Continuous limit and the moments system for the globally coupled phase oscillators

Hayato Chiba

doi:10.3934/dcds.2013.33.1891

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2013, Volume 33, Issue 5: 1891-1903. Doi: 10.3934/dcds.2013.33.1891

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Continuous limit and the moments system for the globally coupled phase oscillators

Hayato Chiba^1,

1.
Institute of Mathematics for Industry, Kyushu University, Fukuoka, 819-0395

Received: November 30, 2011

Revised: June 30, 2012

Published: April 2013

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Abstract

The Kuramoto model, which describes synchronization phenomena, is a system of ordinary differential equations on $N$-torus defined as coupled harmonic oscillators. The order parameter is often used to measure the degree of synchronization. In this paper, the moments systems are introduced for both of the Kuramoto model and its continuous model. It is shown that the moments systems for both systems take the same form. This fact allows one to prove that the order parameter of the $N$-dimensional Kuramoto model converges to that of the continuous model as $N\to \infty$.

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Mathematics Subject Classification: 37L05, 35Q70.

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