Onset of cooperative entrainment in limit-cycle oscillators with uniform all-to-all interactions: bifurcation of the order function

Abstract

The concept of order function was previously proposed as the key to a general theory of macroscopic mutual (or cooperative) entrainment in large populations of limit-cycle oscillators with weak interactions such that every element is linked to all the other, as well as with weakly dispersed intrinsic frequencies, that is, limit-cycle oscillators that can be modeled by globally coupled phase oscillators with distributed frequencies. Following previous work, a bifurcation theory of the order function is developed on the basis of its self-consistent functional equation to elucidate, in particular, generic scaling behavior of such systems at the onset of cooperative entrainment. Among other results, when the onset is not abrupt, the critical exponent of fundamental order parameters turns out to generically differ from the conventional value 1 2 taken by the well-studied sinusoidal coupling model as well as by typical mean-fields models of thermodynamic phase transitions to which coupled-oscillator models investigated here are analogous. The theory also reveals what happens in nongeneric cases. Moreover, a criterion is found of whether the bifurcation is normal or inverted. All these analytical results and predictions are verified not only by numerically solving the equation of the order function, but also by numerical simulations. Although this paper is mainly concerned with the critical behaviors, noncritical regimes are also explored to demonstrate overall power of the order function theory by reproducing simulation results such as average-frequency spectra. The theory, however, keeps some room to be further generalized. A finding which suggests this is put forth.