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

The attractors for 2nd-order stochastic delay lattice systems

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.

Citation: Chengjian Zhang, Lu Zhao. The attractors for 2nd-order stochastic delay lattice systems. Discrete and 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.

[2]

J. BanCh. HsuY. 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.

[3]

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

[4]

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

[5]

T. CaraballoP.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.

[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.

[7]

T. CaraballoF. 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.

[8]

T. CaraballoF. 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.

[9]

T. CaraballoF. 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.

[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.

[11]

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

[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.

[13]

X. HanW. 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.

[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.

[15]

D. LiC. 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.

[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.

[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.

[18]

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

[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.

[20]

X. WangS. 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.

[21]

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

[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.

[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.

[24]

L. ZhaoC. 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.

[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.

[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.

[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.

[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.

[29]

C. ZhaoS. 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.

[30]

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

[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.

[32]

S. ZhouC. 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.

show all references

References:
[1]

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

[2]

J. BanCh. HsuY. 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.

[3]

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

[4]

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

[5]

T. CaraballoP.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.

[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.

[7]

T. CaraballoF. 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.

[8]

T. CaraballoF. 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.

[9]

T. CaraballoF. 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.

[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.

[11]

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

[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.

[13]

X. HanW. 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.

[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.

[15]

D. LiC. 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.

[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.

[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.

[18]

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

[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.

[20]

X. WangS. 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.

[21]

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

[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.

[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.

[24]

L. ZhaoC. 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.

[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.

[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.

[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.

[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.

[29]

C. ZhaoS. 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.

[30]

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

[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.

[32]

S. ZhouC. 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.

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