American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations
  • DCDS-B Home
  • This Issue
  • Next Article
    Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions
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

Zhen Li and  Jicheng Liu ,

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.

Keywords: Synchronization, stationary solutions, multi-scale, stochastic differential equations, moment estimates.
Mathematics Subject Classification: Primary: 60H10; Secondary: 34F05.
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

[1]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024
[2]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277
[3]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192
[4]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115
[5]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175
[6]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[7]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027
[8]

Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190
[9]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024
[10]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018
[11]

Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020210
[12]

Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021004
[13]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088
[14]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203
[15]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037
[16]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[17]

Xiaohu Wang, Dingshi Li, Jun Shen. Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2829-2855. doi: 10.3934/dcdsb.2020207
[18]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[19]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199
[20]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (167)
  • HTML views (320)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences