A systematic path to non-Markovian dynamics II: Probabilistic response of nonlinear multidimensional systems to Gaussian colored noise excitation

TL;DR

The paper tackles non-Markovian dynamics of multidimensional nonlinear systems driven by colored Gaussian noise by deriving a closed, solvable pdf-evolution equation for the first-order response pdf. It leverages the Stochastic Liouville Equation transformed via an extended Novikov-Furutsu theorem and explicit state-transition matrices, with a Magnus-based current-time closure to capture memory effects. The resulting ngFPKE generalizes the FPKE to colored noise, reduces to SCT under certain approximations, and is validated against Monte Carlo simulations on a bistable Duffing oscillator, showing superior accuracy at larger correlation times. The work provides a rigorous, scalable framework for non-Markovian probabilistic modeling in high-dimensional stochastic dynamics and suggests extensions to multiplicative noise. The mathematical machinery hinges on nonlocal diffusion terms and mean-field type decompositions that preserve essential non-Markovian features while enabling tractable computation.

Abstract

The probabilistic characterization of non-Markovian responses to nonlinear dynamical systems under colored excitation is an important issue, arising in many applications. Extending the Fokker-Planck-Kolmogorov equation, governing the first-order response probability density function (pdf), to this case is a complicated task calling for special treatment. In this work, a new pdf-evolution equation is derived for the response of nonlinear dynamical systems under additive colored Gaussian noise. The derivation is based on the Stochastic Liouville equation (SLE), transformed, by means of an extended version of the Novikov-Furutsu theorem, to an exact yet non-closed equation, involving averages over the history of the functional derivatives of the non-Markovian response with respect to the excitation. The latter are calculated exactly by means of the state-transition matrix of variational, time-varying systems. Subsequently, an approximation scheme is implemented, relying on a decomposition of the state-transition matrix in its instantaneous mean value and its fluctuation around it. By a current-time approximation to the latter, we obtain our final equation, in which the effect of the instantaneous mean value of the response is maintained, rendering it nonlinear and non-local in time. Numerical results for the response pdf are provided for a bistable Duffing oscillator, under Gaussian excitation. The pdfs obtained from the solution of the novel equation and a simpler small correlation time (SCT) pdf-evolution equation are compared to Monte Carlo (MC) simulations. The novel equation outperforms the SCT equation as the excitation correlation time increases, keeping good agreement with the MC simulations.

Figures (12)

• Figure 1: Geometrical interpretation of the integral $\underset{\varepsilon \,\,\downarrow \,\,0}{\mathop{\lim }}\,\,\,\int_{{{t}_{0}}}^{t}{{{\delta }_{\varepsilon }}\,(\,t-s\,)\,\,g\,(\,s\,)\,\,ds}$.
• Figure 1: Evolution of response marginal pdfs for the oscillator (\ref{['eqsmm:1']}) configured as described in \ref{['tab_sm:1']}. Analytic marginal pdfs (continuous colored lines) are compared to corresponding ones obtained via the PUFEM solution of the genFPKE (dashed black lines) and MC simulation (marked grey lines), at different times.
• Figure 2: Visual explanation of the notation $\Phi [\mathbf{A}](t;s,\theta )$
• Figure 2: Evolution contours of 2D response pdf for the for the oscillator (\ref{['eqsmm:1']}) configured as described in \ref{['tab_sm:1']}. Contour lines of the analytic pdf (continuous colored lines) are compared to corresponding PUFEM approximations (dashed black lines) and MC simulation (dashed dotted grey lines), at different time instances.
• Figure 3: Evolution of response marginal pdfs for the oscillator (\ref{['eq:6.2a']}), configured as described in \ref{['tab:1']} [$\mathbf{\boldsymbol{m}}_{\mathbf{\boldsymbol{X}}^0} =\mathbf{\boldsymbol{0}} , m_{\Xi}(t) = 0$], for three values of relative correlation time. Marginal pdfs obtained via the PUFEM solutions to the SCT approximation (dashed-dotted lines) and novel genFPKE (dashed lines) are compared to MC simulations (marked continuous lines), in different time instances. The greatest time corresponds to the long-time steady-state regime.