Advertisement

Coupled Map Lattices: at the Age of Maturity

  • L.A. Bunimovich
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 671)

Abstract

Coupled Map Lattices (CML) were simultaneously and independently introduced by K. Kaneko, R. Kapral and S. Kuznetsov in 1983–84 [1, 2, 3, 4, 5, 6]. CML describe the time evolution of fields that can be split into an independent evolution of local systems (elements) of these fields (usually defined by some map of a local phase space) followed by (spatial) interactions of these local systems generated by some operator acting on the entire (global) phase space of CML. A structure of local systems in CML forms a lattice. At any moment of time all values of local variables are defined. These values determine a spatial structure (pattern) of the field. In CML it is assumed that all local dynamical systems are identical and that the spatial interactions between any local system and the rest of CML are the same for all local systems. (In more general Lattice Dynamical Systems (LDS) dynamics is not assumed to be a composition of a local dynamics and spatial interactions and neither to be translationally invariant.)

Keywords

Invariant Measure Coherent Structure Local System Local Dynamic Spatial Interaction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    K. Kaneko, Period doubling of kink-antikink patterns, quasiperiodicity in antiferro-like structures and spatial intermittency in coupled map lattices, Prog. Theor. Phys. 72 (1984) 480-486.Google Scholar
  2. 2.
    K. Kaneko, Spatial period-doubling in open flow, Phys. Lett. A111 (1985) 321-325.Google Scholar
  3. 3.
    I. Waller and R. Kapral, Spatial and temporal structure in systems of coupled nonlinear oscillators, Phys. Rev. A30 (1984) 2047-2055.Google Scholar
  4. 4.
    R. Kapral, Pattern formation in two-dimensional arrays of coupled discrete time oscillators, Phys. Rev. A31 (1985) 3868-3879.Google Scholar
  5. 5.
    S. P. Kuznetsov, On critical behavior of one-dimensional lattices, Pis'ma J. Technich. Fiz. 9 (1983), 94-98 (in Russian).Google Scholar
  6. 6.
    S. P. Kuznetsov, On model description of coupled dynamical systems near the transition point order-disorder, Izv. VUZov-Fizika 27 (1984) 87-96 (in Russian).Google Scholar
  7. 7.
    J. P. Crutchfield and K. Kaneko, Phenomenology of spatio-temporal chaos. In: Directions in Chaos (ed. Hao Bai-lin), World Scientific, Singapore, 1987, 272-353.Google Scholar
  8. 8.
    Theory and Applications of Coupled Map Lattices (ed. by K. Kaneko), John Wiley, Chichester, 1993.Google Scholar
  9. 9.
    Focus Issue on Coupled Map Lattices (ed. by K. Kaneko), Chaos 2 (1993).Google Scholar
  10. 10.
    Lattice Dynamics (ed. by H. Chaté and M. Courbage), Physica D 103 (1997), 1-612.Google Scholar
  11. 11.
    A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms of Lebesgue spaces, Dokl. Akad. Nauk USSR 119 (1958), 861-864.Google Scholar
  12. 12.
    K. Kaneko and I. Tsuda, Chaos and Beyond, Springer, Berlin, 2000.Google Scholar
  13. 13.
    L. A. Bunimovich and Ya. G. Sinai,Space-time chaos in coupled map lattices, Nonlinearity1(1988) 491-519.CrossRefGoogle Scholar
  14. 14.
    Ya. G. Sinai, Gibbs measures in ergodic theory, Russ. Math. Surveys 27 (1972), 21-69.Google Scholar
  15. 15.
    D. Ruelle, Thermodynamic Formalism, Addison-Wesley, London, 1978.Google Scholar
  16. 16.
    R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975) 181-202.CrossRefGoogle Scholar
  17. 17.
    L. A. Bunimovich and Ya. G. Sinai, Statistical mechanics of coupled map lattices. In [8]: pp. 169-190.Google Scholar
  18. 18.
    J. Bricmont and A. Kupiainen, Coupled analytic maps, Nonlinearity 8 (1995) 379-396.CrossRefGoogle Scholar
  19. 19.
    J. Bricmont and A. Kupiainen, High temperature expansions and dynamical systems, Comm. Math. Phys. 178 (1996) 703-732.Google Scholar
