MS14.2: Random Dynamical Systems - Recent Advances and New Directions

Epidemic spread dynamics driven by nonlinearity and noise is a focal research topic post the COVID-19 pandemic. Turing-type instabilities can drastically alter epidemic trajectories leading to unique spread patterns. We study instabilities in epidemic spread using a compartmental, stochastic partial differential equation (SPDE) model, accounting for (1) nonlinear saturation effects and (2) environmental effects represented by additive noise. From second-order perturbations of the disease-free equilibrium we analytically derive a hierarchy of moment equations using Novikov’s theorem. Judiciously truncating this infinite hierarchy of moment equations, we analyze stability by computing the maximal Lyapunov exponent against the wave number that characterizes the perturbations. We analyze stability with respect to the saturation parameter and the noise intensity. Additionally, obtaining stationary solutions of the SPDE, we find distinct spatial patterns of infection spread consistent with the stability results. Specifically, we observe the emergence of noise-induced patterns in cases where diffusion and saturation are insufficient to trigger instabilities. The results are expected to have broader significance beyond epidemic dynamics since reaction-diffusion systems represent a variety of phenomena in the natural sciences and engineering.

This work presents a geometry preserving weak symplectic stochastic integrator for long-time simulation of Hamiltonian systems on S^2 manifold. The proposed algorithm preserves the energy of the system accurately for long-time integration of the system using a multiple time stepping strategy. For the geometric systems under stochastic perturbation, the proposed symplectic scheme is formulated using an Ito-Taylor expansion-based weak integration scheme and exploiting the relation between the manifold and its Lie Algebra through exponential mapping. Since the weak schemes are concerned with accuracy with respect to a function of the true solution rather than pathwise accuracy, the weak schemes have the advantage of considerably simpler and cheaper numerical implementation. Numerical illustrations on non-linear oscillators on the S^2 manifold are demonstrated to show the geometric and symplectic nature of the algorithm.

In the mid-high frequency range, system uncertainties that may arise from manufacture, the environment, or material properties have a far greater influence on the structural response than in the low frequency range. As a result, obtaining estimates of ensemble response statistics become severely expensive when using conventional deterministic methods employing a Monte Carlo analysis. The Hybrid Finite Element - Statistical Energy Analysis (FE-SEA) method, is an alternative approach that employs a statistical description of components to efficiently yield response statistics. A limitation of this technique is that it was developed for linear systems which restricts the applications, since localised nonlinearities are likely to present themselves within practical engineering systems. In this work, a linearisation scheme is developed for an ensemble of random systems with localised deterministic nonlinearities that is based around the technique of Equivalent Linearisation (EL). This accounts for the randomness present in both an ensemble of random structures and the random response exhibited by a deterministic structure under random loading, for which EL was developed. Although the linearisation is completely general, it is used as part of a Hybrid FE-SEA analysis requiring significant extension of existing theory to evaluate statistical distributions of the response concerning the nonlinearity. Benchmark studies are conducted using Monte Carlo simulations with a Lagrange Rayleigh-Ritz model, and this is assessed against the linearised Hybrid FE-SEA method considered for different linearisation techniques.

In this paper, the nonstationary response of nonlinear single-degree-of-freedom systems subject to stochastic excitations is examined. A version of the Path Integral (PI) approach is developed for determining the evolution of the response probability density function (PDF). Specifically, the PI approach is here employed in conjunction with the Laplace’s method of integration. In this manner, an approximate analytical solution of the integral involved in the Chapman-Kolmogorov equation is obtained, thus circumventing the repetitive integrations generally required in the conventional numerical implementation of the procedure. The reliability of the procedure is assessed considering the case of a Duffing nonlinear oscillator, and appropriate comparisons with Monte Carlo simulation data are presented demonstrating the accuracy of the proposed approach.