Large Deviations For Synchronized System

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.