Order function theory of macroscopic phase-locking in globally and weakly coupled limit-cycle oscillators

Macroscopic phase-locking in large populations of coupled nonlinear oscillators is an interesting phenomenon analogous to collective behaviors of conventional cooperative systems, being paid increasing attention in diverse fields of science. The behavior of such populations is known to be well described by what is called phase models, provided that both mutual interactions among oscillators and the dispersion of their intrinsic frequencies are comparatively weak. Here reviewed is one of recent developments in the study of macroscopic phase-locking based on phase models, which is the order function theory enabling us to understand and analyze that phenomenon in a general way for such populations that every element is coupled to all the other. With this theory, generic features of macroscopic phase-locking under such coupling will be clarified, in particular, as to scaling behaviors at the onset of macroscopic phase locking.