Scaling and singularities in the entrainment of globally-coupled oscillators

Abstract

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 $|{\alpha }_{\infty }|\sim \sqrt{\gamma (\gamma +l^{2}D)}$ where $D$ is the diffusion coefficient and $l$ is the mode number of the unstable mode. The unusual $(\gamma +l^{2}D)$ factor arises from a singularity in the cubic term of the amplitude equation.

11 pages (Revtex); paper submitted to Phys. Rev. Lett

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• Amplitude Expansions for Instabilities in Populations of Globally-Coupled Oscillators
• Amplitude Equations for Electrostatic Waves: universal singular behavior in the limit of weak instability
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