Paola, M., Failla, G.: Stochastic response of linear and non-linear systems to \(\alpha \)-stable Lévy white noises. Prob. Eng. Mech. 20, 128–135 (2005)

    Article  Google Scholar 

  33. Di Paola, M., Pirrotta, A., Zingales, M.: Itô calculus extended to systems driven by \(\alpha \)-stable Lévy white noises (a novel clip on the tails of Lévy motion). Prob. Eng. Mech. 42, 1046–1054 (2007)

    MATH  Google Scholar 

  34. Grigoriu, M.: Equivalent linearization for systems driven by Lévy white noise. Prob. Eng. Mech. 15, 185–190 (2000)

    Article  Google Scholar 

  35. Naess, A., Skaug, C.: Extension of the Numerical Path Integration Method to Filtered \(\alpha \)-Stable Levy Noise. Computational Stochastic Mechanics, Santorini (1999)

    MATH  Google Scholar 

  36. Naess, A., Skaug, C.: Path integration methods for calculating response statistics of nonlinear oscillators driven by \(\alpha \)-stable Lévy noise. In: Narayanan, S., Iyengar, R.N. (eds.) IUTAM Symposium on Nonlinear and Stochastic Structural Dynamics, pp. 159–169. Kluwer Academic Publishers, Dordrecht (2001)

    Chapter  Google Scholar 

  37. Chechkin, A., Gonchar, V., Klafter, J., Metzler, R., Tarantov, L.: Stationary state of non-linear oscillator driven by Lévy noise. Chem. Phys. 284, 233–251 (2002)

    Article  Google Scholar 

  38. Risken, H.: The Fokker Planck Equation. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  39. Barone, G., Navarra, G., Pirrotta, A.: Probabilistic response of linear structures equipped with nonlinear dampers devices (PIS method). Prob. Eng. Mech. 2, 125–133 (2008)

    Article  Google Scholar 

  40. Bucher, C., Di Paola, M.: Efficient solution of the first passage problem by Path Integration. Prob. Eng. Mech. 41, 121–128 (2015)

    Article  Google Scholar 

  41. Pirrotta, A.: Non-linear systems under parametric white noise input: digital simulation and response. Int. J. Non-Linear Mech. 40, 1088–1101 (2005)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Building Construction and Technology, Vienna University of Technology, Karlsplatz 13/222, 1040, Vienna, Austria

    Christian Bucher

  2. Dipartimento di Ingegneria Civile, Ambientale Aerospaziale e dei Materiali (DICAM), Università degli Studi di Palermo, Viale delle Scienze, Palermo, 90128, Italy

    Alberto Di Matteo, Mario Di Paola & Antonina Pirrotta

  3. Department of Mathematical Sciences, University of Liverpool, Liverpool, UK

    Antonina Pirrotta

Authors
  1. Christian Bucher
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alberto Di Matteo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mario Di Paola
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Antonina Pirrotta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antonina Pirrotta.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bucher, C., Di Matteo, A., Di Paola, M. et al. First-passage problem for nonlinear systems under Lévy white noise through path integral method. Nonlinear Dyn 85, 1445–1456 (2016). https://doi.org/10.1007/s11071-016-2770-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-2770-9

Keywords

Search

Navigation