Scaling and Singularities in the Entrainment of Globally Coupled Oscillators

Abstract

The onset of collective behavior among oscillators with random frequencies is studied for globally coupled phase dynamical models. A Fokker-Planck equation for the phase distribution describes the dynamics including diffusion due to the noise in the frequencies. We analyze instabilities of the phase-incoherent state using amplitude equations for the unstable modes. In terms of the diffusion coefficient $D$, the linear growth rate γ, and the mode number $l$, the nonlinearly saturated mode amplitude typically scales like $|{\alpha }_{\infty }|\sim \sqrt{\gamma \left(\gamma {+l}^{2}D\right)}$. The unusual $\gamma {+l}^{2}D$ factor arises from a singularity in the cubic term of the amplitude equation.