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Synchronization of coupled stochastic systems with multiplicative noise. (English) Zbl 1203.60077

From the authors’: 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 [Stoch. Dyn. 8, No. 1, 139–154 (2008; Zbl 1149.60036)].

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
37H10 Generation, random and stochastic difference and differential equations

Citations:

Zbl 1149.60036
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References:

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