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October  2019, 24(10): 5709-5736. doi: 10.3934/dcdsb.2019103

## Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense

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

* Corresponding author: Jicheng Liu

Received  May 2018 Revised  September 2018 Published  June 2019

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

We provide a more clear technique to deal with general synchronization problems for SDEs, where the multiplicative noise appears nonlinearly. Moreover, convergence rate of synchronization is obtained. A new method employed here is the techniques of moment estimates for general solutions based on the transformation of multi-scales equations. As a by-product, the relationship between general solutions and stationary solutions is constructed.

Citation: Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5709-5736. doi: 10.3934/dcdsb.2019103
##### References:
 [1] V. S. Afraimovich and H. M. Rodrigues, Uniform dissipativeness and synchronization of nonautonomous equations, International Conference on Differential Equations (Lisboa 1995), (1998), 3–17.  Google Scholar [2] L. Arnold, Random Dynamical Systems, Springer Science and Business Media, 2013. Google Scholar [3] S. A. Azzawi, J. Liu and X. Liu, Convergence rate of synchronization of systems with additive noise, Discrete and Continuous Dynamical Systems-Series B, 22 (2017), 227-245.  doi: 10.3934/dcdsb.2017012.  Google Scholar [4] S. A. Azzawi, J. Liu and X. Liu, The synchronization of stochastic differential equations with linear noise, Stochastics and Dynamics, 18 (2018), 1850049, 31pp. doi: 10.1142/S0219493718500491.  Google Scholar [5] T. Caraballo, I. D. Chueshov and P. E. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM Journal on Mathematical Analysis, 38 (2007), 1489-1507.  doi: 10.1137/050647281.  Google Scholar [6] T. Caraballo and P. E. Kloeden, The persistence of synchronization under environmental noise, Proceedings of the Royal Society of London A, 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, Stochastics and Dynamics, 8 (2008), 139-154.  doi: 10.1142/S0219493708002184.  Google Scholar [8] T. Caraballo, P. E. Kloeden and B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their pertubation, Applied Mathematics and Optimization, 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1.  Google Scholar [9] 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 [10] J. Duan and W. Wei, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, 2014.  Google Scholar [11] R. Z. Khasminskii, Stochastic Stability of Differential Equations, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0.  Google Scholar [12] P. E. Kloeden, Synchronization of nonautonomous dynamical systems, Elect. J. Diff. Eqns., 39 (2003), 1-10.   Google Scholar [13] P. E. Kloeden, Nonautonomous attractors of switching systems, Dynamical Systems, 21 (2006), 209-230.  doi: 10.1080/14689360500446262.  Google Scholar [14] D. Liu, Strong convergence of principle of averaging for multiscale stochastic dynamical systems, Communications in Mathematical Sciences, 8 (2010), 999-1020.   