zbMATH — the first resource for mathematics

Coupled map lattices: one step forward and two steps back. (English) Zbl 0890.58029
Summary: We discuss the general notion of lattice dynamical systems and some recent progress in their studies. We also suggest an approach that allows to extract some information on the dynamics of finitely extended (“real”) systems from the results obtained on the infinitely extended (“artificial”) ones.

37B99 Topological dynamics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Full Text: DOI
[1] ()
[2] ()
[3] L’vov, V.S.; Predtechensky, A.A.; Chernykh, A.I., Bifurcations and chaos in the system of Taylor vortices-laboratory and numerical experiment, (), 238-280
[4] Gaponov-Grekhov, A.V.; Rabinovich, M.I.; Starobinets, I.M., Arising of multidimensional chaos in the active lattices, Soviet. phys. dokl., 292, 64-67, (1984)
[5] Kaneko, K.; Kaneko, K.; Kaneko, K., Overview of coupled map lattices, (), 279-282 · Zbl 1055.37540
[6] Holden, A.V.; Tucker, J.V.; Zhang, H.; Poole, M.J., Coupled map lattices as computational systems, (), 367-376 · Zbl 1055.68544
[7] Bunimovich, L.A.; Sinai, Ya.G., Statistical mechanics of coupled map lattices, (), 169-189 · Zbl 0791.60099
[8] Bunimovich, L.A.; Sinai, Ya.G., Space-time chaos in coupled map lattices, Nonlinearity, 1, 491-504, (1988) · Zbl 0679.58028
[9] Pesin, Ya.B.; Sinai, Ya.G., Space-time chaos in chains of weakly-coupled hyperbolic maps, () · Zbl 0850.70250
[10] Keller, H.; Künzle, M., Transfer operators for coupled map lattices, Ergod. theory dynam. syst., 12, 297-318, (1992) · Zbl 0737.58032
[11] Blank, M.L., Small perturbations of chaotic dynamical systems, Russ. math. surv., 44, 3-28, (1989) · Zbl 0702.58063
[12] Afraimovich, V.S.; Chow, S.-N., Existence of evolution operators group for infinite lattice of coupled ordinary differential equations, Dynam. syst. appl., 3, 155-174, (1994) · Zbl 0802.34013
[13] Gundlach, V.M.; Rand, D.H., Spatio-temporal chaos: 1-3, Nonlinearity, 6, 165-230, (1993) · Zbl 0776.58014
[14] Malyshev, V.A.; Minlos, R.A., Gibbs random fields, (1992), Riedel Dordrecht · Zbl 0731.60099
[15] Afraimovich, V.S.; Bunimovich, L.A., Simplest structures in coupled map lattices and their stability, Rand. comput. dynam., 1, 423-444, (1993) · Zbl 0810.34036
[16] L.A. Bunimovich and E.A. Carlen, To the problem of stability in lattice dynamical systems, J. Diff. Eqns., to be published. · Zbl 0845.58041
[17] Volevich, D.L., Kinetics of coupled map lattices, Nonlinearity, 4, 37-45, (1991) · Zbl 0778.58043
[18] Giberti, C.; Vernia, C., On the presence of normally attracting manifolds containing periodic or quasiperiodic orbits in coupled map lattices, Int. J. bifurc. chaos, 3, 1503-1514, (1993) · Zbl 0890.58076
[19] C. Giberti and C. Vernia, Periodic behavior in 1D and 2D coupled map lattices of small size, Chaos, to be published. · Zbl 1055.37523
[20] Miller, J.; Huse, D.A., Macroscopic equilibrium from microscopic irreversibility in a chaotic coupled map lattice, Phys. rev. E, 48, 2528-2535, (1993)
[21] C. Boldrighini, L.A. Bunimovich, G. Cosimi, A. Frigio and A. Pellegrinotti, Ising-type Phase Transitions in Coupled Map Lattices, J. Stat. Phys., to be published. · Zbl 0999.82052
[22] ()
[23] Eckmann, J.-P.; Procaccia, I., Spatio-temporal chaos, (), 135-172 · Zbl 1082.37505
[24] Rabinovich, M.I.; Fabricant, A.L.; Tsimring, L.Sh., Finite dimensional spatial disorder, (1992), Preprint
[25] V.S. Afraimovich and S.-N. Chow, Criteria of Spatial Chaos in Lattice Dynamical Systems, Japan J. Ind. Appl. Math., to be published. · Zbl 0847.46040
[26] Afraimovich, V.S.; Bunimovich, L.A., Density of defects and spatial entropy in extended systems, Physica D, 80, 277-288, (1995) · Zbl 0888.58015
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.