, Volume 54, Issue 9, pp 1351–1363 | Cite as

Path Integral approach via Laplace’s method of integration for nonstationary response of nonlinear systems

  • Alberto Di MatteoEmail author
Stochastics and Probability in Engineering Mechanics


In this paper the nonstationary response of a class of nonlinear systems subject to broad-band stochastic excitations is examined. A version of the Path Integral (PI) approach is developed for determining the evolution of the response probability density function (PDF). Specifically, the PI approach, utilized for evaluating the response PDF in short time steps based on the Chapman–Kolmogorov equation, is here employed in conjunction with the Laplace’s method of integration. In this manner, an approximate analytical solution of the integral involved in this equation is obtained, thus circumventing the repetitive integrations generally required in the conventional numerical implementation of the procedure. Further, the method is extended to nonlinear oscillators, approximately modeling the amplitude of the system response as a one-dimensional Markovian process. Various nonlinear systems are considered in the numerical applications, including Duffing and Van der Pol oscillators. Appropriate comparisons with Monte Carlo simulation data are presented, demonstrating the efficiency and accuracy of the proposed approach.


Path Integral Laplace’s method Nonstationary response Evolutionary excitation Probability density function 



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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Dipartimento di IngegneriaUniversità degli Studi di PalermoPalermoItaly

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