skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Probabilistic density function method for nonlinear dynamical systems driven by colored noise

Abstract

We present a probability density function (PDF) method for a system of nonlinear stochastic ordinary differential equations driven by colored noise. The method provides an integro-differential equation for the temporal evolution of the joint PDF of the system's state, which we close by means of a modified Large-Eddy-Diffusivity-type closure. Additionally, we introduce the generalized local linearization (LL) approximation for deriving a computable PDF equation in the form of the second-order partial differential equation (PDE). We demonstrate the proposed closure and localization accurately describe the dynamics of the PDF in phase space for systems driven by noise with arbitrary auto-correlation time. We apply the proposed PDF method to the analysis of a set of Kramers equations driven by exponentially auto-correlated Gaussian colored noise to study the dynamics and stability of a power grid.

Authors:
;
Publication Date:
Research Org.:
Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1327085
Report Number(s):
PNNL-SA-114643
Journal ID: ISSN 2470-0045; KJ0401000
DOE Contract Number:  
AC05-76RL01830
Resource Type:
Journal Article
Journal Name:
Physical Review E
Additional Journal Information:
Journal Volume: 93; Journal Issue: 5; Journal ID: ISSN 2470-0045
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
Probability Density Function method; uncertainty quantification; Langevin Equation; power grid

Citation Formats

Barajas-Solano, David A., and Tartakovsky, Alexandre M. Probabilistic density function method for nonlinear dynamical systems driven by colored noise. United States: N. p., 2016. Web. doi:10.1103/PhysRevE.93.052121.
Barajas-Solano, David A., & Tartakovsky, Alexandre M. Probabilistic density function method for nonlinear dynamical systems driven by colored noise. United States. doi:10.1103/PhysRevE.93.052121.
Barajas-Solano, David A., and Tartakovsky, Alexandre M. Sun . "Probabilistic density function method for nonlinear dynamical systems driven by colored noise". United States. doi:10.1103/PhysRevE.93.052121.
@article{osti_1327085,
title = {Probabilistic density function method for nonlinear dynamical systems driven by colored noise},
author = {Barajas-Solano, David A. and Tartakovsky, Alexandre M.},
abstractNote = {We present a probability density function (PDF) method for a system of nonlinear stochastic ordinary differential equations driven by colored noise. The method provides an integro-differential equation for the temporal evolution of the joint PDF of the system's state, which we close by means of a modified Large-Eddy-Diffusivity-type closure. Additionally, we introduce the generalized local linearization (LL) approximation for deriving a computable PDF equation in the form of the second-order partial differential equation (PDE). We demonstrate the proposed closure and localization accurately describe the dynamics of the PDF in phase space for systems driven by noise with arbitrary auto-correlation time. We apply the proposed PDF method to the analysis of a set of Kramers equations driven by exponentially auto-correlated Gaussian colored noise to study the dynamics and stability of a power grid.},
doi = {10.1103/PhysRevE.93.052121},
journal = {Physical Review E},
issn = {2470-0045},
number = 5,
volume = 93,
place = {United States},
year = {2016},
month = {5}
}