zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Long-time behavior for second order lattice dynamical systems. (English) Zbl 1166.37029
Author’s abstract: We prove the existence of a global attractor for a new type of second order lattice dynamical systems in the Hilbert space l 2 ×l 2 . For specific choices of the linear operators this system can be regraded as a spatial discretization of a continuous damped nonlinear Boussinesq equation on m ,m1.
MSC:
37L30Attractors and their dimensions, Lyapunov exponents
37L60Lattice dynamics (infinite-dimensional dissipative systems)
References:
[1]Abdallah, A.Y.: Global attractor for the lattice dynamical system of a nonlinear Boussinesq equation. Abstr. Appl. Anal. 6, 655–671 (2005) · Zbl 1100.37052 · doi:10.1155/AAA.2005.655
[2]Abdallah, A.Y.: Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete Contin. Dyn. Syst. B 5, 899–916 (2005) · Zbl 1095.37041 · doi:10.3934/dcdsb.2005.5.899
[3]Abdallah, A.Y.: Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems. Commun. Pure Appl. Anal. 5, 55–69 (2006) · Zbl 1220.37073 · doi:10.3934/cpaa.2006.5.55
[4]Afraimovich, V.S., Nekorkin, V.I.: Chaos of traveling waves in a discrete chain of diffusively coupled maps. Int. J. Bifurc. Chaos 4, 631–637 (1994) · Zbl 0870.58049 · doi:10.1142/S0218127494000459
[5]Bates, P.W., Chen, X., Chmaj, A.: Traveling waves of bistable dynamics of a lattice. SIAM J. Math. Anal. 35, 520–546 (2003) · Zbl 1050.37041 · doi:10.1137/S0036141000374002
[6]Bates, P.W., Lu, K., Wang, B.: Attractors for lattice dynamical systems. Int. J. Bifurc. Chaos 11, 143–153 (2001) · Zbl 1091.37515 · doi:10.1142/S0218127401002031
[7]Bell, J.: Some threshold results for models of myelinated nerves. Math. Biosci. 54, 181–190 (1981) · Zbl 0454.92009 · doi:10.1016/0025-5564(81)90085-7
[8]Bell, J., Cosner, C.: Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons. Q. Appl. Math. 42, 1–14 (1984)
[9]Boussinesq, J.: Théorie des ondes et de remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement parielles de la surface au fond. J. Math. Pures Appl. Ser. 2 17, 55–108 (1872)
[10]Carrol, T.L., Pecora, L.M.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990) · doi:10.1103/PhysRevLett.64.821
[11]Chate, H., Courbage, M. (eds.): Lattice Systems. Physica D 103(1–4), 1–612 (1997) · Zbl 1194.82048 · doi:10.1016/S0167-2789(96)00249-7
[12]Chow, S.N., Mallet-Paret, J.: Pattern formation and spatial chaos in lattice dynamical systems I. IEEE Trans. Circuits Syst. 42, 746–751 (1995) · doi:10.1109/81.473583
[13]Chow, S.N., Mallet-Paret, J., Shen, W.: Traveling waves in lattice dynamical systems. J. Differ. Equ. 149, 248–291 (1998) · Zbl 0911.34050 · doi:10.1006/jdeq.1998.3478
[14]Chow, S.N., Mallet-Paret, J., Van Vleck, E.S.: Pattern formation and spatial chaos in spatially discrete evolution equations. Rand. Comput. Dyn. 4, 109–178 (1996)
[15]Chua, L.O., Roska, T.: The CNN paradigm. IEEE Trans. Circuits Syst. 40, 147–156 (1993)
[16]Chua, L.O., Yang, Y.: Cellular neural networks: theory. IEEE Trans. Circuits Syst. 35, 1257–1272 (1988) · Zbl 0663.94022 · doi:10.1109/31.7600
[17]Chua, L.O., Yang, Y.: Cellular neural networks: applications. IEEE Trans. Circuits Syst. 35, 1273–1290 (1988) · doi:10.1109/31.7601
[18]Erneux, T., Nicolis, G.: Propagating waves in discrete bistable reaction diffusion systems. Physica D 67, 237–244 (1993) · Zbl 0787.92010 · doi:10.1016/0167-2789(93)90208-I
[19]Kapral, R.: Discrete models for chemically reacting systems. J. Math. Chem. 6, 113–163 (1991) · doi:10.1007/BF01192578
[20]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 · doi:10.1137/0147038
[21]Keener, J.P.: The effects of discrete gap junction coupling on propagation in myocardium. J. Theor. Biol. 148, 49–82 (1991) · doi:10.1016/S0022-5193(05)80465-5
[22]Mallet-Paret, J., Chow, S.N.: Pattern formation and spatial chaos in lattice dynamical systems II. IEEE Trans. Circuits Syst. 42, 752–756 (1995) · doi:10.1109/81.473584
[23]Sell, G.R., You, Y.: Dynamics of Evolutionary Equations. Applied Mathematical Sciences, vol. 143. Springer, New York (2002)
[24]Temam, R.: Infinite-dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Applied Mathematical Sciences, vol. 68. Springer, New York (1997)
[25]You, Y.: Global dynamics of 2D Boussinesq equations. Nonlinear Anal. 30, 4643–4654 (1997) · Zbl 0927.37058 · doi:10.1016/S0362-546X(96)00218-0
[26]Zhou, S.: Attractors for second order lattice dynamical systems. J. Differ. Equ. 179, 605–624 (2002) · Zbl 1002.37040 · doi:10.1006/jdeq.2001.4032
[27]Zhou, S.: Attractors for first order dissipative lattice dynamical systems. Physica D 178, 51–61 (2003) · Zbl 1011.37047 · doi:10.1016/S0167-2789(02)00807-2
[28]Zhou, S.: Attractors and approximations for lattice dynamical systems. J. Differ. Equ. 200, 342–368 (2004) · Zbl 1173.37331 · doi:10.1016/j.jde.2004.02.005
[29]Zinner, B.: Existence of traveling wavefront solutions for the discrete Nagumo equation. J. Differ. Equ. 96, 1–27 (1992) · Zbl 0752.34007 · doi:10.1016/0022-0396(92)90142-A