Most probable escape paths in periodically driven nonlinear oscillators

Abstract

The dynamics of mechanical systems such as turbomachinery with multiple blades are often modeled by arrays of periodically driven coupled nonlinear oscillators. It is known that such systems may have multiple stable vibrational modes, and transitions between them may occur under the influence of random factors. A methodology for finding most probable escape paths and estimating the transition rates in the small noise limit is developed and applied to a collection of arrays of coupled monostable oscillators with cubic nonlinearity, weak damping, and harmonic external forcing. The methodology is built upon the action plot method (Beri et al., 2005) and relies on the large deviation theory, optimal control theory, and Floquet theory. The action plot method is promoted to non-autonomous high-dimensional systems, and a method is proposed for solving the arising optimization problem with discontinuous objective function restricted to a certain manifold. The most probable escape paths between stable vibrational modes in arrays of up to five oscillators and the corresponding quasipotential barriers are computed and visualized. The dependence of the quasipotential barrier of the parameters on the system is discussed.Response transitions from one dynamic state to another can occur due to random perturbations in a variety of systems, including mechanical and structural systems. Some examples are sensor arrays, energy harvesters, and rotating machinery. The aim of the present work is to elucidate these transitions, in particular, when the random perturbations are weak. To that end, a methodology based on the Action Plot Method from Large Deviation Theory and Optimal Control Theory is developed and illustrated for a range of periodically forced systems.