Synchronization of a large number of continuous one-dimensional stochastic elements with time delayed mean field coupling

We study synchronization as a means of control of collective behavior of an ensemble of coupled stochastic units in which oscillations are induced merely by external noise. We determine the boundary of the synchronization domain of a large number of onedimensional continuous stochastic elements with time delayed non-homogeneous mean-field coupling. Exact location of the synchronization threshold is shown to be a solution of the boundary value problem (BVP) which was derived from the linearized Fokker-Planck equation. Here the synchronization threshold is found by solving this BVP numerically. Approximate analytics is obtained by expanding the solution of the linearized Fokker-Planck equation into a series of eigenfunctions of the stationary Fokker-Planck operator. Bistable systems with a polynomial and piece-wise linear potential are considered as examples. Multistability and hysteresis is observed in the Langevin equations for finite noise intensity. In the limit of small noise intensities the critical coupling strength was shown to remain finite.