Nonlinear Dynamics

, Volume 77, Issue 3, pp 729–738 | Cite as

Probabilistic characterization of nonlinear systems under Poisson white noise via complex fractional moments

  • A. Di Matteo
  • M. Di Paola
  • A. PirrottaEmail author
Original Paper


In this paper, the probabilistic characterization of a nonlinear system enforced by Poissonian white noise in terms of complex fractional moments (CFMs) is presented. The main advantage in using such quantities, instead of the integer moments, relies on the fact that, through the CFMs the probability density function (PDF) is restituted in the whole domain. In fact, the inverse Mellin transform returns the PDF by performing integration along the imaginary axis of the Mellin transform, while the real part remains fixed. This ensures that the PDF is restituted in the whole range with exception of the value in zero, in which singularities appear. It is shown that using Mellin transform theorem and related concepts, the solution of the Kolmogorov–Feller equation is obtained in a very easy way by solving a set of linear differential equations. Results are compared with those of Monte Carlo simulation showing the robustness of the solution pursued in terms of CFMs. Further, a very elegant strategy, to give a relationship between the CFMs evaluated for different value of real part, is introduced as well.


Complex fractional moment Mellin transform Poisson white noise Probability density function Kolmogorov–Feller 


  1. 1.
    Caughey, T.K., Dienes, J.K.: Analysis of nonlinear first-order system with a white noise input. J. Appl. Phys. 32, 2476–2483 (1961)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Caughey, T.K.: Non-linear theory of random vibrations. In: Yih, C.S. (ed.) Advances in Applied Mechanics, pp. 209–235. Academic Press, New York (1971)Google Scholar
  3. 3.
    Dimentberg, M.F.: An exact solution to a certain nonlinear random vibration problem. Int. J. Non-Linear Mech. 17, 231–236 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Grigoriu, M.: Applied Non-Gaussian Processes: Examples, Theory, Simulation, Linear Random Vibration, and MATLAB Solutions. Prentice-Hall, Englewood Cliffs (1985)Google Scholar
  5. 5.
    Di Paola, M.: Stochastic differential calculus. In: Casciati, F. (ed.) Dynamic Motion: Chaotic and Stochastic Behaviour, pp. 29–92. Springer, Wien (1993)Google Scholar
  6. 6.
    Iwankiewicz, R.: Dynamic response of non-linear systems to Poisson-distributed pulse trains: Markov approach. In: Casciati, F., Elishakoff, I., Roberts, J.B. (eds.) Nonlinear Structural Systems Under Random Conditions, pp. 223–228. Elsevier, Amsterdam (1990)Google Scholar
  7. 7.
    Tung, C.C.: Random response of highway bridges to vehicle loads. J. Eng. Mech. Div. ASCE 93, 79–94 (1967)Google Scholar
  8. 8.
    Roberts, J.B.: On the response of a simple oscillator to random impulses. J. Sound Vib. 4, 51–61 (1966)CrossRefGoogle Scholar
  9. 9.
    Lin, Y.K.: Application of non-stationary shot noise in the study of system response to a class of non-stationary excitations. J. Appl. Mech. ASME 30, 555–558 (1963)CrossRefzbMATHGoogle Scholar
  10. 10.
    Merchant, D.H.: A stochastic model of wind gusts. Technical Report 48. Stanford University, Stanford, CA (1964)Google Scholar
  11. 11.
    Liepmann, H.W.: On the application of statistical concepts to the buffeting problem. J. Aeronaut. Sci. 19, 793–800 (1952)CrossRefGoogle Scholar
  12. 12.
    Vasta, M.: Exact stationary solution for a class of nonlinear systems driven by a non-normal delta-correlated processes. Int. J. Non-Linear Mech. 30, 407–418 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Proppe, C.: Exact stationary solution for a class of nonlinear systems driven by a non-normal delta-correlated processes. Int. J. Non-Linear Mech. 38, 557–566 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Köylüoglu, H.U., Nielsen, S.R.K., Iwankiewicz, R.: Response and reliability of Poisson-driven systems by path integration. J. Eng. Mech. 121, 117–130 (1995)CrossRefGoogle Scholar
  15. 15.
    Iwankiewicz, R., Nielsen, S.R.K.: Solution techniques for pulse problems in nonlinear stochastic dynamics. Probab. Eng. Mech. 15, 25–36 (2000)Google Scholar
  16. 16.
    Di Paola, M., Santoro, R.: Nonlinear systems under Poisson white noise handled by path integral solution. J. Vib. Control 14, 35–49 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Di Paola, M., Santoro, R.: Path integral solution for non-linear system enforced by Poisson white noise. Probab. Eng. Mech. 23, 164–169 (2008) Google Scholar
  18. 18.
    Pirrotta, A., Santoro, R.: Probabilistic response of nonlinear systems under combined normal and Poisson white noise via path integral method. Probab. Eng. Mech. 26, 26–32 (2011)CrossRefGoogle Scholar
  19. 19.
    Barone, G., Navarra, G., Pirrotta, A.: Probabilistic response of linear structures equipped with nonlinear damper devices (PIS method). Probab. Eng. Mech. 23, 125–133 (2008)CrossRefGoogle Scholar
  20. 20.
    Zhu, W.Q.: Stochastic averaging methods in random vibration. Appl. Mech. Rev. 41, 189–199 (1998)CrossRefGoogle Scholar
  21. 21.
    Zeng, Y., Zhu, W.Q.: Stochastic averaging of strongly nonlinear oscillators under Poisson white noise excitation. In: Zhu, W.Q., Lin, Y.K., Cai, G.Q. (eds.) IUTAM Symposium on Nonlinear Stochastic Dynamics and Control, pp. 147–155. IUTAM Bookseries 29. Springer, Netherlands (2011)Google Scholar
  22. 22.
    Cottone, G., Di Paola, M.: On the use of fractional calculus for the probabilistic characterization of random variables and vectors. Phys. A 389, 909–920 (2009)CrossRefGoogle Scholar
  23. 23.
    Di Paola, M., Pinnola, F.P.: Riesz fractional integral and complex fractional moments for the probabilistic characterization of random variables. Probab. Eng. Mech. 29, 149–156 (2012)CrossRefGoogle Scholar
  24. 24.
    Cottone, G., Di Paola, M., Pirrotta, A.: Path integral solution by fractional calculus. J. Phys. (2008). doi: 10.1088/1742-6596/96/1/012007
  25. 25.
    Di Paola, M.: Stochastic Methods in Structural Dynamics: Stochastic differential calculus challenges and future developments. Vienna Congress on Recent Advances in Earthquake Engineering and Structural Dynamics 2013 (VEESD 2013), 28–30 August 2013, Wien, Austria (2013)Google Scholar
  26. 26.
    Pirrotta, A.: Non-linear systems under delta correlated processes handled by perturbation theory. Probab. Eng. Mech. 13, 283–290 (1998)CrossRefGoogle Scholar
  27. 27.
    Pirrotta, A.: Multiplicative cases from additive cases: extension of Kolmogorov–Feller equation to parametric Poisson white noise processes. Probab. Eng. Mech. 22, 127–135 (2007)CrossRefGoogle Scholar
  28. 28.
    Podlubny, I.: Fractional Differential Equation. An Introduction to Fractional Derivatives, Fractional Differential Equations, some Methods of their Solution and some of their Applications. Academic Press, San Diego (1999)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale, dei Materiali (DICAM)Università degli Studi di PalermoPalermoItaly

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