# American Institute of Mathematical Sciences

March  2017, 22(2): 227-245. doi: 10.3934/dcdsb.2017012

## Convergence rate of synchronization of systems with additive noise

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

* Corresponding author: Jicheng Liu

Received  January 2016 Revised  September 2016 Published  December 2016

Fund Project: The first author is supported by NSF grants of China Nos. 11271013 and 11471340.

The synchronization of stochastic differential equations (SDEs) with additive noise is investigated in pathwise sense, moreover convergence rate of synchronization is obtained. The optimality of the convergence rate is illustrated through examples.

Citation: Shahad Al-azzawi, Jicheng Liu, Xianming Liu. Convergence rate of synchronization of systems with additive noise. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 227-245. doi: 10.3934/dcdsb.2017012
##### References:
 [1] V. Afraimovich and H. Rodrigues, Uniform dissipativeness and synchronization of nonautonomous equations, in international Conference on Differential Equations (Lisboa 1995), 3-17, World Sci. Publ. , River Edge, NJ, 1998. [2] L. Arnold, Random Dynamical Systems Springer-Verlag, Heidelberg, 1998. [3] T. Caraballo, I. Chueshov and P. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal., 38 (2007), 1489-1507.  doi: 10.1137/050647281. [4] T. Caraballo and P. Kloeden, The persistence of synchronization under environmental noise, Proc. Roy. Soc. Lond. A, 461 (2005), 2257-2267.  doi: 10.1098/rspa.2005.1484. [5] T. Caraballo, P. Kloeden and A. Neuenkirch, Synchronization of systems with multiplicative noise, Stoch. Dyn., 8 (2008), 139-154.  doi: 10.1142/S0219493708002184. [6] T. Caraballo, P. Kloeden and B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their pertubation, Appl. Math. Optim., 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1. [7] G. Dimitroff and M. Scheutzow, Attractors and expansion for Brownian flows, Electron. J. Probab., 16 (2011), 1193-1213.  doi: 10.1214/EJP.v16-894. [8] P. Kloeden, Synchronization of nonautonomous dynamical systems, Electron. J. Differential Equations, 39 (2003), 1-10. [9] P. Kloeden, A. Neuenkirch and R. Pavani, Synchronization of noisy dissipative systems under discretization, J. Diff. Equ. Appl., 15 (2009), 785-801.  doi: 10.1080/10236190701754222. [10] X. Mao, Stochastic Differential Equations and Applications 2nd edition, Horwood Publishing Limited, Chichester, 2008. [11] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion 3rd edition, Springer-Verlag, Berlin, 1999. [12] X. Liu, J. Duan, J. Liu and P. E. Kloeden, Synchronization of systems of Marcus canonical equations driven by α-stable noises, Nonlinear Anal. Real World Appl., 11 (2010), 3437-3445.  doi: 10.1016/j.nonrwa.2009.12.004. [13] H. Rodrigues, Abstract methods for synchronization and applications, Appl. Anal., 62 (1996), 263-296.  doi: 10.1080/00036819608840483. [14] S. Strogatz, Sync: The Emerging Science of Spontaneous Order Hyperion Press, New York, 2003.

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##### References:
 [1] V. Afraimovich and H. Rodrigues, Uniform dissipativeness and synchronization of nonautonomous equations, in international Conference on Differential Equations (Lisboa 1995), 3-17, World Sci. Publ. , River Edge, NJ, 1998. [2] L. Arnold, Random Dynamical Systems Springer-Verlag, Heidelberg, 1998. [3] T. Caraballo, I. Chueshov and P. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal., 38 (2007), 1489-1507.  doi: 10.1137/050647281. [4] T. Caraballo and P. Kloeden, The persistence of synchronization under environmental noise, Proc. Roy. Soc. Lond. A, 461 (2005), 2257-2267.  doi: 10.1098/rspa.2005.1484. [5] T. Caraballo, P. Kloeden and A. Neuenkirch, Synchronization of systems with multiplicative noise, Stoch. Dyn., 8 (2008), 139-154.  doi: 10.1142/S0219493708002184. [6] T. Caraballo, P. Kloeden and B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their pertubation, Appl. Math. Optim., 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1. [7] G. Dimitroff and M. Scheutzow, Attractors and expansion for Brownian flows, Electron. J. Probab., 16 (2011), 1193-1213.  doi: 10.1214/EJP.v16-894. [8] P. Kloeden, Synchronization of nonautonomous dynamical systems, Electron. J. Differential Equations, 39 (2003), 1-10. [9] P. Kloeden, A. Neuenkirch and R. Pavani, Synchronization of noisy dissipative systems under discretization, J. Diff. Equ. Appl., 15 (2009), 785-801.  doi: 10.1080/10236190701754222. [10] X. Mao, Stochastic Differential Equations and Applications 2nd edition, Horwood Publishing Limited, Chichester, 2008. [11] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion 3rd edition, Springer-Verlag, Berlin, 1999. [12] X. Liu, J. Duan, J. Liu and P. E. Kloeden, Synchronization of systems of Marcus canonical equations driven by α-stable noises, Nonlinear Anal. Real World Appl., 11 (2010), 3437-3445.  doi: 10.1016/j.nonrwa.2009.12.004. [13] H. Rodrigues, Abstract methods for synchronization and applications, Appl. Anal., 62 (1996), 263-296.  doi: 10.1080/00036819608840483. [14] S. Strogatz, Sync: The Emerging Science of Spontaneous Order Hyperion Press, New York, 2003.
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