MS14.1: Random Dynamical Systems - Recent Advances and New Directions

This work studies an effective and straightforward approach to investigate the response of impact systems driven by both period and random excitations. A single-degree-of-freedom impacting system with a one-sided soft constraint is considered in this study. The critical noise intensity for noise-induced bifurcation is estimated by utilising stochastic sensitivity analysis and confidence ellipses. The jump location of the stochastic attractor is identified based on the maximum eigenvalue evolution of the stochastic sensitivity function. Extensive simulations are provided to validate the effectiveness of the proposed approach. The potential for switching control of coexisting attractors under stochastic perturbation is also investigated.

This study discusses the stochastic behaviour of vibroimpact oscillators featuring dielectric elastomer membranes designed for vibration energy harvesting. In these devices, a ball moves freely within a forced cylindrical capsule, intermittently colliding with both ends, covered with dielectric membranes to harvest mechanical energy. The interaction of the rolling motion, impacts and friction introduces stochastic behaviour, even in case of deterministic forcing and constant parameters. In this talk we compare several stochastic mechanical models of the vibro-imapct oscillators to measurements done on a physical device to find an efficient and concise model that reproduces the observed behaviour with high fidelity.

The distinction between dynamical systems with random parameters and noise is investigated. The transfer operator theory is presented for both cases, showing their differences, and discussing ways to numerically approximate them. It is shown that random parameter systems can be approximated through stochastic systems, given a well-adjusted noise.

It becomes apparent that stochastic partial differential equations driven by white noise are conceptually flawed for modeling purposes. This is because neighboring infinitesimally small spatial regions receive similar random perturbations, which violates the fundamental assumption of white noise—namely, that these perturbations should be independent. To address this issue, a more natural approach is to employ spatially-correlated or colored noise, which not only rectifies the modeling inconsistencies but also introduces regularization to the solutions. Nevertheless, the utilization of colored noise introduces a complication, as it disrupts the fluctuation-dissipation relationship of thermal equilibrium. In this presentation, I will discuss how the continuous spatiotemporal limit of a Metropolis Hastings random walk can be employed to derive a stochastic partial differential equation driven by colored noise that preserves its ability to sample the equilibrium distribution, even for a system of magnetic spins. This introduces an additional geometric constraint and yields non-trivial interactions with the correlated noise, further enhancing our understanding of these intricate systems.

This research is about the non-stationary probabilistic solution of the vibro-impact system under external Gaussian white noise. The unilateral barrier is adopted as the constraint to the system, and the constrained system is transformed into another system without barriers. The relevant Fokker-Planck equation is then formulated and solved by the non-stationary exponential-polynomial-closure (EPC) method. The results from the non-stationary EPC method are compared with those from Monte Carlo simulation (MCS) and equivalent linearization method (EQL). The result comparisons indicate that the non-stationary EPC method is much more efficient than MCS and can give much more accurate solution than EQL.