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

Lifang Feng, Bin Pei, Yong Xu

TL;DR

This paper addresses probabilistic responses of nonlinear SDOF oscillators subject to fractional Gaussian noise carrying memory. It develops a memory-dependent FPK (memFPK) equation via fractional Wick-It\ô-Skorohod calculus and provides a data-driven discretized local mean method (DLMM) to estimate the memory-dependent diffusion, which is then solved with a finite-difference scheme to obtain transient joint PDFs. The approach accurately captures non-Markovian effects, cross-derivative diffusion, and non-Gaussian features across linear and nonlinear examples, with strong validation against analytical solutions and large Monte Carlo simulations. The framework offers a practical tool for non-Markovian stochastic dynamics and can be extended 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.

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

TL;DR

This paper addresses probabilistic responses of nonlinear SDOF oscillators subject to fractional Gaussian noise carrying memory. It develops a memory-dependent FPK (memFPK) equation via fractional Wick-It\ô-Skorohod calculus and provides a data-driven discretized local mean method (DLMM) to estimate the memory-dependent diffusion, which is then solved with a finite-difference scheme to obtain transient joint PDFs. The approach accurately captures non-Markovian effects, cross-derivative diffusion, and non-Gaussian features across linear and nonlinear examples, with strong validation against analytical solutions and large Monte Carlo simulations. The framework offers a practical tool for non-Markovian stochastic dynamics and can be extended 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.
Paper Structure (15 sections, 3 theorems, 61 equations, 12 figures)

This paper contains 15 sections, 3 theorems, 61 equations, 12 figures.

Table of Contents

  1. Introduction
  2. The memFPK equation for SDOF oscillators excited by FGN
  3. Linear SDOF oscillators
  4. Nonlinear SDOF oscillators
  5. Mathematical support for the memFPK equation
  6. The estimation of the memory-dependent diffusion coefficients
  7. Numerical results
  8. Example 1
  9. Example 2
  10. Example 3
  11. Example 4
  12. Discussions and conclusions
  13. Several useful Lemmas
  14. Derivatives for the explicit expression of Malliavin derivatives
  15. Derivation for the memFPK equation (Theorem \ref{['them-1']})

Key Result

Theorem 2

Consider (2dsds-1) with initial condition $\boldsymbol{Y}_0=\boldsymbol{y_0}$ which is a random vector with known distribution. Suppose that the function $\boldsymbol{\hat{f}}$ is differentiable function vector, and $\hat{\Sigma}$ is a non-zero constant matrix. Then, for the two-dimensional stochast where the memory-dependent drift and diffusion coefficients are with

Figures (12)

  • Figure 1: Joint PDF of Example 1 obtained from memFPK equation and an analytical expression via mean and variance: (a)-(c) memFPK equation solution vs (d)-(f) analytical expression solution.
  • Figure 2: Absolute errors plot of the joint PDF of Example 1: memFPK equation solution vs analytical expression solution
  • Figure 3: Transient PDF of $X(t)$ and $V(t)$ of Example 1 obtained from memFPK equation and analytical expression solutions.
  • Figure 4: Joint PDF of Example 2 obtained from memFPK equation and MCS solutions ($6\times10^6$ samples): (a)-(c) memFPK equation solutions vs (d)-(f) MCS solutions.
  • Figure 5: Transient PDF of $X(t)$ and $V(t)$ of Example 2 obtained from memFPK and MCS solutions ($6\times10^6$ samples).
  • ...and 7 more figures

Theorems & Definitions (7)

  • Remark 1
  • Theorem 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma A.1
  • Lemma A.2