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Fully synchronous solutions and the synchronization phase transition for the finite-*N* Kuramoto model

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10.1063/1.4745197

### Abstract

We present a detailed analysis of the stability of phase-locked solutions to the Kuramoto system of oscillators. We derive an analytical expression counting the dimension of the unstable manifold associated to a given stationary solution. From this we are able to derive a number of consequences, including analytic expressions for the first and last frequency vectors to phase-lock, upper and lower bounds on the probability that a randomly chosen frequency vector will phase-lock, and very sharp results on the large *N* limit of this model. One of the surprises in this calculation is that for frequencies that are Gaussian distributed, the correct scaling for full synchrony is not the one commonly studied in the literature; rather, there is a logarithmic correction to the scaling which is related to the extremal value statistics of the random frequency vector.

© 2012 American Institute of Physics

Received 06 January 2012
Accepted 30 May 2012
Published online 28 August 2012

Lead Paragraph:

The Kuramoto model is a fundamental model for the study of phase-locking phenomena and has been proposed as a model for phenomena as diverse as the synchronization of fireflies and the onset of unwanted oscillations in the Millennium bridge. In this work we study the problem of full phase-locking, where all oscillatorsoscillate with the mean frequency. We derive an exact expression for the set of frequency vectors for which phase-locking is possible and use this representation to derive upper and lower bounds on the probability of phase-locking in the case in which the natural frequencies of the oscillators are random. The bounds show rigorously that there is a phase transition in the model; in the limit of a large number of oscillators the probability of phase-locking is zero below a critical coupling value and is identically one above a (larger) critical coupling value. This represents the first rigorous proof of such a phase transition in the finite-*N*model.

Acknowledgments: The authors would like to thank Yulij Baryshnikov and Florian Dörfler for useful discussions, and the two anonymous referees for very useful comments. L.D. was partially supported by NSF grant CMG-0934491. J.C.B. and M.J.P. were partially supported by NSF grant DMS-0807584.

Article outline:

I. INTRODUCTION

A. History of Kuramoto model

B. Problem formulation

II. CHARACTERIZATION OF THE STABLE SET

A. Notation

B. Index theorem

C. Lower bounds on the stable set

III. UPPER AND LOWER BOUNDS ON

IV. LARGE *N* LIMIT

V. EXAMPLES

A. Comprehensive example for three oscillators

B. Example for four oscillators

VI. SCALING AND EXTREME VALUE STATISTICS

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2012-08-28

2014-04-17

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