Scaling and singularities in the entrainment of globally-coupled oscillators

The onset of collective behavior in a population of globally coupled oscillators with randomly distributed frequencies is studied for phase dynamical models with arbitrary coupling. The population is described by a Fokker-Planck equation for the distribution of phases which includes the diffusive effect of noise in the oscillator frequencies. The bifurcation from the phase-incoherent state is analyzed using amplitude equations for the unstable modes with particular attention to the dependence of the nonlinearly saturated mode $|\alpha_\infty|$ on the linear growth rate $\gamma$. In general we find $|\alpha_\infty|\sim \sqrt{\gamma(\gamma+l^2D)}$ where $D$ is the diffusion coefficient and $l$ is the mode number of the unstable mode. The unusual $(\gamma+l^2D)$ factor arises from a singularity in the cubic term of the amplitude equation.