1997/01/10 by John D. Crawford, Crawford, John David, K. T. R. Davies +1 · 1 citation
Computer Science · Neuroscience · Physics and Astronomy · #Condensed Matter (cond-mat) #FOS: Physical sciences #Neural dynamics and brain function #Nonlinear Dynamics and Pattern Formation #Pattern Formation and Solitons (nlin.PS) #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.patt-sol/9701006
openalex publication_date 1997/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 effect of a stochastic temporal variation in the frequencies is also included. The Fokker-Planck equation for the coupled Langevin system is reduced to a kinetic equation for the oscillator distribution function. Instabilities of the phase-incoherent state are studied by center manifold reduction to the amplitude dynamics of the unstable modes. Depending on the coupling, the coefficients in the normal form can be singular in the limit of weak instability when the diffusive effect of the noise is neglected. A detailed analysis of these singularities to all orders in the normal form expansion is presented. Physically, the singularities are interpreted as predicting an altered scaling of the entrained component near the onset of synchronization.