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In this paper, we mainly construct a connection between synchronized systems and multi-scale equations, and then use the averaging principle as an intermediate step to obtain synchronization. This strategy solves the synchronization problem of dissipative stochastic differential equations, regardless of the structure of the noise. Moreover, the averaging principle of stationary solutions is also investigated, which is different from the averaging principle of solutions with the fixed initial values. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the averaging principle for a two-time-scale system of jump-diffusion stochastic differential equations. Under suitable conditions, we expand the weak error in powers of timescale parameter. We prove that the rate of weak convergence to the averaged dynamics is of order 1. This reveals that the rate of weak convergence is essentially twice that of strong convergence. [ABSTRACT FROM AUTHOR]