  We study the synchronisation properties of the Kuramoto model of coupled
phase oscillators on a general network. Here we distinguish the ability of such
a system to self-synchronise from the stability of this behaviour. While
self-synchronisation is a consequence of genuine non-perturbative dynamics, the
stability in dynamical systems is usually accessible by fluctuations about a
fixed point, here taken to be the synchronised solution. We examine this
problem in terms of modes of the graph Laplacian, by which the absolute
Lyapunov stability of the synchronised fixed point is readily demonstrated.
Departures from stability are seen to arise at the next order in fluctuations
where the dynamical equations resemble those for species population models, the
logistic and Lotka-Volterra equations. Methods from these systems are exploited
to analytically derive new critical couplings signalling deviation from
classical stability. We observe in some cases an intermediate regime of
behaviour, between incoherence and synchronisation, where system wide periodic
behaviours are exhibited. We discuss these results in light of simulations.
