American Institute of Mathematical Sciences

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January  2017, 37(1): 575-590. doi: 10.3934/dcds.2017023

The attractors for 2nd-order stochastic delay lattice systems

Chengjian Zhang 1, and  Lu Zhao 1,2,

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
2. 

Department of Mathematics, Henan Normal University, Xinxiang 453007, China

Received  February 2016 Revised  September 2016 Published  November 2016

Fund Project: This work is supported by NSFC (Grant Nos. 11571128,11601133).

This paper deals with the long-time dynamical behavior of a classof 2nd-order stochastic delay lattice systems. It is shown under thedissipative and sublinear growth conditions that such a systempossesses a compact global random attractor within the set oftempered random bounded sets. A numerical example is given toillustrate the obtained theoretical result.

Keywords: Random attractor, long-time dynamics, stochastic delay lattice system, additive white noise, numerical simulation.
Mathematics Subject Classification: Primary:34K31, 35B40;Secondary:60H15, 35B4.
Citation: Chengjian Zhang, Lu Zhao. The attractors for 2nd-order stochastic delay lattice systems. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 575-590. doi: 10.3934/dcds.2017023
References:
[1]

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

[2]

J. Ban, Ch. Hsu, Y. Lin and T. Yang, Pullback and forward attractors for dissipative lattice dynamical systems with additive noises, Dyn. Syst., 24 (2009), 139-155.  doi: 10.1080/14689360802508777.  Google Scholar

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T. Caraballo, P.E. Kloeden and J. Real, Pullback and forward attractors for a damped wave equation with delays, Stoch. Dyn., 4 (2004), 405-423.  doi: 10.1142/S0219493704001139.  Google Scholar

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T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335.  doi: 10.1007/s11464-008-0028-7.  Google Scholar

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T. Caraballo, F. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-lipschitz nonlinearity, J. Differ. Equ. Appl., 17 (2011), 161-184.  doi: 10.1080/10236198.2010.549010.  Google Scholar

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T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-lipschitz nonlinearities, J. Differ. Equ., 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.  Google Scholar

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T. Caraballo, F. Morillas and J. Valero, On differential equations with delay in banach spaces and attractors for retarded lattice dynamical systems, Discrete Cont. Dyn. Syst. Ser. A, 34 (2014), 51-77.  doi: 10.3934/dcds.2014.34.51.  Google Scholar

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T. Caraballo and J. Real, Attractors for 2d-Navier-Stokes models with delays, J. Differ. Equ., 205 (2004), 271-297.  doi: 10.1016/j.jde.2004.04.012.  Google Scholar

[11]

X. Ding and J. Jiang, Random attractors for stochastic retarded lattice dynamical systems, Abstr. Appl. Analy. , 2012 (2012), Art. ID 409282, 27 pp.  Google Scholar

[12]

X. Han, Random attractors for stochastic sine-gordon lattice systems with multiplicative white noise, J. Math. Anal. Appl., 376 (2011), 481-493.  doi: 10.1016/j.jmaa.2010.11.032.  Google Scholar

[13]

X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differ. Equ., 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.  Google Scholar

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X. Han, Random attractors for second order stochastic lattice dynamical systems with multiplicative noise in weighted spaces Stoch. Dyn., 12(2012), 1150024, 20pp. doi: 10.1142/S0219493711500249.  Google Scholar

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D. Li, C. Zhang and W. Wang, Long time behavior of non-Fickian delay reaction-diffusion equations, Nonlinear Anal.: RWA, 13 (2012), 1401-1415.  doi: 10.1016/j.nonrwa.2011.11.005.  Google Scholar

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[19]

B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121-136.  doi: 10.1016/j.jmaa.2006.08.070.  Google Scholar

[20]

X. Wang, S. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal.: TMA, 72 (2010), 483-494.  doi: 10.1016/j.na.2009.06.094.  Google Scholar

[21]

L. Xu and W. Yan, Stochastic Fitzhugh-Nagumo systems with delay, Taiwan. J. Math., 16 (2012), 1079-1103.   Google Scholar

[22]

W. Yan, Y. Li and S. Ji, Random attractors for first order stochastic retarded lattice dynamical systems J. Math. Phys., 51(2010), 032702, 17pp. doi: 10.1063/1.3319566.  Google Scholar

[23]

