Path Integral Method for Nonlinear Systems Under Levy White Noise

[+] Author and Article Information
Alberto Di Matteo

Dipartimento di Ingegneria Civile,
Ambientale Aerospaziale e dei Materiali (DICAM),
Università degli Studi di Palermo,
Viale delle Scienze,
Palermo 90128, Italy
e-mail: alberto.dimatteo@unipa.it

Antonina Pirrotta

Dipartimento di Ingegneria Civile,
Ambientale Aerospaziale e dei Materiali (DICAM),
Università degli Studi di Palermo,
Viale delle Scienze,
Palermo 90128, Italy;
Department of Mathematical Sciences,
University of Liverpool,
Liverpool L69 7ZL, UK
e-mails: antonina.pirrotta@unipa.it;

Manuscript received June 22, 2016; final manuscript received September 21, 2016; published online June 12, 2017. Assoc. Editor: Ioannis Kougioumtzoglou.

ASME J. Risk Uncertainty Part B 3(3), 030905 (Jun 12, 2017) (7 pages) Paper No: RISK-16-1095; doi: 10.1115/1.4036703 History: Received June 22, 2016; Revised September 21, 2016

In this paper, the probabilistic response of nonlinear systems driven by alpha-stable Lévy white noises is considered. The path integral solution is adopted for determining the evolution of the probability density function of nonlinear oscillators. Specifically, based on the properties of alpha-stable random variables and processes, the path integral solution is extended to deal with Lévy white noises input with any value of the stability index alpha. It is shown that at the limit when the time increments tend to zero, the Einstein–Smoluchowsky equation, governing the evolution of the response probability density function, is fully restored. Application to linear and nonlinear systems under different values of alpha is reported. Comparisons with pertinent Monte Carlo simulation data and analytical solutions (when available) demonstrate the accuracy of the results.

