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

2009 ◽  
Vol 5 (1) ◽  
Author(s):  
Pankaj Kumar ◽  
S. Narayanan
Keyword(s):  
Nonlinear Systems ◽  
White Noise ◽  
Path Integral ◽  
Duffing Oscillator ◽  
Planck Equation ◽  
Harmonic Excitation ◽  
Integration Scheme ◽  
Pi Method ◽  
Non Gaussian ◽  
Fokker Planck

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.

scholarly journals GENERALIZED WIENER PROCESS AND KOLMOGOROV'S EQUATION FOR DIFFUSION INDUCED BY NON-GAUSSIAN NOISE SOURCE

2005 ◽  
Vol 05 (02) ◽  
pp. L267-L274 ◽  
Author(s):  
ALEXANDER DUBKOV ◽  
BERNARDO SPAGNOLO
Keyword(s):  
White Noise ◽  
Wiener Process ◽  
Anomalous Diffusion ◽  
Gaussian Noise ◽  
Gaussian White Noise ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Non Gaussian ◽  
Generalized Wiener Process ◽  
Fokker Planck

We show that the increments of generalized Wiener process, useful to describe non-Gaussian white noise sources, have the properties of infinitely divisible random processes. Using functional approach and the new correlation formula for non-Gaussian white noise we derive directly from Langevin equation, with such a random source, the Kolmogorov's equation for Markovian non-Gaussian process. From this equation we obtain the Fokker–Planck equation for nonlinear system driven by white Gaussian noise, the Kolmogorov–Feller equation for discontinuous Markovian processes, and the fractional Fokker–Planck equation for anomalous diffusion. The stationary probability distributions for some simple cases of anomalous diffusion are derived.

Governing equations for probability densities of Marcus stochastic differential equations with Lévy noise

2016 ◽  
Vol 17 (05) ◽  
pp. 1750033 ◽  
Author(s):  
Xu Sun ◽  
Xiaofan Li ◽  
Yayun Zheng
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Probability Density ◽  
Lévy Processes ◽  
Planck Equation ◽  
Levy Processes ◽  
Stochastic Dynamical Systems ◽  
Non Gaussian ◽  
Fokker Planck Equations ◽  
Fokker Planck

Marcus stochastic differential equations (SDEs) often are appropriate models for stochastic dynamical systems driven by non-Gaussian Lévy processes and have wide applications in engineering and physical sciences. The probability density of the solution to an SDE offers complete statistical information on the underlying stochastic process. Explicit formula for the Fokker–Planck equation, the governing equation for the probability density, is well-known when the SDE is driven by a Brownian motion. In this paper, we address the open question of finding the Fokker–Planck equations for Marcus SDEs in arbitrary dimensions driven by non-Gaussian Lévy processes. The equations are given in a simple form that facilitates theoretical analysis and numerical computation. Several examples are presented to illustrate how the theoretical results can be applied to obtain Fokker–Planck equations for Marcus SDEs driven by Lévy processes.

scholarly journals Linking Nonlinearity and Non-Gaussianity of Planetary Wave Behavior by the Fokker–Planck Equation

2005 ◽  
Vol 62 (7) ◽  
pp. 2098-2117 ◽  
Author(s):  
Judith Berner
Keyword(s):  
Stochastic Model ◽  
Probability Density ◽  
Planetary Wave ◽  
Multiplicative Noise ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
State Dependent ◽  
Nonlinear Stochastic Model ◽  
Non Gaussian ◽  
Fokker Planck

Abstract To link prominent nonlinearities in the dynamics of 500-hPa geopotential heights to non-Gaussian features in their probability density, a nonlinear stochastic model of atmospheric planetary wave behavior is developed. An analysis of geopotential heights generated by extended integrations of a GCM suggests that a stochastic model and its associated Fokker–Planck equation call for a nonlinear drift and multiplicative noise. All calculations are carried out in the reduced phase space spanned by the leading EOFs. It is demonstrated that this nonlinear stochastic model of planetary wave behavior captures the non-Gaussian features in the probability density function of atmospheric states to a remarkable degree. Moreover, it not only predicts global temporal characteristics, but also the nonlinear, state-dependent divergence of state trajectories. In the context of this empirical modeling, it is discussed on which time scale a stochastic model is expected to approximate the behavior of a continuous deterministic process. The reduced model is then used to determine the importance of the nonlinearities in the drift and the role of the multiplicative noise. While the nonlinearities in the drift are crucial for a good representation of planetary wave behavior, multiplicative (i.e., state dependent) noise is not absolutely essential. It is found that a major contributor to the stochastic component is the Branstator–Kushnir oscillation, which acts as a fluctuating force for physical processes with even longer time scales, like those that project on the Arctic Oscillation pattern. In this model, the oscillation is represented by strongly correlated noise.

First-passage problem for nonlinear systems under Lévy white noise through path integral method

2016 ◽  
Vol 85 (3) ◽  
pp. 1445-1456 ◽  
Author(s):  
Christian Bucher ◽  
Alberto Di Matteo ◽  
Mario Di Paola ◽  
Antonina Pirrotta
Keyword(s):  
Nonlinear Systems ◽  
White Noise ◽  
Path Integral ◽  
Integral Method ◽  
First Passage ◽  
Lévy White Noise ◽  
Path Integral Method ◽  
First Passage Problem

scholarly journals The influence of nonlinear terms in mechanical systems having two degrees of freedom

2006 ◽  
Vol 28 (3) ◽  
pp. 155-164
Author(s):  
Nguyen Duc Tinh
Keyword(s):  
Nonlinear Systems ◽  
White Noise ◽  
Degrees Of Freedom ◽  
Averaging Method ◽  
Duffing Oscillator ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Higher Order ◽  
White Noise Excitation ◽  
Two Degree Of Freedom

For many years the higher order stochastic averaging method has been widely used for investigating nonlinear systems subject to white and coloured noises to predict approximately the response of the systems. In the paper the method is further developed for two-degree-of-freedom systems subjected to white noise excitation. Application to Duffing oscillator is considered.

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