Advanced Search
Article Contents
Article Contents

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

  • * Corresponding author: Jicheng Liu

    * Corresponding author: Jicheng Liu
The authors are supported by NSFs of China (No.11271013, 11471340) and the Fundamental Research Funds for the Central Universities, HUST: 2016YXMS003.
Abstract Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 60H10; Secondary: 34F05.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [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.
    [2] L. Arnold, Random Dynamical Systems, Springer Science and Business Media, 2013.
    [3] S. A. AzzawiJ. 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.
    [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.
    [5] T. CaraballoI. 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.
    [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.
    [7] T. CaraballoP. E. Kloeden and A. Neuenkirch, Synchronization of systems with multiplicative noise, Stochastics and Dynamics, 8 (2008), 139-154.  doi: 10.1142/S0219493708002184.
    [8] T. CaraballoP. 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.
    [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.
    [10] J. Duan and W. Wei, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, 2014.
    [11] R. Z. Khasminskii, Stochastic Stability of Differential Equations, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0.
    [12] P. E. Kloeden, Synchronization of nonautonomous dynamical systems, Elect. J. Diff. Eqns., 39 (2003), 1-10. 
    [13] P. E. Kloeden, Nonautonomous attractors of switching systems, Dynamical Systems, 21 (2006), 209-230.  doi: 10.1080/14689360500446262.
    [14] D. Liu, Strong convergence of principle of averaging for multiscale stochastic dynamical systems, Communications in Mathematical Sciences, 8 (2010), 999-1020. 
    [15] X. LiuJ. DuanJ. 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.
    [16] Y. LiuX. Wan and E. Wu, et al., Finite time synchronization of Markovian neural networks with proportional delays and discontinuous activations, Nonlinear Analysis: Modeling and Control, 23 (2018), 515-532. 
    [17] J. LuD. 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. 
    [18] J. LuD. 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.
    [19] X. Mao, Stochastic Differential Equations and Applications, 2$^nd$ edition, Horwood, 2008. doi: 10.1533/9780857099402.
    [20] A. PikovskyM. Rosenblum and  J. KurthsSynchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, 12. Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
    [21] H. M. Rodrigues, Abstract methods for synchronization and applications, Appl. Anal., 62 (1996), 263-296.  doi: 10.1080/00036819608840483.
    [22] B. Schmalfuss, Lyapunov functions and non-trivial stationary solutions of stochastic differential equations, Dynamical Systems, 16 (2001), 303-317.  doi: 10.1080/14689360110069439.
    [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.
    [24] S. StrogatzSync: The Emerging Science of Spontaneous Order, Hyperion Press, New York, 2003. 
    [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.
    [26] X. YangJ. Lu and D. W. C. Ho, et al., Synchronization of uncertain hybrid switching and impulsive complex networks, Applied Mathematical Modelling, 59 (2018), 379-392.  doi: 10.1016/j.apm.2018.01.046.
    [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.
    [28] W. ZhangC. Li and T. Huang, et al., 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.
    [29] C. ZhouW. Zhang and X. Yang, et al., Finite-time synchronization of complex-valued neural networks with mixed delays and uncertain perturbations, Neural Processing Letters, 46 (2017), 271-291. 
  • 加载中

Article Metrics

HTML views(385) PDF downloads(247) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint