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Numerical attractors and approximations for stochastic or deterministic sine-Gordon lattice equations. (English) Zbl 07427465

Summary: First, we apply the implicit Euler scheme to discretize the sine-Gordon lattice equation (possessing a global attractor) and prove the existence of a numerical attractor for the time-discrete sine-Gordon lattice system with small step sizes. Second, we establish the upper semi-convergence from the numerical attractor towards the global attractor when the step size tends to zero. Third, we establish the upper semi-convergence from the random attractor of the stochastic sine-Gordon lattice equation to the global attractor when the intensity of noise goes to zero. Fourth, we show the finitely dimensional approximations of the three (numerical, random and global) attractors as the dimension of the state space goes to infinity. In a word, we establish four paths of convergence of finitely dimensional (numerical and random) attractors towards the global attractor.

MSC:

34D45 Attractors of solutions to ordinary differential equations
37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
65L20 Stability and convergence of numerical methods for ordinary differential equations
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[1] Bates, P. W.; Lisei, H.; Lu, K., Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 1, 1-21 (2006) · Zbl 1105.60041
[2] Bates, P. W.; Lu, K.; Wang, B., Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos, 11, 143-153 (2001) · Zbl 1091.37515
[3] Bates, P. W.; Lu, K.; Wang, B., Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246, 845-869 (2009) · Zbl 1155.35112
[4] Bell, J.; Cosner, C., Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Q. Appl. Math., 42, 1-14 (1984) · Zbl 0536.34050
[5] Caraballo, T.; Han, X.; Schmalfuss, B.; Valero, J., Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Anal., 130, 255-278 (2016) · Zbl 1329.60208
[6] Chow, S. N.; Mallet-Paret, J.; Shen, W. X., Traveling waves in lattice dynamical systems, J. Differential Equations, 49, 248-291 (1998) · Zbl 0911.34050
[7] Erneux, T.; Nicolis, G., Propagating waves in discrete bistable reaction diffusion systems, Physica D, 67, 237-244 (1993) · Zbl 0787.92010
[8] 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
[9] Gu, A.; Li, Y.; Li, J., Random attractors on lattice of stochastic FitzHugh-Nagumo systems driven by alpha-stable Levy noises, Internat. J. Bifur. Chaos, 24, 1450123 (2014) · Zbl 1302.34013
[10] Han, X., Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, J. Math. Anal. Appl., 376, 481-493 (2011) · Zbl 1209.60038
[11] Han, X.; Kloeden, P. E.; Sonner, S., Discretisation of global attractors for lattice dynamical systems, J. Dynam. Differ. Equ., 32, 1457-1474 (2020) · Zbl 1466.34021
[12] Han, X.; Kloeden, P. E., Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261, 2986-3009 (2016) · Zbl 1345.34128
[13] Han, X.; Shen, W.; Zhou, S., Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250, 1235-1266 (2011) · Zbl 1208.60063
[14] Jentzen, A.; Kloeden, P. E., Taylor approximations of stochastic partial differential equations, CBMS Lecture series (2011), SIAM: SIAM Philadelphia · Zbl 1240.35001
[15] Kapval, R., Discrete models for chemically reacting systems, J. Math. Chem., 6, 113-163 (1991)
[16] Keener, J. P., Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47, 556-572 (1987) · Zbl 0649.34019
[17] Kloeden, P. E.; Lorenz, J., Stable attracting sets in dynamical systems and in their one-step discretizations, SIAM J. Numer. Anal., 23, 986-995 (1986) · Zbl 0613.65083
[18] Li, F.; Li, Y., Asymptotic behavior of stochastic \(g\)-navier-stokes equations on a sequence of expanding domains, J. Math. Phys., 60, 061505 (2019) · Zbl 1418.35292
[19] Li, Y.; Gu, A.; Li, J., Existence and continuity of bi-spatial random attractors and application to stochastic semilinear laplacian equations, J. Differential Equations, 258, 504-534 (2015) · Zbl 1306.37091
[20] Li, Y.; Yang, S., Backward compact and periodic random attractors for non-autonomous sine-gordon equations with multiplicative noise, Commun. Pure Appl. Anal., 18, 1155-1175 (2019) · Zbl 1411.35048
[21] Li, Y.; Li, F., Limiting dynamics for stochastic FitzHugh-Nagumo equations on large domains, Stoch. Dyn., 19, 1950037 (2019) · Zbl 1423.60104
[22] Sui, M.; Wang, Y., Upper semicontinuity of pullback attractors for lattice nonclassical diffusion delay equations under singular perturbations, Appl. Math. Comput., 242, 315-327 (2014) · Zbl 1334.37015
[23] Wang, B., Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14, 1450009 (2014) · Zbl 1304.35789
[24] Wang, G.; Zhu, S., Dimension of the global attractor for the discretized damped sine-Gordon equation, Appl. Math. Comput., 117, 257-265 (2001) · Zbl 1026.37065
[25] Wang, R.; Li, Y., Regularity and backward compactness of attractors for non-autonomous lattice systems with random coefficients, Appl. Math. Comput., 354, 86-102 (2019) · Zbl 1428.37074
[26] Yang, S.; Li, Y., Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain, Evol. Equ. Control Theory, 9, 581-604 (2020) · Zbl 1456.35043
[27] Yang, S.; Li, Y., Asymptotic autonomous attractors for a stochastic lattice model with random viscosity, J. Difference Equ. Appl., 4, 540-560 (2020) · Zbl 1455.37063
[28] Zhao, W., Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space, Appl. Math. Comput., 291, 226-243 (2016) · Zbl 1410.37054
[29] Zhou, S.; Wang, Z., Random attractors for stochastic retarded lattice systems, J. Differ. Equ. Appl., 19, 1523-1543 (2013) · Zbl 1309.37074
[30] Zhou, S.; Han, X., Pullback exponential attractors for non-autonomous lattice systems, J. Dyn. Differ. Equ., 24, 601-631 (2012) · Zbl 1267.34020
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