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Attractors for stochastic lattice dynamical systems with a multiplicative noise. (English) Zbl 1155.60324
Summary: In this paper, we consider a stochastic lattice differential equation with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and multiplicative white noise at each node. We prove the existence of a compact global random attractor which, pulled back, attracts tempered random bounded sets.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
34B45 Boundary value problems on graphs and networks for ordinary differential equations
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[1] Afraimovich V S, Nekorkin V I. Chaos of traveling waves in a discrete chain of diffusively coupled maps. Int J Bifur Chaos, 1994, 4: 631–637 · Zbl 0870.58049
[2] Arnold L. Random Dynamical Systems. Berlin: Springer-Verlag, 1998 · Zbl 0906.34001
[3] Bates P W, Chmaj A. A discrete convolution model for phase transitions. Arch Ration Mech Anal, 1999, 150(4): 281–305 · Zbl 0956.74037
[4] Bates P W, Lisei H, Lu K. Attractors for stochastic lattice dynamical systems. Stochastics & Dynamics, 2006, 6(1): 1–21 · Zbl 1105.60041
[5] Bates P W, Lu K, Wang B. Attractors for lattice dynamical systems. Int J Bifur Chaos, 2001, 11: 143–153 · Zbl 1091.37515
[6] Bell J. Some threshhold results for models of myelinated nerves. Mathematical Biosciences, 1981, 54: 181–190 · Zbl 0454.92009
[7] Bell J, Cosner C. Threshold behaviour and propagation for nonlinear differentialdifference systems motivated by modeling myelinated axons. Quarterly Appl Math, 1984, 42: 1–14 · Zbl 0536.34050
[8] Caraballo T, Kloeden P E, Schmalfuß B. Exponentially stable stationary solutions for stochastic evolution equations and their perturbation. Applied Mathematics and Optimization, 2004, 50: 183–207 · Zbl 1066.60058
[9] Caraballo T, Lukaszewicz G, Real J. Pullback attractors for asymptotically compact nonautonomous dynamical systems. Nonlinear Analysis TMA, 2006, 64(3): 484–498 · Zbl 1128.37019
[10] Chow S-N, Mallet-Paret J. Pattern formulation and spatial chaos in lattice dynamical systems: I. IEEE Trans Circuits Syst, 1995, 42: 746–751
[11] Chow S-N, Mallet-Paret J, Shen W. Traveling waves in lattice dynamical systems. J Diff Eq, 1998, 149: 248–291 · Zbl 0911.34050
[12] Chow S-N, Mallet-Paret J, Van Vleck E S. Pattern formation and spatial chaos in spatially discrete evolution equations. Random Computational Dynamics, 1996, 4: 109–178 · Zbl 0883.58020
[13] Chow S-N, Shen W. Dynamics in a discrete Nagumo equation: Spatial topological chaos. SIAM J Appl Math, 1995, 55: 1764–1781 · Zbl 0840.34012
[14] Chua L O, Roska T. The CNN paradigm. IEEE Trans Circuits Syst, 1993, 40: 147–156 · Zbl 0800.92041
[15] Chua L O, Yang L. Cellular neural networks: Theory. IEEE Trans Circuits Syst, 1988, 35: 1257–1272 · Zbl 0663.94022
[16] Chua L O, Yang L. Cellular neural networks: Applications. IEEE Trans Circuits Syst, 1988, 35: 1273–1290
[17] Crauel H. Random point attractors versus random set attractors. J London Math Soc, 2002, 63: 413–427 · Zbl 1011.37032
[18] Crauel H, Debussche A, Flandoli F. Random Attractors. J Dyn Diff Eq, 1997, 9: 307–341 · Zbl 0884.58064
[19] Crauel H, Flandoli F. Attractors for random dynamical systems. Probab Theory Relat Fields, 1994, 100: 365–393 · Zbl 0819.58023
[20] Dogaru R, Chua L O. Edge of chaos and local activity domain of Fitz-Hugh-Nagumo equation. Int J Bifurcation and Chaos, 1988, 8: 211–257 · Zbl 0933.37042
[21] Erneux T, Nicolis G. Propagating waves in discrete bistable reaction diffusion systems. Physica D, 1993, 67: 237–244 · Zbl 0787.92010
[22] Flandoli F, Lisei H. Stationary conjugation of flows for parabolic SPDEs with multiplicative noise and some applications. Stoch Anal Appl, 2004, 22 1385–1420 · Zbl 1063.60089
[23] Flandoli F, Schmalfuß B. Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise. Stochastics and Stochastic Rep, 1996, 59: 21–45 · Zbl 0870.60057
[24] Imkeller P, Schmalfuß B. The conjugacy of stochastic and random differential equations and the existence of global attractors. J Dyn Diff Eq, 2001, 13: 215–249 · Zbl 1004.37034
[25] Kapval R. Discrete models for chemically reacting systems. J Math Chem, 1991, 6: 113–163
[26] Keener J P. Propagation and its failure in coupled systems of discrete excitable cells. SIAM J Appl Math, 1987, 47: 556–572 · Zbl 0649.34019
[27] Keener J P. The effects of discrete gap junction coupling on propagation in myocardium. J Theor Biol, 1991, 148: 49–82
[28] Laplante J P, Erneux T. Propagating failure in arrays of coupled bistable chemical reactors. J Phys Chem, 1992, 96: 4931–4934
[29] Mallet-Paret J. The global structure of traveling waves in spatially discrete dynamical systems. J Dynam Differential Equations, 1999, 11(1): 49–127 · Zbl 0921.34046
[30] Pérez-Muñuzuri A, Pérez-Muñuzuri V, Pérez-Villar V, et al. Spiral waves on a 2-d array of nonlinear circuits. IEEE Trans Circuits Syst, 1993, 40: 872–877 · Zbl 0844.93056
[31] Rashevsky N. Mathematical Biophysics. Vol 1. New York: Dover Publications, Inc, 1960 · JFM 64.1148.01
[32] Ruelle D. Characteristic exponents for a viscous fluid subjected to time dependent forces. Commu Math Phys, 1984, 93: 285–300 · Zbl 0565.76031
[33] Scheutzow M. Comparison of various concepts of a random attractor: A case study. Arch Math, 2002, 78: 233–240 · Zbl 1100.37032
[34] Scott A C. Analysis of a myelinated nerve model. Bull Math Biophys, 1964, 26: 247–254
[35] Shen W. Lifted lattices, hyperbolic structures, and topological disorders in coupled map lattices. SIAM J Appl Math, 1996, 56: 1379–1399 · Zbl 0868.58059
[36] Zinner B. Existence of traveling wavefront solutions for the discrete Nagumo equation. J Diff Eq, 1992, 96: 1–27 · Zbl 0752.34007
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