  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.

Adaptation, Noise, and Self-Organizing Systems;Adaptation and Self-Organizing Systems