  20. 20.
    J. Bricmont and A. Kupiainen, Infinite-dimensional SRB measures, Phys. D. 103 (1997) 18-33.CrossRefGoogle Scholar
  21. 21.
    V. Baladi, M. Degli Esposti, S. Isola, E. Järvenpää and A. Kupiainen, The spectrum of weakly coupled map lattices, J. Math. Pures Appl. 77 (1998) 539-584.CrossRefGoogle Scholar
  22. 22.
    Ya. B. Pesin and Ya. G. Sinai, Space-time chaos in chains of weakly interacting hyperbolic mappings, Advances in Soviet Math. 3 (1991) 165-198.Google Scholar
  23. 23.
    M. Jiang and Ya. B. Pesin, Equilibrium measures for coupled map lattices, Comm. Math. Phys. 193 (1998) 675-711.CrossRefGoogle Scholar
  24. 24.
    M. Jiang, Equilibrium states for lattice models of hyperbolic type, Nonlinearity 8 (1995) 631-659.CrossRefGoogle Scholar
  25. 25.
    M. Jiang, SRB measures for lattice dynamical systems, J. Stat. Phys. 111 (2003) 863-902.CrossRefGoogle Scholar
  26. 26.
    G. Keller and R. Zweimüller, Unidirectionally coupled interval maps: Between dynamics and statistical mechanics, Nonlinearity 15 (2002) 1-24.CrossRefGoogle Scholar
  27. 27.
    D. L. Volevich, Kinetics of coupled map lattices, Nonlinearity 4 (1991) 37-45.CrossRefGoogle Scholar
  28. 28.
    D. L. Volevich, The SRB measure for a multi-dimensional lattice of interacting hyperbolic mappings, Dokl. Akad. Nauk USSR 47 (1993) 117-121.Google Scholar
  29. 29.
    D. L. Volevich, Construction of an analogue of SRB measure for a multidimensional lattice of interacting hyperbolic mappings, Russ. Acad. Math. Sbornik 79 (1994) 347-363.Google Scholar
  30. 30.
    E. Järvenpää, An SRB-measure for globally coupled circle maps, Nonlinearity 10 (1997) 1435-1469.CrossRefGoogle Scholar
  31. 31.
    E. Järvenpää and M. Järvenpää, On the definition of SRB measures for coupled map lattices, Comm. Math. Phys. 220 (2001) 109-143.Google Scholar
  32. 32.
    L. A. Bunimovich, Lattice dynamical systems. In: From Finite to Infinite Dimensional Dynamical Systems (ed. by J. C. Robinson and P. A. Glendinning), Kluwer, Dordrecht, 2001.Google Scholar
  33. 33.
    P. Grassberger and T. Schreiber, Phase transitions in coupled map lattices, Physica D 50 (1991) 171-188.CrossRefGoogle Scholar
  34. 34.
    S. Sakaguchi, Phase transitions in coupled Bernoulli maps, Progr. Theor. Phys. 80 (1992) 7-13.Google Scholar
  35. 35.
    H. Chaté and P. Manneville, Collective behavior in spatially extended systems with local interactions and synchronous updating, Progr. Theor. Phys. 87 (1992) 1-60.Google Scholar
  36. 36.
    J. Miller and D. A. Huse, Macroscopic equilibrium from microscopic irreversibility in a chaotic coupled map lattice, Phys. Rev. F 48 (1993) 2528-2534.Google Scholar
  37. 37.
    C. Boldrighini, L. A. Bunimovich, G. Cosimi, S. Frigio and A. Pellegrinotti, Ising type transitions in coupled map lattices, J. Stat. Phys. 80 (1995) 1185-1205.CrossRefGoogle Scholar
  38. 38.
    L. A. Bunimovich and E. Carlen, On the problem of stability in lattice dynamical systems, J. Diff. Eqs. 123 (1995) 213-229.CrossRefGoogle Scholar
  39. 39.
    P. Marcq and H. Chaté, Early-time critical dynamics in lattices of coupled chaotic maps, Phys. Rev. E 57 (1998) 1591-1603.CrossRefGoogle Scholar
  40. 40.
    M. L. Blank, Generalized phase transitions in finite coupled map lattices, Physica D 103 (1997) 34-50.CrossRefGoogle Scholar
  41. 41.
    M. L. Blank, Stability and Localization in Chaotic Dynamics, Moscow, MCCME, 2001.Google Scholar
  42. 42.
    G. Keller and M. Künzle, Transfer operators for coupled map lattices, Erg. Th. and Dyn. Syst. 12 (1992) 297-318.Google Scholar
  43. 43.
    G. Gielis and R. S. MacKay, Coupled map lattices with phase transitions, Nonlinearity 13 (2000) 867-888.CrossRefGoogle Scholar
  44. 44.
    M. L. Blank and L. A. Bunimovich, Multicomponent dynamical systems: SRB measures and phase transitions, Nonlinearity 16 (2003) 387-401.CrossRefGoogle Scholar