C. Zhang and S. Li, Dissipativity and exponentially asymptotic stability of the solutions for nonlinear neutral functional-differential equations, Appl. Math. comput., 119 (2001), 109-115.  doi: 10.1016/S0096-3003(99)00264-7.  Google Scholar

[24]

L. Zhao, C. Zhang and D. Li, Global attractor for a class of retarded lattice dynamical systems, J. Math. Anal. Appl., 425 (2015), 178-193.  doi: 10.1016/j.jmaa.2014.12.026.  Google Scholar

[25]

C. Zhao and S. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl., 354 (2009), 78-95.  doi: 10.1016/j.jmaa.2008.12.036.  Google Scholar

[26]

C. Zhao and S. Zhou, Attractors of retarded first order lattice systems, Nonlinearity, 20 (2007), 1987-2006.  doi: 10.1088/0951-7715/20/8/010.  Google Scholar

[27]

C. Zhao and S. Zhou, Compact uniform attractors for dissipative lattice dynamical systems with delays, Discrete Cont. Dyn. Syst. Ser. A, 21 (2008), 643-663.  doi: 10.3934/dcds.2008.21.643.  Google Scholar

[28]

X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 763-785.  doi: 10.3934/dcdsb.2008.9.763.  Google Scholar

[29]

C. Zhao, S. Zhou and W. Wang, Compact kernel sections for lattice systems with delays, Nonlinear Anal.: TMA, 70 (2009), 1330-1348.  doi: 10.1016/j.na.2008.02.015.  Google Scholar

[30]

S. Zhou, Attractors for second order lattice dynamical systems, J. Differ. Equ., 179 (2002), 605-624.  doi: 10.1006/jdeq.2001.4032.  Google Scholar

[31]

S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, J. Differ. Equ., 224 (2006), 172-204.  doi: 10.1016/j.jde.2005.06.024.  Google Scholar

[32]

S. Zhou, C. Zhao and X. Liao, Compact uniform attractors for dissipative non-autonomous lattice dynamical systems, Commun. Pur. Appl. Anal., 6 (2007), 1087-1111.  doi: 10.3934/cpaa.2007.6.1087.  Google Scholar

show all references

References:
[1]

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

[2]

J. Ban, Ch. Hsu, Y. Lin and T. Yang, Pullback and forward attractors for dissipative lattice dynamical systems with additive noises, Dyn. Syst., 24 (2009), 139-155.  doi: 10.1080/14689360802508777.  Google Scholar

[3]

P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.  Google Scholar

[4]

P.W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Int. J. Bifurcat. Chaos, 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.  Google Scholar

[5]

T. Caraballo, P.E. Kloeden and J. Real, Pullback and forward attractors for a damped wave equation with delays, Stoch. Dyn., 4 (2004), 405-423.  doi: 10.1142/S0219493704001139.  Google Scholar

[6]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335.  doi: 10.1007/s11464-008-0028-7.  Google Scholar

[7]

T. Caraballo, F. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-lipschitz nonlinearity, J. Differ. Equ. Appl., 17 (2011), 161-184.  doi: 10.1080/10236198.2010.549010.  Google Scholar

[8]

T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-lipschitz nonlinearities, J. Differ. Equ., 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.  Google Scholar

[9]

T. Caraballo, F. Morillas and J. Valero, On differential equations with delay in banach spaces and attractors for retarded lattice dynamical systems, Discrete Cont. Dyn. Syst. Ser. A, 34 (2014), 51-77.  doi: 10.3934/dcds.2014.34.51.  Google Scholar

[10]

T. Caraballo and J. Real, Attractors for 2d-Navier-Stokes models with delays, J. Differ. Equ., 205 (2004), 271-297.  doi: 10.1016/j.jde.2004.04.012.  Google Scholar

[11]

X. Ding and J. Jiang, Random attractors for stochastic retarded lattice dynamical systems, Abstr. Appl. Analy. , 2012 (2012), Art. ID 409282, 27 pp.  Google Scholar

[12]

X. Han, Random attractors for stochastic sine-gordon lattice systems with multiplicative white noise, J. Math. Anal. Appl., 376 (2011), 481-493.  doi: 10.1016/j.jmaa.2010.11.032.  Google Scholar

[13]

X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differ. Equ., 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.  Google Scholar

[14]

