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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Random attractor for stochastic non-autonomous damped wave equation with critical exponent
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 |
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 and Continuous Dynamical Systems,
2017, 37
(1)
: 575-590.
doi: 10.3934/dcds.2017023
![]()
References:
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Random Dynamical Systems, Springer-Verlag, Berlin, 1998.
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Pullback and forward attractors for dissipative lattice dynamical systems with additive noises, Dyn. Syst., 24 (2009), 139-155.
doi: 10.1080/14689360802508777. |
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P. W. Bates, H. Lisei and K. Lu,
Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.
doi: 10.1142/S0219493706001621. |
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P.W. Bates, K. Lu and B. Wang,
Attractors for lattice dynamical systems, Int. J. Bifurcat. Chaos, 11 (2001), 143-153.
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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. |
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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. |
| [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. |
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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. |
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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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
show all references
References:
| [1] |
L. Arnold,
Random Dynamical Systems, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-12878-7. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. 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. |
| [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. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
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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