  The Kuramoto model captures various synchronization phenomena in biological
and man-made systems of coupled oscillators. It is well-known that there exists
a critical coupling strength among the oscillators at which a phase transition
from incoherency to synchronization occurs. This paper features four
contributions. First, we characterize and distinguish the different notions of
synchronization used throughout the literature and formally introduce the
concept of phase cohesiveness as an analysis tool and performance index for
synchronization. Second, we review the vast literature providing necessary,
sufficient, implicit, and explicit estimates of the critical coupling strength
for finite and infinite-dimensional, and for first and second-order Kuramoto
models. Third, we present the first explicit necessary and sufficient condition
on the critical coupling to achieve synchronization in the finite-dimensional
Kuramoto model for an arbitrary distribution of the natural frequencies. The
multiplicative gap in the synchronization condition yields a practical
stability result determining the admissible initial and the guaranteed ultimate
phase cohesiveness as well as the guaranteed asymptotic magnitude of the order
parameter. Fourth and finally, we extend our analysis to multi-rate Kuramoto
models consisting of second-order Kuramoto oscillators with inertia and viscous
damping together with first-order Kuramoto oscillators with multiple time
constants. We prove that the multi-rate Kuramoto model is locally topologically
conjugate to a first-order Kuramoto model with scaled natural frequencies, and
we present necessary and sufficient conditions for almost global phase
synchronization and local frequency synchronization. Interestingly, these
conditions do not depend on the inertiae which contradicts prior observations
on the role of inertiae in synchronization of second-order Kuramoto models.
