×

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
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[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
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.