On the critical coupling strength for Kuramoto oscillators

Abstract

Abstract-The celebrated Kuramoto model captures various synchronization phenomena in biological and man-made dynamical 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 three 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 in the finite and infinitedimensional case. Finally, we present the first explicit necessary and sufficient condition on the critical coupling strength 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. I. THE KURAMOTO MODEL OF COUPLED OSCILLATORSA classic model for the synchronization of coupled oscillators is due to Kuramoto [1]. The Kuramoto model considers n ≥ 2 coupled oscillators each represented by a phase variable θ i ∈ T 1 , the 1-tours, and a natural frequency ω i ∈ R. The system of coupled oscillators obeys the dynamicṡwhere K > 0 is the coupling strength among the oscillators. The Kuramoto model (1) finds application in various biological synchronization phenomena, and we refer the reader to the excellent reviews Kuramoto himself analyzed the model (1) based on the order parameter re iψ 1 n n j=1 e iθj , which corresponds the centroid of all oscillators when represented as points on the unit circle in C 1 . The magnitude r of the order parameter can be understood as a measure of synchronization: if all oscillators are perfectly synchronized with identical angles θ i (t), then r = 1, and if all oscillators are spaced equally on the unit circle, then r = 0. With the order parameter, the Kuramoto model (1) can be written in the insightful forṁThis work was supported in part by NSF grants IIS-0904501 and CNS-0834446.Florian Dörfler and Francesco Bullo are with the Center for Control, Dynamical Systems and Computation, University of California at Santa Barbara, Santa Barbara, CA 93106, {dorfler, bullo}@engineering.ucsb.edu Equation (2) gives the intuition that the oscillators synchronize by coupling to a mean field represented by the order parameter re iψ . Intuitively, for small coupling strength K each oscillator rotates with its natural frequency ω i , whereas for large coupling strength K all angles θ i (t) will be entrained by the mean field re iψ . This phase transition from incoherency to synchronization occurs for some critical coupling K critical and has been the source of nume...

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