  The present paper introduces a linear reformulation of the Kuramoto model
describing a self-synchronizing phase transition in a system of globally
coupled oscillators that in general have different characteristic frequencies.
The reformulated model provides an alternative coherent framework through which
one can analytically tackle synchronization problems that are not amenable to
the original Kuramoto analysis. It allows one to solve explicitly for the
synchronization order parameter and the critical point of 1) the full
phase-locking transition for a system with a finite number of oscillators
(unlike the original Kuramoto model, which is solvable implicitly only in the
mean-field limit) and 2) a new class of continuum systems. It also makes it
possible to probe the system's dynamics as it moves towards a steady state.
While discussion in this paper is restricted to systems with global coupling,
the new formalism introduced by the linear reformulation also lends itself to
solving systems that exhibit local or asymmetric coupling.
