×

zbMATH — the first resource for mathematics

Coupled map lattices: some topological and ergodic properties. (English) Zbl 1194.82048
Summary: Lattice dynamical systems (LDSs) form the class of extended systems that is the intermediate one between partial differential equations (PDEs) and cellular automata. The most popular class of LDSs is formed by coupled map lattices (CMLs). While being introduced rather recently LDSs allowed already to clarify and to define exactly some notions that form the basis of the modern phenomenological theory of spatio-temporal dynamics and to obtain some new, and even rigorous results on space-time chaos, intermittency and pattern formation. We discuss the rigorous results that were obtained in this area, but eventually give also some applications.

MSC:
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
37Axx Ergodic theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aubry, S., The concept of anti-integrability: definitions, theorems and applications, in: twist mappings and their applications, (), 7-54
[2] Afraimovich, V.S.; Bunimovich, L.A., Simplest structures in coupled map lattices and their stabilities, Random comput. dynamics, 1, 423-444, (1993) · Zbl 0810.34036
[3] Afraimovich, V.S.; Bunimovich, L.A., Density of defects and spatial entropy in extended systems, Physica D, 80, 277-288, (1995) · Zbl 0888.58015
[4] Afraimovich, V.A.; Chow, S.-N., Spatial chaos and homoclinic points of \(Z\^{}\{d\}- action\) in lattice dynamical systems, Japan J. indust. appl. math., 12, 1-17, (1995)
[5] V.S. Afraimovich, S.-N. Chow and W. Shen, Hyperbolic homoclinic points of \(Z\^{}\{d\}- actions\) in lattice dynamical system, Int. J. Bifurc . and Chaos, to appear. · Zbl 0874.58064
[6] Afraimovich, V.S.; Glebsky, L.Y.; Nekorkin, V.I., Stability of stationary states and topological spatial chaos in lattice dynamical systems, Random comput. dynamics, 2, 287-303, (1994) · Zbl 0827.58027
[7] Afraimovich, V.S.; Nekorkin, V.I., Chaos of travelling waves in a discrete chain of diffusively coupled maps, Int. J. bifurc. and chaos, 4, 631-637, (1994) · Zbl 0870.58049
[8] Afraimovich, V.S.; Pesin, Ya.B.; Afraimovich, V.S.; Pesin, Ya.B., Traveling waves in lattice models of multi-dimensional and multi-component media, II. ergodic properties and dimension, Nonlinearity, Chaos, 3, 233-241, (1993) · Zbl 1055.37584
[9] Bowen, R., Equilibrium states and the ergodic theory of Anosov diffeomorphisms, () · Zbl 0308.28010
[10] Bourgain, J., On the Cauchy and invariant measure problem for the periodic zakhavov system, Duke math. J., 76, 175-202, (1994) · Zbl 0821.35120
[11] Bourgain, J., Periodic nonlinear Schrödinger equations and invariant measures, Comm. math. phys., 166, 1-26, (1994) · Zbl 0822.35126
[12] Bourgain, J., Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, (), preprint · Zbl 0852.35131
[13] Bunimovich, L.A., Coupled map lattices: one step forward and two steps back, Physica D, 86, 248-255, (1995) · Zbl 0890.58029
[14] Boldrighini, C.; Bunimovich, L.A.; Cosimi, G.; Frigio, A.; Pellegrinotti, A., Ising-type transitions in coupled map lattices, J. statisc. phys., 80, 1185-1205, (1995) · Zbl 1081.82516
[15] Bunimovich, L.A.; Carlen, E.A., On the problem of stability in lattice dynamical systems, J. differential equqtions, 123, 213-229, (1995) · Zbl 0845.58041
[16] L.A. Bunimovich and E.A. Carlen, On stability of structures in lattice dynamical systems in progress. · Zbl 0845.58041
[17] Bunimovich, L.A.; Franceschini, V.; Giberti, C.; Vernia, C., On stability of structures and patterns in extended systems, Physica D, 103, 412-418, (1997) · Zbl 1194.34112
[18] Brickmont, J.; Kupiainen, A., Coupled analytic maps, () · Zbl 0836.58027
[19] Brickmont, J.; Kupiainen, A., High temperature expansions and dynamical systems, (1995), preprint
[20] Brickmont, J.; Kupiainen, A., Infinite dimensional SBR measures, (1995), preprint
[21] Bunimovich, L.A.; Lambert, A.; Lima, R., The emergence of coherent structures in coupled map lattices, J. statist. phys., 61, 253-262, (1990)
[22] Bunimovich, L.A.; Livi, R.; Martinez-Mekler, G.; Ruffo, S., Coupled trivial maps, Chaos, 2, (1992) · Zbl 1055.37560
[23] Bunimovich, L.A.; Sinai, Ya.G., Space-time chaos in coupled map lattices, Nonlinearity, 1, 491-518, (1988) · Zbl 0679.58028
[24] Bunimovich, L.A.; Sinai, Ya.G., Statistical mechanics of coupled map lattices, () · Zbl 0791.60099
[25] Bunimovich, L.A.; Venkatagiri, S., Onset of chaos in coupled map lattices via peak-crossing bifurcations, Nonlinearity, 9, (1996) · Zbl 0895.58037
[26] ()
[27] Crutchfield, J.; Kaneko, K., Phenomenology of the space-time chaos, (), 272-353
