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Long-term fluctuations in globally coupled phase oscillators with general coupling: Finite size effects

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10.1063/1.3692966

### Abstract

We investigate the diffusion coefficient of the time integral of the Kuramoto order parameter in globally coupled nonidentical phase oscillators. This coefficient represents the deviation of the time integral of the order parameter from its mean value on the sample average. In other words, this coefficient characterizes long-term fluctuations of the order parameter. For a system of *N* coupled oscillators, we introduce a statistical quantity *D*, which denotes the product of *N* and the diffusion coefficient. We study the scaling law of *D* with respect to the system size *N*. In other well-known models such as the Ising model, the scaling property of *D* is for both coherent and incoherent regimes except for the transition point. In contrast, in the globally coupled phase oscillators, the scaling law of *D* is different for the coherent and incoherent regimes: with a certain constant in the coherent regime and in the incoherent regime. We demonstrate that these scaling laws hold for several representative coupling schemes.

© 2012 American Institute of Physics

Received 13 September 2011
Accepted 14 February 2012
Published online 21 March 2012

Lead Paragraph:

In real-world systems, large populations of coupled oscillators often experience global synchronous oscillations. To elucidate the general properties of such phenomena, considerable research has been conducted on simple models of globally coupled phase oscillators. The Kuramoto order parameter has been widely used to measure the degree of synchronization and to characterize the synchronization transition in the phase oscillatormodel. When the system size is infinite, fluctuations of the order parameter vanish after a transient period; the scaling law for this parameter has been well studied for a general coupling function. However, when the system is large but of finite size, the fluctuations in the order parameter do not vanish and their behavior has not been fully understood. It is not clear whether the conventional standard statistical quantities such as the variance and the correlation time of the order parameter can fully characterize its fluctuation behavior. Further, the dependence of the statistical properties of the order parameter on the coupling scheme is still not completely understood. As a step toward understanding these problems, we focus on a statistical quantity that characterizes long-term fluctuations in the order parameter. In other well-known models such as the Ising model, the scaling property of the statistical quantity with respect to the system size is the same for coherent and incoherent regimes except for the transition point. In contrast, in the globally coupled phase oscillators, the decay speed of the statistical quantity in the coherent regime is faster than that in the incoherent regime. This difference is caused by a difference in the correlations among the phases of the oscillators at different times. We show that the scaling laws hold for a large class of general coupling schemes.

Acknowledgments: We would like to thank K. Ouchi, Y. Takahashi, and Y. Sento for their fruitful discussions. This research is supported by Grant-in-Aid for Scientific Research (A) (20246026) from MEXT of Japan, and by the Aihara Innovative Mathematical Modelling Project, the Japan Society for the Promotion of Science (JSPS) through the “Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program),” initiated by the Council for Science and Technology Policy (CSTP).

Article outline:

I. INTRODUCTION

II. A STATISTICAL QUANTITY *D* CHARACTERIZING LONG-TERM FLUCTUATIONS OF THE ORDER PARAMETER

III. SCALING LAW OF *D* WITH SYSTEM SIZE FOR THE KURAMOTO MODEL

A. Coherent regime

B. Incoherent regime

IV. SCALING LAW OF *D* WITH SYSTEM SIZE FOR MORE GENERAL COUPLINGS

A. Coherent regime

B. Incoherent regime

V. DERIVATION OF *D* = 0 FOR THE KURAMOTO MODEL

VI. DERIVATION OF *D* = 0 FOR A GENERAL COUPLING FUNCTION

A. Transformation of the phase oscillatormodel

B. The self-consistent equation of fluctuations

C. The Fourier transform of the self-consistent equation

VII. SUMMARY AND DISCUSSION

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2012-03-21

2014-04-23

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