ONSET OF COOPERATIVE ENTRAINMENT IN LIMIT-CYCLE OSCILLATORS WITH UNIFORM ALL-TO-ALL INTERACTIONS - BIFURCATION OF THE ORDER FUNCTION

The concept of order function was previously proposed as the key to a general theory of macroscopic mutual (or cooperative) entrainment in l arge populations of limit-cycle oscillators with weak interactions suc h that every element is linked to all the other, as well as with weakl y dispersed intrinsic frequencies, that is, limit-cycle oscillators th at can be modeled by globally coupled phase oscillators with distribut ed frequencies. Following previous work, a bifurcation theory of the o rder function is developed on the basis of its self-consistent functio nal equation to elucidate, in particular, generic scaling behavior of such systems at the onset of cooperative entrainment. Among other resu lts, when the onset is not abrupt, the critical exponent of fundamenta l order parameters turns out to generically differ from the convention al value 1/2 taken by the well-studied sinusoidal coupling model as we ll as by typical mean-fields models of thermodynamic phase transitions to which coupled-oscillator models investigated here are analogous. T he theory also reveals what happens in nongeneric cases. Moreover, a c riterion is found of whether the bifurcation is normal or inverted. Al l these analytical results and predictions are verified not only by nu merically solving the equation of the order function, but also by nume rical simulations. Although this paper is mainly concerned with the cr itical 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.