  The Kuramoto model is a standard model for the dynamics of coupled oscillator
networks. In particular, it is used to study long time behavior such as
phase-locking where all oscillators rotate at a common frequency with fixed
angle differences. It has been observed that phase-locking occurs if the
natural frequencies of the oscillators are sufficiently close to each other,
and doesn't if they aren't. Intuitively, the closer the natural frequencies are
to each other the more stable the system is. In this paper, we study the
Kuramoto model on ring networks with positive coupling between the oscillators
and derive a criterion for a bifurcation in which two branches of phase-locked
solutions collide as we increase the coupling strengths. Furthermore, we state
a criterion for when one of these branches consists of stable phase-locked
solutions. In this case stability is lost as the positive coupling is
increased. We then apply our criteria to show that for any size of the ring
network there always exists choices of the natural frequencies and coupling
strengths so that this bifurcation occurs. (We require at least five
oscillators for one branch to consist of stable phase-locked solutions.)
Finally, we conjecture that generically our bifurcation is locally a
subcritical bifurcation and globally an $S$-curve. (In the case where one
branch consists of phase-locked solutions, an $S$-curve results in bistability
i.e. the existence of two distinct stable phase-locked solutions.) Finally, we
note that our methods are constructive.
