Memory-Dependent FPK Equations for Nonlinear SDOF Oscillators Under Fractional Gaussian Noise Excitation

TL;DR

This paper develops a novel memory-dependent FPK equation framework to analyze the probabilistic responses of nonlinear SDOF oscillators driven by fractional Gaussian noise, capturing memory effects absent in traditional methods.

Contribution

It introduces a memFPK equation based on fractional calculus and a discretized local mean method for the first time to handle non-Markovian FGN-driven systems.

Findings

Accurate probabilistic response predictions for nonlinear oscillators under FGN.

Validation shows excellent agreement with analytical and Monte Carlo solutions.

Framework extends to multidimensional and parametric FGN problems.

Abstract

This paper investigates the probabilistic responses of nonlinear single-degree-of-freedom oscillators under fractional Gaussian noise (FGN) excitation. Unlike Gaussian white noise, FGN exhibits persistent correlations and memory effects, making conventional Fokker-Planck-Kolmogorov (FPK) equation methods inapplicable. To address this, we develop memory-dependent FPK (memFPK) equations based on fractional Wick-Ito-Skorohod calculus, capable of capturing the joint probability of system responses. A discretized local mean method (DLMM) is proposed to estimate the memory-dependent diffusion coefficient, and a finite difference scheme solves the memFPK equation numerically. Validation through linear and nonlinear examples shows excellent agreement with analytical or Monte Carlo solutions. This framework provides a practical tool for analyzing non-Markovian stochastic dynamics, with potential extensions to multidimensional and parametric FGN problems.