Google Scholar [15] X. Liu, J. Duan, J. Liu and P. E. Kloeden, Synchronization of systems of Marcus canonical equations driven by $\alpha$-stable noises, Nonlinear Anal: Real World Appl., 11 (2010), 3437-3445.  doi: 10.1016/j.nonrwa.2009.12.004.  Google Scholar [16] Y. Liu, X. Wan and E. Wu, Finite time synchronization of Markovian neural networks with proportional delays and discontinuous activations, Nonlinear Analysis: Modeling and Control, 23 (2018), 515-532.   Google Scholar [17] J. Lu, D. W. C. Ho and Z. Wang, Pinning stabilization of linearly coupled stochastic neural networks via minimum number of controllers, IEEE Transactions on Neural Networks, 20 (2009), 1617-1629.   Google Scholar [18] J. Lu, D. W. C. Ho and L. Wu, Exponential stabilization of switched stochastic dynamical networks, Nonlinearity, 22 (2009), 889-911.  doi: 10.1088/0951-7715/22/4/011.  Google Scholar [19] X. Mao, Stochastic Differential Equations and Applications, 2$^nd$ edition, Horwood, 2008. doi: 10.1533/9780857099402.  Google Scholar [20] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, 12. Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.  Google Scholar [21] H. M. Rodrigues, Abstract methods for synchronization and applications, Appl. Anal., 62 (1996), 263-296.  doi: 10.1080/00036819608840483.  Google Scholar [22] B. Schmalfuss, Lyapunov functions and non-trivial stationary solutions of stochastic differential equations, Dynamical Systems, 16 (2001), 303-317.  doi: 10.1080/14689360110069439.  Google Scholar [23] B. Schmalfuss and R. Schneider, Invariant manifolds for random dynamical systems with slow and fast variables, Journal of Dynamics and Differential Equations, 20 (2008), 133-164.  doi: 10.1007/s10884-007-9089-7.  Google Scholar [24] S. Strogatz, Sync: The Emerging Science of Spontaneous Order, Hyperion Press, New York, 2003.   Google Scholar [25] T. Su and X. Yang, Finite-time synchronization of competitive neural networks with mixed delays, Discrete and Continuous Dynamical System-B, 21 (2016), 3655-3667.  doi: 10.3934/dcdsb.2016115.  Google Scholar [26] X. Yang, J. Lu and D. W. C. Ho, Synchronization of uncertain hybrid switching and impulsive complex networks, Applied Mathematical Modelling, 59 (2018), 379-392.  doi: 10.1016/j.apm.2018.01.046.  Google Scholar [27] W. Zhang, X. Yang, C. Li, Fixed-time stochastic synchronization of complex networks via continuous control, IEEE Transactions on Cybernetics, 2018, 1–6. doi: 10.1109/TCYB.2018.2839109.  Google Scholar [28] W. Zhang, C. Li and T. Huang, Fixed-time synchronization of complex networks with nonidentical nodes and stochastic noise perturbations, Physica A: Statistical Mechanics and its Applications, 492 (2018), 1531-1542.  doi: 10.1016/j.physa.2017.11.079.  Google Scholar [29] C. Zhou, W. Zhang and X. Yang, Finite-time synchronization of complex-valued neural networks with mixed delays and uncertain perturbations, Neural Processing Letters, 46 (2017), 271-291.   Google Scholar

