Frequency-entrainment measures in coupled-oscillator systems

We study the frequency entrainment behavior in systems of coupled oscillators. In particular, we pay attention to the measures for detecting the frequency entrainment. We consider a system of coupled oscillators in three dimensions (3D) as one example. Whereas the system does not exhibit the phase synchrony at any finite coupling strength in the thermodynamic limit, the frequencies of the oscillators are known to be entrained in 3D. We illustrate several measures that have been used in the previous studies to detect the frequency entrainment and compare their scaling behaviors near criticality. Via a finite-size scaling analysis combined with the recently reported critical exponents (β = 0 and ν = 2 in 3D, we reveal that the Edward-Anderson-type order parameter popularly used in the spin glass systems can, indeed, be used as one good order parameter to detect frequency entrainment.