X. Han, Random attractors for second order stochastic lattice dynamical systems with multiplicative noise in weighted spaces Stoch. Dyn., 12(2012), 1150024, 20pp. doi: 10.1142/S0219493711500249.  Google Scholar

[15]

D. Li, C. Zhang and W. Wang, Long time behavior of non-Fickian delay reaction-diffusion equations, Nonlinear Anal.: RWA, 13 (2012), 1401-1415.  doi: 10.1016/j.nonrwa.2011.11.005.  Google Scholar

[16]

Y. Lv and J. Sun, Dynamical behavior for stochastic lattice systems, Chaos Solitons Fractals, 27 (2006), 1080-1090.  doi: 10.1016/j.chaos.2005.04.089.  Google Scholar

[17]

J. Mallet-Paret and G.R. Sell, Systems of differential delay equations: Floquet multipliers and discrete lyapunov functions, J. Differ. Equ., 125 (1996), 385-440.  doi: 10.1006/jdeq.1996.0036.  Google Scholar

[18]

B. Wang, Dynamics of systems on infinite lattices, J. Differ. Equ., 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003.  Google Scholar

[19]

B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121-136.  doi: 10.1016/j.jmaa.2006.08.070.  Google Scholar

[20]

X. Wang, S. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal.: TMA, 72 (2010), 483-494.  doi: 10.1016/j.na.2009.06.094.  Google Scholar

[21]

L. Xu and W. Yan, Stochastic Fitzhugh-Nagumo systems with delay, Taiwan. J. Math., 16 (2012), 1079-1103.   Google Scholar

[22]

W. Yan, Y. Li and S. Ji, Random attractors for first order stochastic retarded lattice dynamical systems J. Math. Phys., 51(2010), 032702, 17pp. doi: 10.1063/1.3319566.  Google Scholar

[23]

C. Zhang and S. Li, Dissipativity and exponentially asymptotic stability of the solutions for nonlinear neutral functional-differential equations, Appl. Math. comput., 119 (2001), 109-115.  doi: 10.1016/S0096-3003(99)00264-7.  Google Scholar

[24]

L. Zhao, C. Zhang and D. Li, Global attractor for a class of retarded lattice dynamical systems, J. Math. Anal. Appl., 425 (2015), 178-193.  doi: 10.1016/j.jmaa.2014.12.026.  Google Scholar

[25]

C. Zhao and S. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl., 354 (2009), 78-95.  doi: 10.1016/j.jmaa.2008.12.036.  Google Scholar

[26]

C. Zhao and S. Zhou, Attractors of retarded first order lattice systems, Nonlinearity, 20 (2007), 1987-2006.  doi: 10.1088/0951-7715/20/8/010.  Google Scholar

[27]

C. Zhao and S. Zhou, Compact uniform attractors for dissipative lattice dynamical systems with delays, Discrete Cont. Dyn. Syst. Ser. A, 21 (2008), 643-663.  doi: 10.3934/dcds.2008.21.643.  Google Scholar

[28]

X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 763-785.  doi: 10.3934/dcdsb.2008.9.763.  Google Scholar

[29]

C. Zhao, S. Zhou and W. Wang, Compact kernel sections for lattice systems with delays, Nonlinear Anal.: TMA, 70 (2009), 1330-1348.  doi: 10.1016/j.na.2008.02.015.  Google Scholar

[30]

S. Zhou, Attractors for second order lattice dynamical systems, J. Differ. Equ., 179 (2002), 605-624.  doi: 10.1006/jdeq.2001.4032.  Google Scholar

[31]

S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, J. Differ. Equ., 224 (2006), 172-204.  doi: 10.1016/j.jde.2005.06.024.  Google Scholar

[32]

S. Zhou, C. Zhao and X. Liao, Compact uniform attractors for dissipative non-autonomous lattice dynamical systems, Commun. Pur. Appl. Anal., 6 (2007), 1087-1111.  doi: 10.3934/cpaa.2007.6.1087.  Google Scholar

Figure 1.  Numerical simulation for the equation (5.1) with $u_i(t)=\frac{\partial}{\partial t}u_i(t)=\exp(t)\cos(\frac{i\pi }{50})$
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Figure 2.  Numerical simulation for the equation (5.1) with $u_i(t)=\frac{\partial}{\partial t}u_i(t)=\exp(t)\sin(\frac{i\pi }{50})$
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Figure 3.  Numerical solutions with different initial conditions at time $t=10$
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