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

Journal Article · Sun May 01 00:00:00 EDT 2016 · Physical Review E

DOI:https://doi.org/10.1103/PhysRevE.93.052121· OSTI ID:1327085

Barajas-Solano, David A.; Tartakovsky, Alexandre M.

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.

• Research Organization: Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
• Sponsoring Organization: USDOE
• DOE Contract Number: AC05-76RL01830
• OSTI ID: 1327085
• Report Number(s): PNNL-SA-114643; KJ0401000
• Journal Information: Physical Review E, Vol. 93, Issue 5; ISSN 2470-0045
• Publisher: American Physical Society (APS)
• Country of Publication: United States
• Language: English

Related Subjects

Probability Density Function method
uncertainty quantification
Langevin Equation
power grid