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 |α∞| on the linear growth rate γ. In general we find |α∞| ∼ √ γ(γ + l2D) where D is the diffusion coefficient and l is the mode number of the unstable mode. The unusual (γ + l2D) factor arises from a singularity in the cubic term of the amplitude equation. Typeset using REVTEX 1 The onset of collective oscillations is a multi-faceted phenomenon of interest in physics, chemistry, biology and most recently neuroscience. [1–5] One important class of models describes a collection of N dissipative (limit cycle) oscillators that have some weak mutual interaction. For sufficiently weak couplings, the basic form of the uncoupled cycle persists, and the fundamental effect of the interaction is to alter the frequencies and evolving phases of the oscillators. When the coupling between two oscillators, f(θj − θi), is uniform for all oscillator pairs, then the evolution of the phases is given by θ̇i = ωi + K N N