Synchronization and fluctuation of a stochastic coupled systems with additive noise

The synchronization and fluctuation of a stochastic coupled system with additive noise were investigated in this paper. According to the relationship between the stochastic coupled system and multi-scale system, an averaging principle in which the multi-scale system with singular coefficients was established, thereby the synchronization of stochastic coupled systems was obtained. Then the convergence rate of synchronization was also obtained. In addition, to prove fluctuation of multi-scale system, the martingale approach method was used. And then the fluctuation of the stochastic coupled systems was got. In the end, we give an example to illustrate the utility of our results.