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Dynamics of systems on infinite lattices. (English) Zbl 1085.37056

Summary: The dynamics of infinite-dimensional lattice systems is studied. A necessary and sufficient condition for asymptotic compactness of lattice dynamical systems is introduced. It is shown that a lattice system has a global attractor if and only if it has a bounded absorbing set and is asymptotically null. As an application, it is proved that the lattice reaction-diffusion equation has a global attractor in a weighted \(l^{2}\) space, which is compact as well as contains traveling waves. The upper semicontinuity of global attractors is also obtained when the lattice reaction-diffusion equation is approached by finite-dimensional systems.

MSC:

37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
35B41 Attractors
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
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[1] Afraimovich, V. S.; Nekorkin, V. I., Chaos of traveling waves in a discrete chain of diffusively coupled maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4, 631-637 (1994) · Zbl 0870.58049
[2] Babin, A. V.; Vishik, M. I., Attractors of Evolution Equations (1992), North-Holland: North-Holland Amsterdam · Zbl 0778.58002
[3] Ball, J. M., Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7, 475-502 (1997) · Zbl 0903.58020
[4] Bates, P. W.; Chen, X.; Chmaj, A., Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35, 520-546 (2003) · Zbl 1050.37041
[5] Bates, P. W.; Chmaj, A., On a discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150, 281-305 (1999) · Zbl 0956.74037
[6] P.W. Bates, H. Lisei, K. Lu, Attractors for stochastic lattice dynamical systems, Preprint.; P.W. Bates, H. Lisei, K. Lu, Attractors for stochastic lattice dynamical systems, Preprint. · Zbl 1105.60041
[7] Bates, P. W.; Lu, K.; Wang, B., Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11, 143-153 (2001) · Zbl 1091.37515
[8] Bell, J., Some threshold results for models of myelinated nerves, Math. Biosci., 54, 181-190 (1981) · Zbl 0454.92009
[9] Bell, J.; Cosner, C., Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quart. Appl. Math., 42, 1-14 (1984) · Zbl 0536.34050
[10] Beyn, W. J.; Pilyugin, S. Y., Attractors of reaction diffusion systems on infinite lattices, J. Dynam. Differential Equations, 15, 485-515 (2003) · Zbl 1041.37040
[11] Carrol, T. L.; Pecora, L. M., Synchronization in chaotic systems, Phys. Rev. Lett., 64, 821-824 (1990) · Zbl 0938.37019
[12] Chow, S. N.; Mallet-Paret, J., Pattern formation and spatial chaos in lattice dynamical systems, I, II, IEEE Trans. Circuits Systems, 42, 746-751 (1995)
[13] Chow, S. N.; Mallet-Paret, J.; Shen, W., Traveling waves in lattice dynamical systems, J. Differential Equations, 49, 248-291 (1998) · Zbl 0911.34050
[14] Chow, S. N.; Mallet-Paret, J.; Van Vleck, E. S., Pattern formation and spatial chaos in spatially discrete evolution equations, Random Comput. Dynam., 4, 109-178 (1996) · Zbl 0883.58020
[15] Chow, S. N.; Shen, W., Dynamics in a discrete Nagumo equationspatial topological chaos, SIAM J. Appl. Math., 55, 1764-1781 (1995) · Zbl 0840.34012
[16] Chua, L. O.; Roska, T., The CNN paradigm, IEEE Trans. Circuits Systems, 40, 147-156 (1993) · Zbl 0800.92041
[17] Chua, L. O.; Yang, Y., Cellular neural networkstheory, IEEE Trans. Circuits Systems, 35, 1257-1272 (1988) · Zbl 0663.94022
[18] Chua, L. O.; Yang, Y., Cellular neural networksapplications, IEEE Trans. Circuits Systems, 1, 35, 1273-1290 (1988)
[19] Elmer, C. E.; Van Vleck, E. S., Analysis and computation of traveling wave solutions of bistable differential-difference equations, Nonlinearity, 12, 771-798 (1999) · Zbl 0945.35046
[20] Elmer, C. E.; Van Vleck, E. S., Traveling waves solutions for bistable differential-difference equations with periodic diffusion, SIAM J. Appl. Math., 61, 1648-1679 (2001) · Zbl 0981.35020
[21] Erneux, T.; Nicolis, G., Propagating waves in discrete bistable reaction diffusion systems, Physica D, 67, 237-244 (1993) · Zbl 0787.92010
[22] Hale, J. K., Asymptotic behavior of dissipative systems (1988), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0642.58013
[23] Hale, J. K., Numerical dynamics, (Chaotic Numerics, Contemporary Mathematics, vol. 172 (1994), American Mathematical Society: American Mathematical Society Providence, RI), 1-30 · Zbl 0808.34061
[24] Kapval, R., Discrete models for chemically reacting systems, J. Math. Chem., 6, 113-163 (1991)
[25] 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
[26] Keener, J. P., The effects of discrete gap junction coupling on propagation in myocardium, J. Theoret. Biol., 148, 49-82 (1991)
[27] Ladyzhenskaya, O., Attractors for Semigroups and Evolution Equations (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0755.47049
[28] Sell, R.; You, Y., Dynamics of Evolutionary Equations (2002), Springer: Springer New York · Zbl 1254.37002
[29] Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics (1997), Springer: Springer New York · Zbl 0871.35001
[30] Wang, B., Attractors for reaction diffusion equations in unbounded domains, Physica D, 128, 41-52 (1999) · Zbl 0953.35022
[31] Zinner, B., Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Differential Equations, 96, 1-27 (1992) · Zbl 0752.34007
[32] Zhou, S., Attractors for second order lattice dynamical systems, J. Differential Equations, 179, 605-624 (2002) · Zbl 1002.37040
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