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February  2022, 27(2): 1029-1054. doi: 10.3934/dcdsb.2021079

Normal deviation of synchronization of stochastic coupled systems

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

* Corresponding author: Meiling Zhao

Received  October 2020 Revised  January 2021 Published  February 2022 Early access  March 2021

Fund Project: The authors are supported by NSFs of China (No.11271013, 11471340, 10901065) and the Fundamental Research Funds for the Central Universities, HUST: 2016YXMS003, 2014TS066

This paper will prove the normal deviation of the synchronization of stochastic coupled system. According to the relationship between the stationary solution and the general solution, the martingale method is used to prove the normal deviation of the fixed initial value of the multi-scale system, thereby obtaining the normal deviation of the stationary solution. At the same time, with the relationship between the synchronized system and the multi-scale system, the normal deviation of the synchronization is obtained.

Citation: Jicheng Liu, Meiling Zhao. Normal deviation of synchronization of stochastic coupled systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1029-1054. doi: 10.3934/dcdsb.2021079
References:
[1]

V. S. AfraimovichS. N. Chow and J. K. Hale, Synchronization in lattices of coupled oscillators, Lattice dynamics (Paris, 1995). Phys. D, 103 (1997), 442-451.  doi: 10.1016/S0167-2789(96)00276-X.

[2]

V. S. Afraimovich and W.-W. Lin, Synchronization in lattices of coupled oscillators with neumann/periodic boundary conditions, Dynam. Stability Systems, 13 (1998), 237-264.  doi: 10.1080/02681119808806263.

[3]

S. Al-AzzawiJ. Liu and X. Liu, Convergence rate of synchronization of systems with additive noise, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 227-245.  doi: 10.3934/dcdsb.2017012.

[4]

L. Arnold, Random Dynamical Systems, Springer monographs in mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[5]

H. Bessaih, M. J. Garrido-Atienza, V. Köpp, B. Schmalfußand M. Yang, Synchronization of stochastic lattice equations, NoDEA Nonlinear Differential Equations Appl., 27 (2020), Paper No. 36, 25 pp. doi: 10.1007/s00030-020-00640-0.

[6]

T. Caraballo and P. E. Kloeden, The persistence of synchronization under environmental noise, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 2257-2267.  doi: 10.1098/rspa.2005.1484.

[7]

T. CaraballoP. E. Kloeden and A. Neuenkirch, Synchronization of systems with multiplicative noise, Stoch. Dyn., 8 (2008), 139-154.  doi: 10.1142/S0219493708002184.

[8]

S. Cerrai, Normal deviations from the averaged motion for some reaction-diffusion equations with fast oscillating perturbation, J. Math. Pures Appl., 91 (2009), 614-647.  doi: 10.1016/j.matpur.2009.04.007.

[9] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, 1996.  doi: 10.1017/CBO9780511662829.
[10]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4684-0176-9.

[11]

H. Kesten and G. C. Papanicolaou, A limit theorem for turbulent diffusion, Comm. Math. Phys., 65 (1979), 97-128.  doi: 10.1007/BF01225144.

[12]

R. Z. Khasminskii, On stochastic processes defined by differential equations with a small parameter, Theory of Probability and Its Applications, 11 (1966), 118-125. 

[13]

Z. Li and J. Liu, Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5709-5736.  doi: 10.3934/dcdsb.2019103.

[14]

Z. Li and J. Liu, Synchronization and averaging principle of stationary solutions for stochastic differential equations, Potential Analysis, 6 (2020).

[15]

D. Liu, Strong convergence of principle of averaging for multiscale stochastic dynamical systems, Commun. Math. Sci., 8 (2010), 999-1020.  doi: 10.4310/CMS.2010.v8.n4.a11.

[16]

B. Øksendal, Stochastic Differential Equations, sixth ed., Springer, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.

[17]

M. Rosenblatt, A central limit theorem and a strong mixing condition, Proc. Nat. Acad. Sci. U.S.A., 42 (1956), 43-47.  doi: 10.1073/pnas.42.1.43.

[18]

B. Schmalfuss and K. R. Schneider, Invariant manifolds for random dynamical systems with slow and fast variables, J. Dynam. Differential Equations, 20 (2008), 133-164.  doi: 10.1007/s10884-007-9089-7.

[19]

Z. ShenS. Zhou and X. Han, Synchronization of coupled stochastic systems with multiplicative noise, Stoch. Dyn., 10 (2010), 407-428.  doi: 10.1142/S0219493710003029.

[20]

A. N. Shiryaev, Probability, second ed., Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-2539-1.

[21]

S. R. S. Varadhan, Stochastic Processes. Courant Lecture Notes in Mathematics, 16., Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2007. doi: 10.1090/cln/016.

