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The onset of collective behavior in a population of globally coupledoscillators with randomly distributed frequencies is studied for phasedynamical models with arbitrary coupling. The population is described by aFokker-Planck equation for the distribution of phases which includes thediffusive effect of noise in the oscillator frequencies. The bifurcation fromthe phase-incoherent state is analyzed using amplitude equations for theunstable modes with particular attention to the dependence of the nonlinearlysaturated mode $|\alpha_\infty|$ on the linear growth rate $\gamma$. In generalwe find $|\alpha_\infty|\sim \sqrt{\gamma(\gamma+l^2D)}$ where $D$ is thediffusion coefficient and $l$ is the mode number of the unstable mode. Theunusual $(\gamma+l^2D)$ factor arises from a singularity in the cubic term ofthe amplitude equation.Comment: 11 pages (Revtex); paper submitted to Phys. Rev. Let

Scaling and Singularities in the Entrainment of Globally Coupled Oscillators