Finite-size scaling in globally coupled phase oscillators with a general coupling scheme

I Nishikawa,K Iwayama,K Aihara,T Horita,G Tanaka

doi:10.1093/ptep/ptu015

I Nishikawa, K Iwayama + Show 3 more

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https://doi.org/10.1093/ptep/ptu015

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Abstract

We investigate a critical exponent related to synchronization transition in globally coupled nonidentical phase oscillators. The critical exponents of susceptibility, correlation time, and correlation size are significant quantities to characterize fluctuations in coupled oscillator systems of large but finite size and understand a universal property of synchronization. These exponents have been identified for the sinusoidal coupling but not fully studied for other coupling schemes. Herein, for a general coupling function including a negative second harmonic term in addition to the sinusoidal term, we numerically estimate the critical exponent of the correlation size, denoted by $\nu_+$, in a synchronized regime of the system by employing a non-conventional statistical quantity. First, we confirm that the estimated value of $\nu_+$ is approximately 5/2 for the sinusoidal coupling case, which is consistent with the well-known theoretical result. Second, we show that the value of $\nu_+$ increases with an increase in the strength of the second harmonic term. Our result implies that the critical exponent characterizing synchronization transition largely depends on the coupling function.

Highlights

Populations of coupled rhythmic elements can exhibit synchronization and collective behavior via mutual interactions [1]
We investigate a critical exponent related to synchronization transition in globally coupled nonidentical phase oscillators
For a general coupling function including a negative second harmonic term in addition to the sinusoidal term, we numerically estimate the critical exponent of the correlation size, denoted by ν+, in a synchronized regime of the system by employing a non-conventional statistical quantity

Summary

INTRODUCTION

Populations of coupled rhythmic elements can exhibit synchronization and collective behavior via mutual interactions [1]. The critical exponents in the phase oscillator model (1) with finite large N have been obtained for the sinusoidal coupling function [11,12,13,14,15,16], it is known that the critical exponent of the order parameter depends on the coupling scheme [8, 9]. We employ a non-conventional statistical quantity to evaluate the critical exponent of correlation size, ν+, in the synchronized regime of the phase oscillator model (1) with finite large N. This is because it is difficult to compute the value of ν+ using the critical exponent of the order parameter [16]. Our result highlights a new relation between the couping schemes and universal structures in the globally coupled phase oscillators (1)

STATISTICAL QUANTITY FOR THE FINITE-SIZE SCALING ANALYSIS σ2

THE FINITE-SIZE SCALING ANALYSIS

DISCUSSION