  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.
