SCALING AND SINGULARITIES IN THE ENTRAINMENT OF GLOBALLY COUPLED OSCILLATORS

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, and the mode number l, the nonlinearly saturated mode amplitude typically scales like (+l2D). The unusual +l2D factor arises from a singularity in the cubic term of the amplitude equation. © 1995 The American Physical Society.