The persistence of synchronization under $\alpha$-stable noise

Abstract

This work is about the synchronization of nonlinear coupled dynamical systems driven by $\alpha$-stable noise. Firstly, we provide a novel technique to construct the relationship between synchronized system and slow-fast system. Secondly, we show that the slow component of original systems converges to the mild solution of the averaging equation under $L^{p}(1<p<\alpha)$ sense. Finally, using the results of averaging principle for stochastic dynamical system with two-time scales, we show that the synchronization effect is persisted provided equilibria are replaced by stationary random solutions.