Linear reformulation of the Kuramoto model of self-synchronizing coupled oscillators

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 toward a steady state. While discussion in this paper is restricted to systems with global coupling, the formalism introduced by the linear reformulation also lends itself to solving systems that exhibit local or asymmetric coupling.