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

TL;DR

This study explores how the critical exponent for synchronization transition in globally coupled oscillators varies with different coupling schemes, especially when including a second harmonic term, revealing dependence on the coupling function.

Contribution

It numerically estimates the critical exponent for a generalized coupling function, extending understanding beyond the sinusoidal case and showing its dependence on coupling parameters.

Findings

The critical exponent $ u_+$ is approximately 5/2 for sinusoidal coupling.

The exponent $ u_+$ increases with the strength of the second harmonic term.

Critical exponents depend significantly on the form of the coupling function.

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.