Uniform global attractors for first order non-autonomous lattice dynamical systems

Abdallah, Ahmed

doi:10.1090/S0002-9939-10-10440-7

Uniform global attractors for first order non-autonomous lattice dynamical systems
HTML articles powered by AMS MathViewer

by Ahmed Y. Abdallah PDF

Proc. Amer. Math. Soc. 138 (2010), 3219-3228 Request permission

Abstract:

Recently, many authors investigated the existence of global attractors for different types of autonomous lattice dynamical systems. Within this work, we carefully study the existence of a uniform global attractor for a new class of first order non-autonomous lattice dynamical system in the Hilbert space $l^{2}$.

References

Ahmed Y. Abdallah, Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems, Commun. Pure Appl. Anal. 5 (2006), no. 1, 55–69. MR 2190776, DOI 10.3934/cpaa.2006.5.55
Ahmed Y. Abdallah, Upper semicontinuity of the attractor for a second order lattice dynamical system, Discrete Contin. Dyn. Syst. Ser. B 5 (2005), no. 4, 899–916. MR 2170215, DOI 10.3934/dcdsb.2005.5.899
Ahmed Y. Abdallah, Upper semicontinuity of the attractor for lattice dynamical systems of partly dissipative reaction-diffusion systems, J. Appl. Math. 3 (2005), 273–288. MR 2201975, DOI 10.1155/JAM.2005.273
V. S. Afraĭmovich and V. I. Nekorkin, Chaos of traveling waves in a discrete chain of diffusively coupled maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 4 (1994), no. 3, 631–637. MR 1298079, DOI 10.1142/S0218127494000459
Jonathan Bell, Some threshold results for models of myelinated nerves, Math. Biosci. 54 (1981), no. 3-4, 181–190. MR 630848, DOI 10.1016/0025-5564(81)90085-7
Jonathan Bell and Chris Cosner, Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quart. Appl. Math. 42 (1984), no. 1, 1–14. MR 736501, DOI 10.1090/S0033-569X-1984-0736501-0
Peter W. Bates, Xinfu Chen, and Adam J. J. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal. 35 (2003), no. 2, 520–546. MR 2001111, DOI 10.1137/S0036141000374002
Peter W. Bates, Kening Lu, and Bixiang Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 11 (2001), no. 1, 143–153. MR 1815532, DOI 10.1142/S0218127401002031
Louis M. Pecora and Thomas L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990), no. 8, 821–824. MR 1038263, DOI 10.1103/PhysRevLett.64.821
V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl. (9) 73 (1994), no. 3, 279–333. MR 1273705
Shui-Nee Chow and John Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems. I, II, IEEE Trans. Circuits Systems I Fund. Theory Appl. 42 (1995), no. 10, 746–751, 752–756. MR 1363311, DOI 10.1109/81.473583
Shui-Nee Chow, John Mallet-Paret, and Wenxian Shen, Traveling waves in lattice dynamical systems, J. Differential Equations 149 (1998), no. 2, 248–291. MR 1646240, DOI 10.1006/jdeq.1998.3478
Shui-Nee Chow, John Mallet-Paret, and Erik S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Comput. Dynam. 4 (1996), no. 2-3, 109–178. MR 1402415
L.O. Chua, T. Roska, The CNN paradigm, IEEE Trans. Circuits and Systems 40 $\left ( 1993\right )$, 147-156.
Leon O. Chua and Lin Yang, Cellular neural networks: theory, IEEE Trans. Circuits and Systems 35 (1988), no. 10, 1257–1272. MR 960777, DOI 10.1109/31.7600
Leon O. Chua and Lin Yang, Cellular neural networks: applications, IEEE Trans. Circuits and Systems 35 (1988), no. 10, 1273–1290. MR 960778, DOI 10.1109/31.7601
Thomas Erneux and Grégoire Nicolis, Propagating waves in discrete bistable reaction-diffusion systems, Phys. D 67 (1993), no. 1-3, 237–244. MR 1234443, DOI 10.1016/0167-2789(93)90208-I
Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371, DOI 10.1090/surv/025
Raymond Kapral, Discrete models for chemically reacting systems, J. Math. Chem. 6 (1991), no. 2, 113–163. MR 1101758, DOI 10.1007/BF01192578
James P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math. 47 (1987), no. 3, 556–572. MR 889639, DOI 10.1137/0147038
J.P. Keener, The effects of discrete gap junction coupling on propagation in myocardium, J. Theor. Biol. 148 $\left ( 1991\right )$, 49-82.
B. M. Levitan and V. V. Zhikov, Almost periodic functions and differential equations, Cambridge University Press, Cambridge-New York, 1982. Translated from the Russian by L. W. Longdon. MR 690064
Qingfeng Ma, Shouhong Wang, and Chengkui Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J. 51 (2002), no. 6, 1541–1559. MR 1948459, DOI 10.1512/iumj.2002.51.2255
Shui-Nee Chow and John Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems. I, II, IEEE Trans. Circuits Systems I Fund. Theory Appl. 42 (1995), no. 10, 746–751, 752–756. MR 1363311, DOI 10.1109/81.473583
Bixiang Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl. 331 (2007), no. 1, 121–136. MR 2305992, DOI 10.1016/j.jmaa.2006.08.070
Shengfan Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations 179 (2002), no. 2, 605–624. MR 1885681, DOI 10.1006/jdeq.2001.4032
Shengfan Zhou, Attractors for first order dissipative lattice dynamical systems, Phys. D 178 (2003), no. 1-2, 51–61. MR 1983188, DOI 10.1016/S0167-2789(02)00807-2
Shengfan Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations 200 (2004), no. 2, 342–368. MR 2052618, DOI 10.1016/j.jde.2004.02.005
Bertram Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Differential Equations 96 (1992), no. 1, 1–27. MR 1153307, DOI 10.1016/0022-0396(92)90142-A