A linear reformulation of the Kuramoto model of self-synchronizing oscillators

Abstract

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.

arXiv commentAccepted to Phys. Rev. E. v5: Further clarified terminology; expanded discussion; added references

References in corpus (2)

• The Spectrum of the Partially Locked State for the Kuramoto Model
• Selfsynchronization and dissipation-induced threshold in collective atomic recoil lasing

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• Synchrony for weak coupling in the complexified Kuramoto model
• Continuum limits of coupled oscillator networks depending on multiple sparse graphs
• Complexified Synchrony
• A linear path toward synchronization: Anomalous scaling in a new class of exactly solvable Kuramoto models