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 & 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.Google Scholar

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T. CaraballoI. 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. Google Scholar

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T. CaraballoP. Kloeden and A. Neuenkirch, Synchronization of systems with multiplicative noise, Stoch. Dyn., 8 (2008), 139-154. doi: 10.1142/S0219493708002184. Google Scholar

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T. CaraballoP. 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. Google Scholar

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P. Kloeden, Synchronization of nonautonomous dynamical systems, Electron. J. Differential Equations, 39 (2003), 1-10. Google Scholar

[9]

P. KloedenA. Neuenkirch and R. Pavani, Synchronization of noisy dissipative systems under discretization, J. Diff. Equ. Appl., 15 (2009), 785-801. doi: 10.1080/10236190701754222. Google Scholar

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X. Mao, Stochastic Differential Equations and Applications 2nd edition, Horwood Publishing Limited, Chichester, 2008.Google Scholar

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D. Revuz and M. Yor, Continuous Martingales and Brownian Motion 3rd edition, Springer-Verlag, Berlin, 1999.Google Scholar

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X. LiuJ. DuanJ. 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. Google Scholar

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H. Rodrigues, Abstract methods for synchronization and applications, Appl. Anal., 62 (1996), 263-296. doi: 10.1080/00036819608840483. Google Scholar

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S. Strogatz, Sync: The Emerging Science of Spontaneous Order Hyperion Press, New York, 2003.Google Scholar

show all references

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.Google Scholar

[2]

L. Arnold, Random Dynamical Systems Springer-Verlag, Heidelberg, 1998.Google Scholar

[3]

T. CaraballoI. 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. Google Scholar

[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. Google Scholar

[5]

T. CaraballoP. Kloeden and A. Neuenkirch, Synchronization of systems with multiplicative noise, Stoch. Dyn., 8 (2008), 139-154. doi: 10.1142/S0219493708002184. Google Scholar

[6]

T. CaraballoP. 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. Google Scholar

[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. Google Scholar

[8]

P. Kloeden, Synchronization of nonautonomous dynamical systems, Electron. J. Differential Equations, 39 (2003), 1-10. Google Scholar

[9]

P. KloedenA. Neuenkirch and R. Pavani, Synchronization of noisy dissipative systems under discretization, J. Diff. Equ. Appl., 15 (2009), 785-801. doi: 10.1080/10236190701754222. Google Scholar

[10]

X. Mao, Stochastic Differential Equations and Applications 2nd edition, Horwood Publishing Limited, Chichester, 2008.Google Scholar

[11]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion 3rd edition, Springer-Verlag, Berlin, 1999.Google Scholar

[12]

X. LiuJ. DuanJ. 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. Google Scholar

[13]

H. Rodrigues, Abstract methods for synchronization and applications, Appl. Anal., 62 (1996), 263-296. doi: 10.1080/00036819608840483. Google Scholar

[14]

S. Strogatz, Sync: The Emerging Science of Spontaneous Order Hyperion Press, New York, 2003.Google Scholar

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