Probabilistic solution of nonlinear oscillators excited by combined Gaussian and Poisson white noises

Zhu, Hongtao; Er, Guo-Kang; Iu, Vai Pan; Kou, K. P.

doi:10.1016/j.jsv.2011.01.005

Cited by 47 publications

(9 citation statements)

References 37 publications

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“…Besides computing the first statistical moments of the response or performing a stability analysis of systems under stochastic vibrations, we must emphasize that the computation of the finite distribution (usually termed "fidis") associated to the stationary solution, and particularly of the stationary PDF, is also a major goal in the realm of vibratory systems with uncertainties. Some interesting contributions in this regard include [25,26]. In [25], the authors first present a complete overview of methods and techniques available to determine the stationary PDF of nonlinear oscillators excited by random functions.…”

Section: Introductionmentioning

confidence: 99%

Approximating the Density of Random Differential Equations with Weak Nonlinearities via Perturbation Techniques

et al. 2021

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We combine the stochastic perturbation method with the maximum entropy principle to construct approximations of the first probability density function of the steady-state solution of a class of nonlinear oscillators subject to small perturbations in the nonlinear term and driven by a stochastic excitation. The nonlinearity depends both upon position and velocity, and the excitation is given by a stationary Gaussian stochastic process with certain additional properties. Furthermore, we approximate higher-order moments, the variance, and the correlation functions of the solution. The theoretical findings are illustrated via some numerical experiments that confirm that our approximations are reliable.

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Section: Introductionmentioning

confidence: 99%

Approximating the Density of Random Differential Equations with Weak Nonlinearities via Perturbation Techniques

et al. 2021

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“…However, exact analytical solutions for the PDF of the response are only available for very restricted classes of nonlinear systems [6][7][8][9][10][11]. Various approximate and numerical procedures have been developed to solve PDF solutions for the FPK equation or the KF equation [12][13][14][15][16][17][18][19][20][21][22][23][24][25], such as stochastic averaging method, path integral method, cell mapping method and exponential-polynomial closure method. Among these methods, the cell mapping method has been demonstrated to be a very efficient tool due to its ability of global analysis of the strongly nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic response of dynamical systems based on the global attractor with the compatible cell mapping method

Yue

et al. 2019

Physica A: Statistical Mechanics and its Applications

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A generalized compatible cell mapping (CCM) method is proposed in this paper to take advantages of the simple cell mapping (SCM) method, the generalized cell mapping (GCM) method together with a subdivision procedure. A coarse cell partition is first used to obtain a covering set of the global attractor. Then, a finer global attractor is obtained by the subdivision process. The probabilistic response of stochastic dynamic systems is obtained by the sparse matrix analysis algorithm applied to the covering set of the global attractor. Because the computational domain is the covering set of the global attractor rather than the whole state space, the numerical efficiency of the proposed method can be greatly improved as compared to the GCM. A three-dimensional and a four-dimensional dynamical system under Poisson white noise excitation are studied to demonstrate the effectiveness of the proposed method for the probabilistic response analysis. Monte Carlo simulations show a good agreement with the proposed method.

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“…The undetermined coefficients of the polynomial are determined by solving simultaneous quadratic algebraic equations derived from the method of weighted residue. Recently, the EPC method has been extended to the nonlinear systems excited by a Poisson white noise [39] or by a combined Gaussian and Poisson white noises [40]. They have studied the system with multiple peaks in the PDF [41], and a nonlinear impact oscillator [42,43].…”

Section: Introductionmentioning

confidence: 99%

The closed-form solution of the reduced Fokker–Planck–Kolmogorov equation for nonlinear systems

Chen

Sun

2016

Communications in Nonlinear Science and Numerical Simulation

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In this paper, we propose a new method to obtain the closed-form solution of the reduced Fokker-Planck-Kolmogorov equation for single degree of freedom nonlinear systems under external and parametric Gaussian white noise excitations. The assumed stationary probability density function consists of an exponential polynomial with a logarithmic term to account for parametric excitations. The undetermined coefficients in the assumed solution are computed with the help of an iterative method of weighted residue. We have found that the iterative process generates a sequence of solutions that converge to the exact solutions if they exist. Three examples with known exact steady-state probability density functions are used to demonstrate the convergence of the proposed method.

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Probabilistic solution of nonlinear oscillators excited by combined Gaussian and Poisson white noises

Cited by 47 publications

References 37 publications

Approximating the Density of Random Differential Equations with Weak Nonlinearities via Perturbation Techniques

Approximating the Density of Random Differential Equations with Weak Nonlinearities via Perturbation Techniques

Probabilistic response of dynamical systems based on the global attractor with the compatible cell mapping method

The closed-form solution of the reduced Fokker–Planck–Kolmogorov equation for nonlinear systems

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