Transient probability in basins of noise influenced responses of mono and coupled Duffing oscillators

Abstract

Duffing systems excited by harmonic excitations and subjected to noise additions are considered, and it is examined whether the noise addition can be used to steer the response from one stable mode to another. To help with this examination, the authors propose a methodology for estimating the probability that a short duration Gaussian white noise can be used to generate or destroy stable modes of a single nonlinear oscillator as well as a set of coupled nonlinear oscillators. This estimation is done by using the path integral method to find the transient joint probability density function at discrete points in time and then integrating the probability density function over the basins of attraction of the responses of the deterministic system. Results are provided and discussed for the single Duffing oscillator and two coupled Duffing oscillators forced by a near resonance harmonic excitation and noise addition. This work can form a basis for carrying out noise influenced energy movement or localization in the arrays of nonlinear oscillators and have relevance for applications in sensors, energy harvesting devices, and more.