Most probable escape paths in periodically driven nonlinear oscillators

TL;DR

This paper develops a methodology combining large deviation, optimal control, and Floquet theories to identify the most probable escape paths and transition rates between vibrational modes in periodically driven nonlinear oscillator arrays.

Contribution

It extends the action plot method to non-autonomous high-dimensional systems and introduces a new approach for solving related optimization problems with discontinuities.

Findings

Computed escape paths and quasipotential barriers for systems with up to five oscillators.

Analyzed how system parameters influence transition barriers.

Visualized transition pathways in complex oscillator arrays.

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, small 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 the Floquet theory. The action plot method is promoted to non-autonomous high-dimensional systems, and a method for solving the arising optimization problem with discontinuous objective function restricted to a certain manifold is proposed. 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 on the parameters of the system is discussed.