Modified Path Integral Solution of Fokker–Planck Equation: Response and Bifurcation of Nonlinear Systems

Pankaj Kumar · S. Narayanan

29 citations0/0 files storedJournal of Computational and Nonlinear Dynamics

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Probabilistic and Robust Engineering Design Vibration and Dynamic Analysis stochastic dynamics and bifurcation

Abstract

Response of nonlinear systems subjected to harmonic, parametric, and random excitations is of importance in the field of structural dynamics. The transitional probability density function (PDF) of the random response of nonlinear systems under white or colored noise excitation (delta correlated) is governed by both the forward Fokker–Planck (FP) and the backward Kolmogorov equations. This paper presents a new approach for efficient numerical implementation of the path integral (PI) method in the solution of the FP equation for some nonlinear systems subjected to white noise, parametric, and combined harmonic and white noise excitations. The modified PI method is based on a non-Gaussian transition PDF and the Gauss–Legendre integration scheme. The effects of white noise intensity, amplitude, and frequency of harmonic excitation and the level of nonlinearity on stochastic jump and bifurcation behaviors of a hardening Duffing oscillator are also investigated.

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References

1990Random vibration and statistical linearization 1965On the relation between ordinary and stochastic differential equations 1992Random Vibration of Mechanical and Structural Systems 1993Random vibration of mechanical and structural systems 1983Introduction to Random Vibrations 1993On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems 2000Efficient path integration methods for nonlinear dynamic systems 2006Solution of Fokker-Planck equation by finite element and finite difference methods for nonlinear systems

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2024Efficient path integral approach via analytical asymptotic expansion for nonlinear systems under Gaussian white noise

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