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Risken, H. , 1996, The Fokker Planck Equation, Springer, Berlin.
Wehner, M. F. , and Wolfer, W. G. , 1983, “ Numerical Evaluation of Path-Integral Solutions to Fokker-Planck Equations,” Phys. Rev. A, 27(5), pp. 2663–2670. [CrossRef]
Barone, G. , Navarra, G. , and Pirrotta, A. , 2008, “ Probabilistic Response of Linear Structures Equipped With Nonlinear Damper Devices (PIS Method),” Probab. Eng. Mech., 23(2–3), pp. 125–133. [CrossRef]
Naess, A. , and Moe, V. , 2000, “ Efficient Path Integration Method for Nonlinear Dynamic Systems,” Probab. Eng. Mech., 15(2), pp. 221–231. [CrossRef]
Iourtchenko, D. V. , Mo, E. , and Naess, A. , 2006, “ Response Probability Density Functions of Strongly Non-Linear Systems by the Path Integration Method,” Int. J. Nonlinear Mech., 41(5), pp. 693–705. [CrossRef]
Di Paola, M. , and Santoro, R. , 2008, “ Non-Linear Systems Under Poisson White Noise Handled by Path Integral Solution,” J. Vib. Control, 14(1–2), pp. 35–49. [CrossRef]
Di Paola, M. , and Santoro, R. , 2008, “ Path Integral Solution for Non-Linear System Enforced by Poison White Noise,” Probab. Eng. Mech., 23(2–3), pp. 164–169. [CrossRef]
Pirrotta, A. , and Santoro, R. , 2011, “ Probabilistic Response of Nonlinear Systems Under Combined Normal and Poisson White Noise Via Path Integral Method,” Probab. Eng. Mech., 26(1), pp. 26–32. [CrossRef]
Di Matteo, A. , Di Paola, M. , and Pirrotta, A. , 2016, “ Path Integral Solution for Nonlinear Systems Under Parametric Poissonian White Noise Input,” Probab. Eng. Mech., 44, pp. 89–98. [CrossRef]
Naess, A. , Iourtchenko, D. V. , and Batsevych, O. , 2011, “ Reliability of Systems With Randomly Varying Parameters Via a Path Integration Method,” Probab. Eng. Mech., 26(1), pp. 5–9. [CrossRef]
Bucher, C. , Di Matteo, A. , Di Paola, M. , and Pirrotta, A. , 2016, “ First-Passage Problem for Nonlinear Systems Under Lévy White Noise Through Path Integral Method,” Nonlinear Dyn., 85(3), pp. 1445–1456. [CrossRef]
Kougioumtzoglou, I. A. , and Spanos, P. D. , 2014, “ Stochastic Response Analysis of Softening Duffing Oscillator and Ship Capsizing Probability Determination Via Numerical Path Integral Approach,” Probab. Eng. Mech., 35, pp. 67–74. [CrossRef]
Bucher, C. , and Di Paola, M. , 2015, “ Efficient Solution of the First Passage Problem by Path Integration for Normal and Poissonian White Noise,” Probab. Eng. Mech., 41, pp. 121–128. [CrossRef]
Kougioumtzoglou, I. A. , and Spanos, P. D. , 2013, “ Response and First-Passage Statistics of Nonlinear Oscillators Via a Numerical Path Integral Approach,” J. Eng. Mech., 139(9), pp. 1207–1217. [CrossRef]
Kougioumtzoglou, I. A. , and Spanos, P. D. , 2012, “ An Analytical Wiener Path Integral Technique for Non-Stationary Response Determination of Nonlinear Oscillators,” Probab. Eng. Mech., 28, pp. 125–131. [CrossRef]
Di Matteo, A. , Kougioumtzoglou, I. A. , Pirrotta, A. , Spanos, P. D. , and Di Paola, M. , 2014, “ Stochastic Response Determination of Nonlinear Oscillators With Fractional Derivatives Elements Via the Wiener Path Integral,” Probab. Eng. Mech., 38, pp. 127–135. [CrossRef]
Kougioumtzoglou, I. A. , Di Matteo, A. , Spanos, P. D. , Pirrotta, A. , and Di Paola, M. , 2015, “ An Efficient Wiener Path Integral Technique Formulation for Stochastic Response Determination of Nonlinear MDOF Systems,” ASME J. Appl. Mech., 82(10), p. 101005. [CrossRef]
Samorodnitsky, G. , and Taqqu, S. M. , 2000, Stable Non-Gaussian Random Processes: Stochastic Models With Infinite Variance, Chapman and Hall/CRC Press, Boca Raton, FL.
Chandrasekhar, S. , 1943, “ Stochastic Problems in Physics and Astronomy,” Rev. Mod. Phys., 15(1), pp. 1–89. [CrossRef]
Mandelbrot, B. , 1960, “ The Pareto–Lévy Law and Distribution of Income,” Int. Econ. Rev., 1(2), pp. 79–106. [CrossRef]
Stedinger, J. , 1983, “ Design Events With Specific Flood Risk,” Water Resour. Res., 19(2), pp. 511–522. [CrossRef]
Grigoriu, M. , 1986, “ Structural Response to Uncertain Seismic Excitations,” J. Struct. Eng., 112(6), pp. 1355–1365. [CrossRef]
Frendal, M. , and Rychlick, I. , 1992, “ Rainfall Analysis. Markov Method,” Department of Mathematical Statistics, University of Lund, Sweden, Technical Report No. 6.
Grigoriu, M. , 1995, “ Linear Systems Subjected to Non-Gaussian α-Stable Processes,” Probab. Eng. Mech., 10(1), pp. 23–34. [CrossRef]
Di Paola, M. , and Failla, G. , 2005, “ Stochastic Response of Linear and Non-Linear Systems to α-Stable Lévy White Noises,” Probab. Eng. Mech., 20(2), pp. 128–135. [CrossRef]
Di Paola, M. , Pirrotta, A. , and Zingales, M. , 2007, “ Itô Calculus Extended to Systems Driven by α-Stable Lévy White Noises (A Novel Clip on the Tails of Lévy Motion),” Int. J. Nonlinear Mech., 42(8), pp. 1046–1054.
Grigoriu, M. , 2000, “ Equivalent Linearization for Systems Driven by Lévy White Noise,” Probab. Eng. Mech., 15(2), pp. 185–190. [CrossRef]
Alotta, G. , and Di Paola, M. , 2015, “ Probabilistic Characterization of Nonlinear Systems Under α-Stable Via Complex Fractional Moments,” Physica A, 420, pp. 265–276. [CrossRef]
Chechkin, A. , Gonchar, V. , Klafter, J. , Metzler, R. , and Tarantov, L. , 2002, “ Stationary State of Non-Linear Oscillator Driven by Lévy Noise,” Chem. Phys., 284(1–2), pp. 233–251. [CrossRef]
Naess, A. , and Skaug, C. , 1999, “ Extension of the Numerical Path Integration Method to Filtered α-Stable Levy Noise,” Computational Stochastic Mechanics, P.D. Spanos, ed., A.A. Balkema Publishers, Rotterdam, The Netherlands, pp. 391–396.
Naess, A. , and Skaug, C. , 2001, “ Path Integration Methods for Calculating Response Statistics of Nonlinear Oscillators Driven by α-Stable Lévy Noise,” IUTAM Symposium on Nonlinear and Stochastic Structural Dynamics, Chennai, India, Jan. 4–8, pp. 159–169.
Pirrotta, A. , 2005, “ Non-Linear Systems Under Parametric White Noise Input: Digital Simulation and Response,” Int. J. Nonlinear Mech., 40(8), pp. 1088–1101. [CrossRef]
Metzler, R. , and Klafter, J. , 2000, “ The Random Walk's Guide to Anomalous Diffusion: A Fractional Dynamics Approach,” Phys. Rep., 339(1), pp. 1–77. [CrossRef]
Di Paola, M. , and Pinnola, F. P. , 2011, “ Riesz Fractional Integrals and Complex Fractional Moments for the Probabilistic Characterization of Random Variables,” Probab. Eng. Mech., 29, pp. 149–156. [CrossRef]


Grahic Jump Location
Fig. 1

Evolution of the PDF for α=0.5: linear case (γ=1)

Grahic Jump Location
Fig. 2

Evolution of the PDF for α=1: (a) linear case (γ=1) and (b) nonlinear case (γ=1, ε=1, ρ=1.5)

Grahic Jump Location
Fig. 3

Evolution of the PDF for α=1.5: (a) linear case (γ=1) and (b) nonlinear case (γ=1, ε=1, ρ=1.5)




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