Phase and Amplitude dynamics of nonlinearly coupled oscillators

TL;DR

This paper investigates the phase and amplitude dynamics of large systems of nonlinearly coupled, non-identical damped harmonic oscillators, revealing conditions for synchronization and bifurcations influenced by nonlinear coupling.

Contribution

It provides a geometric framework to analyze how nonlinear coupling affects synchronization and bifurcations in large oscillator systems, including creation and destruction of solutions.

Findings

Synchronization occurs with regular structure related to oscillator frequency distribution.

Nonlinear coupling induces pitchfork and saddle-node bifurcations affecting stability.

In-phase and anti-phase solutions emerge in systems with bimodal frequency distributions.

Abstract

This paper addresses the amplitude and phase dynamics of a large system non-linear coupled, non-identical damped harmonic oscillators, which is based on recent research in coupled oscillation in optomechanics. Our goal is to investigate the existence and stability of collective behaviour which occurs due to a play-off between the distribution of individual oscillator frequency and the type of nonlinear coupling. We show that this system exhibits synchronisation, where all oscillators are rotating at the same rate, and that in the synchronised state the system has a regular structure related to the distribution of the frequencies of the individual oscillators. Using a geometric description we show how changes in the non-linear coupling function can cause pitchfork and saddle-node bifurcations which create or destroy stable and unstable synchronised solutions. We apply these results to show how in-phase and anti-phase solutions are created in a system with a bi-modal distribution of frequencies.