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We develop the large deviations principle for synchronized system with small noise. Depending on the interaction between the intensity of the noise with coupling strength, we get different behavior. By simple transformations, the original synchronized system is equivalently converted into the slow-fast system, then we derive the representations for the action functional of the slow variables via weak convergence methods. Therefore, the large deviation properties corresponding to the original synchronized system are derived. In particular, we present a large deviation principle for a particular system in view of Smoluchowski–Kramers arguments and study the synchronization of the quasipotential for a linear system. [ABSTRACT FROM AUTHOR]

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]