Wang, Xiaohu; Li, Shuyong; Xu, Daoyi Random attractors for second-order stochastic lattice dynamical systems. (English) Zbl 1181.60103 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 1, 483-494 (2010). Summary: The asymptotic behavior of second-order stochastic lattice dynamical systems is considered. We firstly show the existence of an absorbing set. Then an estimate on tails of the solutions is derived when the time is large enough, which ensures the asymptotic compactness of the random dynamical system. Finally, the existence of the random attractor is provided. Cited in 61 Documents MSC: 60H15 Stochastic partial differential equations (aspects of stochastic analysis) Keywords:stochastic lattice dynamical systems; asymptotically compact; random attractors PDFBibTeX XMLCite \textit{X. Wang} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 1, 483--494 (2010; Zbl 1181.60103) Full Text: DOI References: [1] Hale, J. K., Numerical dynamics, Chaotic Numerics, Contemp. Math., vol. 172 (1994), American Mathematical Society: American Mathematical Society Providence, RI [2] Chow, S.-N.; Mallet-Paret, J., Pattern formulation and spatial chaos in lattice dynamical systems: I, IEEE Trans. Circuits Syst., 42, 746-751 (1995) [3] Chow, S.-N.; Shen, W., Dynamics in a discrete Nagumo equation: Spatial topological chaos, SIAM J. Appl. Math., 55, 1764-1781 (1995) · Zbl 0840.34012 [4] Shen, W., Lifted lattices, hyperbolic structures, and topological disorders in coupled map lattices, SIAM J. Appl. Math., 56, 1379-1399 (1996) · Zbl 0868.58059 [5] Chow, S.-N.; Mallet-Paret, J.; Shen, W., Traveling waves in lattice dynamical systems, J. Differential Equations, 149, 248-291 (1998) · Zbl 0911.34050 [6] Bates, P. W.; Chmaj, A., A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150, 281-305 (1999) · Zbl 0956.74037 [7] Zinner, B., Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Differential Equations, 96, 1-27 (1992) · Zbl 0752.34007 [8] Bates, P. W.; Lu, K. N.; Wang, B. X., Attractor for lattice dynamical systems, Internat. J. Bifur. Chaos, 1, 143-153 (2001) · Zbl 1091.37515 [9] Zhou, S., Attractors for second order lattice dynamical systems, J. Differential Equations, 179, 605-624 (2002) · Zbl 1002.37040 [10] Zhou, S., Attractors for second-order lattice dynamical systems with damping, J. Math. Phys., 43, 452-465 (2002) · Zbl 1059.37063 [11] Zhou, S., Attractors for first order dissipative lattice dynamical systems, Physica D, 178, 51-61 (2003) · Zbl 1011.37047 [12] Wang, B., Dynamics of systems on infinite lattices, J. Diff. Eqns, 221, 224-245 (2006) · Zbl 1085.37056 [13] Wang, B., Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331, 121-136 (2007) · Zbl 1112.37076 [14] Fan, X.; Wang, Y., Attractors for a second order nonautonomous lattice dynamical system with nonlinear damping, Phys. Lett. A, 365, 17-27 (2007) · Zbl 1203.37122 [15] Bates, P. W.; Lisei, H.; Lu, K. N., Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6, 1-21 (2006) · Zbl 1105.60041 [16] Crauel, H.; Flandoli, F., Attractor for random dynamical systems, Probab. Theory Related Fields, 100, 365-393 (1994) · Zbl 0819.58023 [17] Crauel, H.; Debussche, A.; Flandoli, F., Random attractors, J. Dynam. Differential Equations, 9, 307-341 (1997) · Zbl 0884.58064 [18] B. Schmalfuss, Backward cocycle and attractors of stochastic differential equations, in: V. Reitmann, T. Riedrich, N. Koksch (Eds.), International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Technische Universität, Dresden, 1992, pp. 185-192; B. Schmalfuss, Backward cocycle and attractors of stochastic differential equations, in: V. Reitmann, T. Riedrich, N. Koksch (Eds.), International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Technische Universität, Dresden, 1992, pp. 185-192 [19] Fan, X., Attractors for a damped stochastic wave equation of Sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl., 24, 767-793 (2006) · Zbl 1103.37053 [20] Fan, X.; Wang, Y., Fractal dimension of attractors for a stochastic wave equation with nonlinear damping and white noise, Stoch. Anal. Appl., 25, 381-396 (2007) · Zbl 1115.37066 [21] Zhou, S.; Yin, F.; Ouyang, Z., Random attractor for damped nonlinear wave equations with white noise, SIAM J. Appl. Dyn. Syst., 4, 883-903 (2005) · Zbl 1094.35154 [22] Lv, Y.; Sun, J., Dynamical behavior for stochastic lattice systems, Chaos Solitons Fractals, 27, 1080-1090 (2006) · Zbl 1134.37350 [23] Lv, Y.; Sun, J., Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations, Physica D, 221, 157-169 (2006) · Zbl 1130.37382 [24] Huang, J., The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with white noises, Physica D, 233, 83-94 (2007) · Zbl 1126.37048 [25] Wang, B., Attractors for reaction diffusion equations in unbounded domains, Physica D, 128, 41-52 (1999) · Zbl 0953.35022 [26] Xu, D.; Yang, Z.; Huang, Y., Existence-uniqueness and continuation theorems for stochastic functional differential equations, J. Diff. Eqns, 245, 1681-1703 (2008) · Zbl 1161.34055 [27] Arnold, L., Random Dynamical System (1998), Springer-Verlag: Springer-Verlag New York, Berlin This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.