show all references

##### References:
 [1] V. S. Afraimovich and H. M. Rodrigues, Uniform dissipativeness and synchronization of nonautonomous equations, International Conference on Differential Equations (Lisboa 1995), (1998), 3–17.  Google Scholar [2] L. Arnold, Random Dynamical Systems, Springer Science and Business Media, 2013. Google Scholar [3] S. A. Azzawi, J. Liu and X. Liu, Convergence rate of synchronization of systems with additive noise, Discrete and Continuous Dynamical Systems-Series B, 22 (2017), 227-245.  doi: 10.3934/dcdsb.2017012.  Google Scholar [4] S. A. Azzawi, J. Liu and X. Liu, The synchronization of stochastic differential equations with linear noise, Stochastics and Dynamics, 18 (2018), 1850049, 31pp. doi: 10.1142/S0219493718500491.  Google Scholar [5] T. Caraballo, I. D. Chueshov and P. E. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM Journal on Mathematical Analysis, 38 (2007), 1489-1507.  doi: 10.1137/050647281.  Google Scholar [6] T. Caraballo and P. E. Kloeden, The persistence of synchronization under environmental noise, Proceedings of the Royal Society of London A, 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, Stochastics and Dynamics, 8 (2008), 139-154.  doi: 10.1142/S0219493708002184.  Google Scholar [8] T. Caraballo, P. E. Kloeden and B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their pertubation, Applied Mathematics and Optimization, 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1.  Google Scholar [9] 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 [10] J. Duan and W. Wei, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, 2014.  Google Scholar [11] R. Z. Khasminskii, Stochastic Stability of Differential Equations, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0.  Google Scholar [12] P. E. Kloeden, Synchronization of nonautonomous dynamical systems, Elect. J. Diff. Eqns., 39 (2003), 1-10.   Google Scholar [13] P. E. Kloeden, Nonautonomous attractors of switching systems, Dynamical Systems, 21 (2006), 209-230.  doi: 10.1080/14689360500446262.  Google Scholar [14] D. Liu, Strong convergence of principle of averaging for multiscale stochastic dynamical systems, Communications in Mathematical Sciences, 8 (2010), 999-1020.   Google Scholar [15] X. Liu, J. Duan, J. Liu and P. E. Kloeden, Synchronization of systems of Marcus canonical equations driven by $\alpha$-stable noises, Nonlinear Anal: Real World Appl., 11 (2010), 3437-3445.  doi: 10.1016/j.nonrwa.2009.12.004.  Google Scholar [16] Y. Liu, X. Wan and E. Wu, Finite time synchronization of Markovian neural networks with proportional delays and discontinuous activations, Nonlinear Analysis: Modeling and Control, 23 (2018), 515-532.   Google Scholar [17] J. Lu, D. W. C. Ho and Z. Wang, Pinning stabilization of linearly coupled stochastic neural networks via minimum number of controllers, IEEE Transactions on Neural Networks, 20 (2009), 1617-1629.   Google Scholar [18] J. Lu, D. W. C. Ho and L. Wu, Exponential stabilization of switched stochastic dynamical networks, Nonlinearity, 22 (2009), 889-911.  doi: 10.1088/0951-7715/22/4/011.  Google Scholar [19] X. Mao, Stochastic Differential Equations and Applications, 2$^nd$ edition, Horwood, 2008. doi: 10.1533/9780857099402.  Google Scholar [20] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, 12. Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.  Google Scholar [21] H. M. Rodrigues, Abstract methods for synchronization and applications, Appl. Anal., 62 (1996), 263-296.  doi: 10.1080/00036819608840483.  Google Scholar [22] B. Schmalfuss, Lyapunov functions and non-trivial stationary solutions of stochastic differential equations, Dynamical Systems, 16 (2001), 303-317.  doi: 10.1080/14689360110069439.  Google Scholar [23] B. Schmalfuss and R. Schneider, Invariant manifolds for random dynamical systems with slow and fast variables, Journal of Dynamics and Differential Equations, 20 (2008), 133-164.  doi: 10.1007/s10884-007-9089-7.  Google Scholar [24] S. Strogatz, Sync: The Emerging Science of Spontaneous Order, Hyperion Press, New York, 2003.   Google Scholar [25] T. Su and X. Yang, Finite-time synchronization of competitive neural networks with mixed delays, Discrete and Continuous Dynamical System-B, 21 (2016), 3655-3667.  doi: 10.3934/dcdsb.2016115.  Google Scholar [26] X. Yang, J. Lu and D. W. C. Ho, Synchronization of uncertain hybrid switching and impulsive complex networks, Applied Mathematical Modelling, 59 (2018), 379-392.  doi: 10.1016/j.apm.2018.01.046.  Google Scholar [27] W. Zhang, X. Yang, C. Li, Fixed-time stochastic synchronization of complex networks via continuous control, IEEE Transactions on Cybernetics, 2018, 1–6. doi: 10.1109/TCYB.2018.2839109.  Google Scholar [28] W. Zhang, C. Li and T. Huang, Fixed-time synchronization of complex networks with nonidentical nodes and stochastic noise perturbations, Physica A: Statistical Mechanics and its Applications, 492 (2018), 1531-1542.  doi: 10.1016/j.physa.2017.11.079.  Google Scholar [29] C. Zhou, W. Zhang and X. Yang, Finite-time synchronization of complex-valued neural networks with mixed delays and uncertain perturbations, Neural Processing Letters, 46 (2017), 271-291.   Google Scholar
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