# 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 & 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 [2] L. Arnold, Random Dynamical Systems Springer-Verlag, Heidelberg, 1998.Google Scholar [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. 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. Caraballo, P. Kloeden and A. Neuenkirch, Synchronization of systems with multiplicative noise, Stoch. Dyn., 8 (2008), 139-154. doi: 10.1142/S0219493708002184. Google Scholar [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. 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. 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. 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. 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. 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

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. 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. 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. Caraballo, P. Kloeden and A. Neuenkirch, Synchronization of systems with multiplicative noise, Stoch. Dyn., 8 (2008), 139-154. doi: 10.1142/S0219493708002184. Google Scholar [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. 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. 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. 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. 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. 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
 [1] Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409 [2] Yongjiang Guo, Yuantao Song. The (functional) law of the iterated logarithm of the sojourn time for a multiclass queue. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-28. doi: 10.3934/jimo.2018192 [3] Piermarco Cannarsa, Vilmos Komornik, Paola Loreti. One-sided and internal controllability of semilinear wave equations with infinitely iterated logarithms. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 745-756. doi: 10.3934/dcds.2002.8.747 [4] Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 581-587. doi: 10.3934/dcds.2004.10.581 [5] Luis Barreira, Davor Dragičević, Claudia Valls. From one-sided dichotomies to two-sided dichotomies. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2817-2844. doi: 10.3934/dcds.2015.35.2817 [6] Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247 [7] Kengo Matsumoto. K-groups of the full group actions on one-sided topological Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3753-3765. doi: 10.3934/dcds.2013.33.3753 [8] Tadahisa Funaki, Yueyuan Gao, Danielle Hilhorst. Convergence of a finite volume scheme for a stochastic conservation law involving a $Q$-brownian motion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1459-1502. doi: 10.3934/dcdsb.2018159 [9] Rogério Martins. One-dimensional attractor for a dissipative system with a cylindrical phase space. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 533-547. doi: 10.3934/dcds.2006.14.533 [10] Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157 [11] Yong Ren, Wensheng Yin, Dongjin Zhu. Exponential stability of SDEs driven by $G$-Brownian motion with delayed impulsive effects: average impulsive interval approach. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3347-3360. doi: 10.3934/dcdsb.2018248 [12] Yong Ren, Huijin Yang, Wensheng Yin. Weighted exponential stability of stochastic coupled systems on networks with delay driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3379-3393. doi: 10.3934/dcdsb.2018325 [13] Yong Ren, Wensheng Yin. Quasi sure exponential stabilization of nonlinear systems via intermittent $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5871-5883. doi: 10.3934/dcdsb.2019110 [14] María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473 [15] Monia Karouf. Reflected solutions of backward doubly SDEs driven by Brownian motion and Poisson random measure. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5571-5601. doi: 10.3934/dcds.2019245 [16] Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015 [17] Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210 [18] Ken Shirakawa, Hiroshi Watanabe. Energy-dissipative solution to a one-dimensional phase field model of grain boundary motion. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 139-159. doi: 10.3934/dcdss.2014.7.139 [19] Hassan Allouba. Brownian-time Brownian motion SIEs on $\mathbb{R}_{+}$ × $\mathbb{R}^d$: Ultra regular direct and lattice-limits solutions and fourth order SPDEs links. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 413-463. doi: 10.3934/dcds.2013.33.413 [20] Fabrice Baudoin, Camille Tardif. Hypocoercive estimates on foliations and velocity spherical Brownian motion. Kinetic & Related Models, 2018, 11 (1) : 1-23. doi: 10.3934/krm.2018001

2018 Impact Factor: 1.008

## Metrics

• PDF downloads (12)
• HTML views (61)
• Cited by (2)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]