Huang, Jianhua The random attractor of stochastic Fitzhugh-Nagumo equations in an infinite lattice with white noises. (English) Zbl 1126.37048 Physica D 233, No. 2, 83-94 (2007). Summary: The present paper is devoted to the existence of a random attractor for the stochastic FitzHugh-Nagumo equations in an infinite lattice with additive white noise. Using the Ornstein-Uhlenbeck transform, we firstly show the existence of an absorbing set, then we prove that the random dynamical system is asymptotically compact. Finally, the existence of the random attractor is provided. Cited in 1 ReviewCited in 39 Documents MSC: 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 35K57 Reaction-diffusion equations 35B41 Attractors 37H10 Generation, random and stochastic difference and differential equations Keywords:random attractor; additive white noise; stochastic Fitzhugh-Nagumo systems PDFBibTeX XMLCite \textit{J. Huang}, Physica D 233, No. 2, 83--94 (2007; Zbl 1126.37048) Full Text: DOI References: [1] Arnold, L., Random Dynamical System (1998), Springer-Verlag: Springer-Verlag New York, Berlin [2] Bates, P.; Lisei, H.; Lu, K., Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6, 1-21 (2006) · Zbl 1105.60041 [3] Crauel, H.; Flandoli, F., Attractor for random dynamical systems, Probab. Theory Related Fields, 100, 365-393 (1994) · Zbl 0819.58023 [4] Crauel, H.; Debussche, A.; Flandoli, F., Random attractors, J. Dynam. Differential Equations, 9, 307-341 (1997) · Zbl 0884.58064 [5] Crauel, H.; Flandoli, F., Hausdorff dimensional of invariant sets for random dynamical systems, J. Dynam. Differential Equations, 10, 449-474 (1998) · Zbl 0927.37031 [6] Elmer, C.; Vleck, E., Spatially discrete FitzHugh-Nagumo equations, SIAM J. Appl. Math., 12 (2004) · Zbl 1089.34052 [7] Gao, W.; Wang, J., Existence of wavefronts and impulses to FitzHugh-Nagumo equations, Nonlinear Anal., 57, 667-676 (2004) · Zbl 1137.35391 [8] Jones, C., Stability of the traveling wave solution of the FitzHugh-Nagumo systems, Trans. Amer. Math. Soc., 286, 431-469 (1984) · Zbl 0567.35044 [9] Kitajima, H.; Kurths, J., Synchronized firing of FitzHugh-Nagumo neurons by noise, Chaos, 15, 023704:1-023704:5 (2005) · Zbl 1080.92020 [10] Lu, Y.; Sun, J., Dynamical behavior for stochastic lattice systems, Chaos Solitons Fractals, 27, 1080-1090 (2006) · Zbl 1134.37350 [11] Chow, S.; Mallet-Paret, John; Shen, W., Traveling waves in lattice dynamical systems, J. Differential Equations, 149, 248-291 (1998) · Zbl 0911.34050 [12] Teman, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics (1988), Springer-Verlag: Springer-Verlag New York [13] Vleck, E.; Wang, B., Attractors for lattice FitzHugh-Nagumo systems, Physica D, 212, 317-336 (2005) · Zbl 1086.34047 [14] Zhou, S., Attractors for first order dissipative lattice dynamical systems, Physica D, 178, 51-61 (2003) · Zbl 1011.37047 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.