# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021079
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## Normal deviation of synchronization of stochastic coupled systems

 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

* Corresponding author: Meiling Zhao

Received  October 2020 Revised  January 2021 Early access March 2021

Fund Project: The authors are supported by NSFs of China (No.11271013, 11471340, 10901065) and the Fundamental Research Funds for the Central Universities, HUST: 2016YXMS003, 2014TS066

This paper will prove the normal deviation of the synchronization of stochastic coupled system. According to the relationship between the stationary solution and the general solution, the martingale method is used to prove the normal deviation of the fixed initial value of the multi-scale system, thereby obtaining the normal deviation of the stationary solution. At the same time, with the relationship between the synchronized system and the multi-scale system, the normal deviation of the synchronization is obtained.

Citation: Jicheng Liu, Meiling Zhao. Normal deviation of synchronization of stochastic coupled systems. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021079
##### References:
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##### References:
 [1] V. S. Afraimovich, S. N. Chow and J. K. Hale, Synchronization in lattices of coupled oscillators, Lattice dynamics (Paris, 1995). Phys. D, 103 (1997), 442-451.  doi: 10.1016/S0167-2789(96)00276-X.  Google Scholar [2] V. S. Afraimovich and W.-W. Lin, Synchronization in lattices of coupled oscillators with neumann/periodic boundary conditions, Dynam. Stability Systems, 13 (1998), 237-264.  doi: 10.1080/02681119808806263.  Google Scholar [3] S. Al-Azzawi, J. Liu and X. Liu, Convergence rate of synchronization of systems with additive noise, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 227-245.  doi: 10.3934/dcdsb.2017012.  Google Scholar [4] L. Arnold, Random Dynamical Systems, Springer monographs in mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar [5] H. Bessaih, M. J. Garrido-Atienza, V. Köpp, B. Schmalfußand M. Yang, Synchronization of stochastic lattice equations, NoDEA Nonlinear Differential Equations Appl., 27 (2020), Paper No. 36, 25 pp. doi: 10.1007/s00030-020-00640-0.  Google Scholar [6] T. Caraballo and P. E. Kloeden, The persistence of synchronization under environmental noise, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 2257-2267.  doi: 10.1098/rspa.2005.1484.  Google Scholar [7] T. Caraballo, P. E. Kloeden and A. Neuenkirch, Synchronization of systems with multiplicative noise, Stoch. Dyn., 8 (2008), 139-154.  doi: 10.1142/S0219493708002184.  Google Scholar [8] S. Cerrai, Normal deviations from the averaged motion for some reaction-diffusion equations with fast oscillating perturbation, J. Math. Pures Appl., 91 (2009), 614-647.  doi: 10.1016/j.matpur.2009.04.007.  Google Scholar [9] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, 1996.  doi: 10.1017/CBO9780511662829.  Google Scholar [10] M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4684-0176-9.  Google Scholar [11] H. Kesten and G. C. Papanicolaou, A limit theorem for turbulent diffusion, Comm. Math. Phys., 65 (1979), 97-128.  doi: 10.1007/BF01225144.  Google Scholar [12] R. Z. Khasminskii, On stochastic processes defined by differential equations with a small parameter, Theory of Probability and Its Applications, 11 (1966), 118-125.   Google Scholar [13] Z. Li and J. Liu, Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5709-5736.  doi: 10.3934/dcdsb.2019103.  Google Scholar [14] Z. Li and J. Liu, Synchronization and averaging principle of stationary solutions for stochastic differential equations, Potential Analysis, 6 (2020). Google Scholar [15] D. Liu, Strong convergence of principle of averaging for multiscale stochastic dynamical systems, Commun. Math. Sci., 8 (2010), 999-1020.  doi: 10.4310/CMS.2010.v8.n4.a11.  Google Scholar [16] B. Øksendal, Stochastic Differential Equations, sixth ed., Springer, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar [17] M. Rosenblatt, A central limit theorem and a strong mixing condition, Proc. Nat. Acad. Sci. U.S.A., 42 (1956), 43-47.  doi: 10.1073/pnas.42.1.43.  Google Scholar [18] B. Schmalfuss and K. R. Schneider, Invariant manifolds for random dynamical systems with slow and fast variables, J. Dynam. Differential Equations, 20 (2008), 133-164.  doi: 10.1007/s10884-007-9089-7.  Google Scholar [19] Z. Shen, S. Zhou and X. Han, Synchronization of coupled stochastic systems with multiplicative noise, Stoch. Dyn., 10 (2010), 407-428.  doi: 10.1142/S0219493710003029.  Google Scholar [20] A. N. Shiryaev, Probability, second ed., Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-2539-1.  Google Scholar [21] S. R. S. Varadhan, Stochastic Processes. Courant Lecture Notes in Mathematics, 16., Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2007. doi: 10.1090/cln/016.  Google Scholar [22] V. A. Volkonski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$ and Yu. A. Rozanov, Some limit theorems for random functions, Theor. Probability Appl., 4 (1959), 178-197.  doi: 10.1137/1104015.  Google Scholar [23] W. Wang and A. J. Roberts, Average and deviation for slow-fast stochastic partial differential equations, J. Differential Equations, 253 (2012), 1265-1286.  doi: 10.1016/j.jde.2012.05.011.  Google Scholar
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