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Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions
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 |
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.
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. |
[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. |
[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. 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. |
[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. Caraballo, P. E. Kloeden and A. Neuenkirch,
Synchronization of systems with multiplicative noise, Stochastics and Dynamics, 8 (2008), 139-154.
doi: 10.1142/S0219493708002184. |
[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. |
[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. 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. |
[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.
|
[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. |
[19] |
X. Mao, Stochastic Differential Equations and Applications, 2$^nd$ edition, Horwood, 2008.
doi: 10.1533/9780857099402. |
[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.![]() ![]() |
[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. Strogatz, Sync: 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. 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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. Caraballo, P. E. Kloeden and A. Neuenkirch,
Synchronization of systems with multiplicative noise, Stochastics and Dynamics, 8 (2008), 139-154.
doi: 10.1142/S0219493708002184. |
[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. |
[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. 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. |
[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.
|
[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. |
[19] |
X. Mao, Stochastic Differential Equations and Applications, 2$^nd$ edition, Horwood, 2008.
doi: 10.1533/9780857099402. |
[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.![]() ![]() |
[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. Strogatz, Sync: 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. 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. |
[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. 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. |
[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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