Asymptotic Dynamics of a Class of Coupled Oscillators Driven by White Noises

TL;DR

This paper investigates the long-term behavior of coupled second order oscillators driven by white noise, showing conditions for frequency locking and the existence of a global random attractor, extending previous results to coupled systems.

Contribution

It generalizes existing results on single oscillators to coupled systems, demonstrating the asymptotic dynamics and frequency locking under large damping and coupling.

Findings

Existence of a global random attractor for the system.

Frequency locking occurs with large damping and coupling.

Coupled oscillators behave like first order systems as damping increases.

Abstract

This paper is devoted to the study of the asymptotic dynamics of a class of coupled second order oscillators driven by white noises. It is shown that any system of such coupled oscillators with positive damping and coupling coefficients possesses a global random attractor. Moreover, when the damping and the coupling coefficients are sufficiently large, the global random attractor is a one-dimensional random horizontal curve regardless of the strength of the noises, and the system has a rotation number, which implies that the oscillators in the system tend to oscillate with the same frequency eventually and therefore the so called frequency locking is successful. The results obtained in this paper generalize many existing results on the asymptotic dynamics for a single second order noisy oscillator to systems of coupled second order noisy oscillators. They show that coupled damped second order oscillators with large damping have similar asymptotic dynamics as the limiting coupled first order oscillators as the damping goes to infinite and also that coupled damped second order oscillators have similar asymptotic dynamics as their proper space continuous counterparts, which are of great practical importance.