Probabilistic responses of dynamical systems subjected to Gaussian coloured noise excitation. Foundations of a non-Markovian theory

TL;DR

This thesis develops a new non-Markovian probabilistic framework for analyzing dynamical systems under Gaussian coloured noise, providing accurate evolution equations that outperform existing models especially in high noise and correlation scenarios.

Contribution

It introduces a novel closure scheme for nonlocal pdf evolution equations, enabling tractable and accurate modeling of non-Markovian responses in stochastic dynamical systems.

Findings

New nonlinear pdf evolution equations match Monte Carlo results.

Equations perform well even with high noise intensity and correlation.

Computational effort is comparable to classical Fokker-Planck equations.

Abstract

The topic of this PhD thesis is the derivation of evolution equations for probability density functions (pdfs) describing the non-Markovian response to dynamical systems under Gaussian coloured (smoothly-correlated) noise. These pdf evolution equations are derived from the stochastic Liouville equations (SLEs), which are formulated by representing the pdfs as averaged random delta functions. SLEs are exact yet non-closed, since they contain averaged terms that are expressed via higher-order pdfs. These averaged terms are further evaluated by employing generalizations of the Novikov-Furutsu (NF) theorem. After the NF theorem, SLE averages are expressed equivalently as nonlocal terms depending on the whole history of the response (in some cases, on the history of excitation too). Then, nonlocal terms are approximated by a novel closure scheme, employing the history of appropriate moments of the response (or joint response-excitation moments). Application of this scheme results in a family of novel pdf evolution equations. These equations are nonlinear and retain a tractable amount of the original nonlocality of SLEs, being also in closed form and solvable. Last, the new evolution equations for the one-time response pdf are solved numerically and their results are compared to Monte Carlo (MC) simulations, for the case of a scalar bistable random differential equation under Ornstein-Uhlenbeck excitation. The results show that the novel evolution equations are in very good agreement with the MC simulations, even for high noise intensities and large correlation times of the excitation, that is, away from the white noise limit, where the existing pdf evolution equations found in literature fail. It should be noted that the computational effort for solving the new pdf evolution equations is comparable to the effort required for solving the respective classical Fokker-Planck-Kolmogorov equation.