Synchronization of coupled stochastic systems with multiplicative noise

We consider the synchronization of solutions to coupled systems of the conjugate random ordinary differential equations (RODEs) for the $N$-Stratronovich stochastic ordinary differential equations (SODEs) with linear multiplicative noise ($N\in \mathbb{N}$). We consider the synchronization between two solutions and among different components of solutions under one-sided dissipative Lipschitz conditions. We first show that the random dynamical system generated by the solution of the coupled RODEs has a singleton sets random attractor which implies the synchronization of any two solutions. Moreover, the singleton sets random attractor determines a stationary stochastic solution of the equivalently coupled SODEs. Then we show that any solution of the RODEs converge to a solution of the averaged RODE within any finite time interval as the coupled coefficient tends to infinity. Our results generalize the work of two Stratronovich SODEs in \cite{9}.