[28] Chaté, H.; Manneville, P., Spatiotemporal intermittency in coupled map lattices, Physica D, 32, 409-423, (1988) · Zbl 0656.76055
[29] Chaté, H.; Manneville, P., Role of defects in the transition to turbulence via spatiotemporal intermittency, Physica D, 37, 33-41, (1989)
[30] Chaté, H.; Manneville, P., Coupled map lattices as cellular automata, J. statist. phys., 56, 357-370, (1989)
[31] Chaté, H.; Manneville, P., Collective behaviors in spatially extended systems, Prog. theoret. phys., 87, 1-60, (1992)
[32] Chow, S.-N.; Mallet-Paret, J.; Chow, S.-N.; Mallet-Paret, J., Pattern formation and spatial chaos in lattice dynamical systems II, IEEE trans. circuits and systems, IEEE trans. circuits and systems, 42, 1-5, (1995)
[33] Chow, S.-N.; Shen, W., Stability and bifurcation of travelling wave solutions in coupled map lattices, Dynamic systems and appl., 4, 1-25, (1995) · Zbl 0821.34046
[34] Dobrushin, R.L., The problem of uniqueness of a Gibbsian random field and the problem of phase transitions, Functional anal. appl., 2, 302-312, (1968) · Zbl 0192.61702
[35] H. Greenside, preprint (1995) (see the paper in these proceedings).
[36] C. Giberti, Calculation of density of defects in lattice systems, Random and Computational Dynamics, to be published. · Zbl 0871.58033
[37] Gundlach, V.M.; Rand, D.A.; Gundlach, V.M.; Rand, D.A.; Gundlach, V.M.; Rand, D.A., Spatial-temporal chaos: 3. natural spatial-temporal measures for coupled circle map lattices, Nonlinearity, Nonlinearity, Nonlinearity, 6, 215-230, (1993) · Zbl 0776.58014
[38] Giberti, C.; Vernia, C., On the presence of normally attracting manifolds containing periodic or quasiperiodic orbits in coupled map lattices, Int. J. bifurc. and chaos, 3, 103-1514, (1993) · Zbl 0890.58076
[39] Giberti, C.; Vernia, C., Periodic behavior in 1D and 2D coupled map lattices of small size, Chaos, 4, 651-664, (1994)
[40] Jiang, M.; Jiang, M., Equilibrium states for lattice models of hyperbolic type, (), 8, 631-659, (1995), see also · Zbl 0836.58032
[41] Jiang, M.; Mazel, A., Uniqueness of Gibbs states and exponential decay of correlations for some lattice models, J. statist. phys., (1995)
[42] M. Jiang and Ya.B. Pesin, Equilibrium measures for coupled map lattices: Existence, uniqueness and finite-dimensional approximations, in preparation. · Zbl 0944.37005
[43] ()
[44] ()
[45] Katok, A.B.; Hasselblatt, B., Introduction to the modern theory of dynamical systems, (1995), Cambridge University Press Cambridge · Zbl 0878.58020
[46] Keller, G.; Künzle, M., Transfer operators for coupled map lattices, Ergodic theory dynamical systems, 12, 297-318, (1992) · Zbl 0737.58032
[47] ()
[48] Lebowitz, J.; Rose, R.; Speer, E., Statistical mechanics of the nonlinear Schrödinger equation, J. statist. phys., 60, 657-687, (1988) · Zbl 1084.82506
[49] Mackay, R.S., Dynamics of networks: features which persist from the uncoupled limit, (1995), preprint · Zbl 0981.37034
[50] Miller, J.; Huse, D.A., Macroscopic equilibrium from microscopic irreversibility in a chaotic coupled-map lattice, Phys. rev. E, 48, 2528-2535, (1993)
[51] Malyshev, V.A.; Minlos, R.A., Gibbs random fields, clusters expansions, (1991), Kluwer Academic Publishers Dordrecht · Zbl 0731.60099
[52] McKean, H.; Vaninski, K., Statistical mechanics of nonlinear wave equations and Brownian motion with restoring drift, Comm. math. phys., 160, 615-630, (1994) · Zbl 0792.60077
[53] Pesin, Ya.B.; Sinai, Ya.G., Space-time chaos in chains of weakly coupled hyperbolic maps, () · Zbl 0850.70250
[54] Ruelle, D., Thermodynamic formalism, (1978), Addison-Wesley Reading, MA
[55] Shen, W., Lifted lattices, hyperbolic structures and topological disorders of coupled map lattices, (1995), Auburn University, preprint
[56] Simon, B., ()
[57] Sinai, Ya.G., Gibbs measures in ergodic theory, Russian math. surveys, 27, 21-64, (1972) · Zbl 0246.28008
[58] Sinai, Ya.G., A remark concerning the thermodynamical limit of Lyapunov spectrum, (1995), preprint · Zbl 0876.34058
[59] Sinai, Ya.G., Topics in ergodic theory, (1994), Princeton University Press Princeton, NJ · Zbl 0805.58005
[60] Teman, R., Infinite dimensional dynamical systems in mechanics and physics, (1988), Springer New York
[61] Volevich, D.L., Kinetics of coupled map lattices, Nonlinearity, 4, 37-45, (1991) · Zbl 0778.58043
[62] Volevich, D.L., The Sinai-Bowen-Ruelle measure for a multidimensional lattice of interacting hyperbolic mappings, Russian acad. dokl. math., 47, 117-121, (1993) · Zbl 0823.58025
[63] Volevich, D.L., Construction of an analogue of Sinai-Bowen-Ruelle measure for a multidimensional lattice of interacting hyperbolic mappings, Russian acad. math. sbornik, 79, 347-363, (1994)
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.