POTOTSKY, A. and JANSON, N.B., 2009. Synchronization of a large number of continuous one-dimensional stochastic elements with time delayed mean field coupling. Physica D: Nonlinear Phenomena, 238 (2), pp.175-183.
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.
This article was accepted for publication in the journal Physica D: Nonlinear Phenomena, and the definitive version can be found at: http://dx.doi.org/10.1016/j.physd.2008.09.010