[22]

V. A. Volkonski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$ and Yu. A. Rozanov, Some limit theorems for random functions, Theor. Probability Appl., 4 (1959), 178-197.  doi: 10.1137/1104015.

[23]

W. Wang and A. J. Roberts, Average and deviation for slow-fast stochastic partial differential equations, J. Differential Equations, 253 (2012), 1265-1286.  doi: 10.1016/j.jde.2012.05.011.

show all references

References:
[1]

V. S. AfraimovichS. N. Chow and J. K. Hale, Synchronization in lattices of coupled oscillators, Lattice dynamics (Paris, 1995). Phys. D, 103 (1997), 442-451.  doi: 10.1016/S0167-2789(96)00276-X.

[2]

V. S. Afraimovich and W.-W. Lin, Synchronization in lattices of coupled oscillators with neumann/periodic boundary conditions, Dynam. Stability Systems, 13 (1998), 237-264.  doi: 10.1080/02681119808806263.

[3]

S. Al-AzzawiJ. Liu and X. Liu, Convergence rate of synchronization of systems with additive noise, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 227-245.  doi: 10.3934/dcdsb.2017012.

[4]

L. Arnold, Random Dynamical Systems, Springer monographs in mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[5]

H. Bessaih, M. J. Garrido-Atienza, V. Köpp, B. Schmalfußand M. Yang, Synchronization of stochastic lattice equations, NoDEA Nonlinear Differential Equations Appl., 27 (2020), Paper No. 36, 25 pp. doi: 10.1007/s00030-020-00640-0.

[6]

T. Caraballo and P. E. Kloeden, The persistence of synchronization under environmental noise, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 2257-2267.  doi: 10.1098/rspa.2005.1484.

[7]

T. CaraballoP. E. Kloeden and A. Neuenkirch, Synchronization of systems with multiplicative noise, Stoch. Dyn., 8 (2008), 139-154.  doi: 10.1142/S0219493708002184.

[8]

S. Cerrai, Normal deviations from the averaged motion for some reaction-diffusion equations with fast oscillating perturbation, J. Math. Pures Appl., 91 (2009), 614-647.  doi: 10.1016/j.matpur.2009.04.007.

[9] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, 1996.  doi: 10.1017/CBO9780511662829.
[10]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4684-0176-9.

[11]

H. Kesten and G. C. Papanicolaou, A limit theorem for turbulent diffusion, Comm. Math. Phys., 65 (1979), 97-128.  doi: 10.1007/BF01225144.

[12]

R. Z. Khasminskii, On stochastic processes defined by differential equations with a small parameter, Theory of Probability and Its Applications, 11 (1966), 118-125. 

[13]

Z. Li and J. Liu, Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5709-5736.  doi: 10.3934/dcdsb.2019103.

[14]

Z. Li and J. Liu, Synchronization and averaging principle of stationary solutions for stochastic differential equations, Potential Analysis, 6 (2020).

[15]

D. Liu, Strong convergence of principle of averaging for multiscale stochastic dynamical systems, Commun. Math. Sci., 8 (2010), 999-1020.  doi: 10.4310/CMS.2010.v8.n4.a11.

[16]

B. Øksendal, Stochastic Differential Equations, sixth ed., Springer, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.

[17]

M. Rosenblatt, A central limit theorem and a strong mixing condition, Proc. Nat. Acad. Sci. U.S.A., 42 (1956), 43-47.  doi: 10.1073/pnas.42.1.43.

[18]

B. Schmalfuss and K. R. Schneider, Invariant manifolds for random dynamical systems with slow and fast variables, J. Dynam. Differential Equations, 20 (2008), 133-164.  doi: 10.1007/s10884-007-9089-7.

[19]

Z. ShenS. Zhou and X. Han, Synchronization of coupled stochastic systems with multiplicative noise, Stoch. Dyn., 10 (2010), 407-428.  doi: 10.1142/S0219493710003029.

[20]

A. N. Shiryaev, Probability, second ed., Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-2539-1.

[21]

S. R. S. Varadhan, Stochastic Processes. Courant Lecture Notes in Mathematics, 16., Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2007. doi: 10.1090/cln/016.

[22]

V. A. Volkonski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$ and Yu. A. Rozanov, Some limit theorems for random functions, Theor. Probability Appl., 4 (1959), 178-197.  doi: 10.1137/1104015.

[23]

W. Wang and A. J. Roberts, Average and deviation for slow-fast stochastic partial differential equations, J. Differential Equations, 253 (2012), 1265-1286.  doi: 10.1016/j.jde.2012.05.011.

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