  45. 45.
    C. Boldrighini, L. A. Bunimovich, G. Cosimi, S. Frigio and A. Pellegrinotti, Ising-type and other transitions in one-dimensional coupled map lattices with sign symmetry, J. Stat. Phys. 102 (2001) 1271-1283.CrossRefGoogle Scholar
  46. 46.
    W. Just, Globally coupled maps: phase transitions and synchronization, Physica D 81 (1995) 317-340.CrossRefGoogle Scholar
  47. 47.
    S. Lepri and W. Just, Mean field theory of critical coupled map lattices, J. Phys. A 31 (1998) 6175-6181.CrossRefGoogle Scholar
  48. 48.
    W. Just, Equilibrium phase transitions in coupled map lattices: a pedestrian approach, J. Stat. Phys. 105 (2001) 133-142.CrossRefGoogle Scholar
  49. 49.
    S. Aubry, The concept of anti-integrability: definitions, theorems and applications. In Twist Mappings and Their Applications (ed. by R. McGehee and K. E. Meyer), Springer, 1992, 7-54.Google Scholar
  50. 50.
    R. S. MacKay, Dynamics of networks: features which persist from the uncoupled limit. In Stochastic and Spatial Structures of Dynamical Systems (ed. by S. J. van Strien and S. M. Verduyn Lunel), North Holland, 1996, 81-104.Google Scholar
  51. 51.
    S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization, Physica D 103 (1997) 201-250.CrossRefGoogle Scholar
  52. 52.
    I. S. Aranson, V. S. Afraimovich and M. I. Rabinovich, Stability of spatially homogeneous chaotic regimes in unidirectional chains, Nonlinearity 3 (1990) 639-651.CrossRefGoogle Scholar
  53. 53.
    V. S. Afraimovich and L. A. Bunimovich, Simplest structures in coupled map lattices and their stability, Random and Comp. Dynamics 1 (1993) 423-444.Google Scholar
  54. 54.
    V. S. Afraimovich and Ya. B. Pesin, Traveling waves in lattice models of multidimensional and multi-component media, Nonlinearity 6 (1993) 429-455.CrossRefGoogle Scholar
  55. 55.
    V. S. Afraimovich and V. I. Nekorkin, Chaos of traveling waves in a discrete chain of diffusively coupled maps, Int. J. of Bif. and Chaos 4 (1994) 631-637.CrossRefGoogle Scholar
  56. 56.
    A. Pikovsky, M. Rosenblum and J. Kurts, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Univ. Press, 2002.Google Scholar
  57. 57.
    A. L. Toom, Stable and attractive trajectories in multicomponent systems. In: Multicomponent Random Systems (ed. by R. L. Dobrushin and Ya. G. Sinai), Dekker, NY, 1980.Google Scholar
  58. 58.
    J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57 (1985) 617-656.CrossRefGoogle Scholar
  59. 59.
    D. J. Farmer, E. Ott and J. A. Yorke, The dimension of chaotic attractors, Physica D 7 (1983) 153-180.CrossRefGoogle Scholar
  60. 60.
    T. Inoue, Sojourn times in small neighborhoods of indifferent fixed points of one-dimensional dynamical systems, Erg. Th. Dyn. Sys. 20 (2000) 241-258.CrossRefGoogle Scholar
  61. 61.
    G. Keller, Completely mixing maps without limit measure, Colloq. Math. 100 (2004), no. 1, 73-76.Google Scholar
  62. 62.
    M. Misiurewicz and A. Zdunik, Convergence of images of certain measures. In Statistical Physics and Dynamical Systems (ed. by J. Fritz, A. Jaffe and D. Szasz), Progr. Phys. Vol. 10, Birkhäuser, Boston, 1985, 203-219.Google Scholar
  63. 63.
    L.-S. Young, What are SRB measures and which dynamical systems have them, J. Stat. Phys. 108 (2002) 733-754.CrossRefGoogle Scholar
  64. 64.
    M. Viana, Dynamical systems: moving into the next century. In: Mathematics Unlimited: 2001 and Beyond, Springer-Verlag, Berlin, 2001, 1167-1178.Google Scholar
  65. 65.
    L. A. Bunimovich and S. Venkatagiri, Onset of chaos in coupled map lattices via the peak-crossing bifurcation, Nonlinearity 9 (1996) 1281-1296.CrossRefGoogle Scholar
  66. 66.
    L. A. Bunimovich and D. Turaev, Localized solutions in lattice systems and bifurcations caused by spatial interactions, Nonlinearity 11 (1998) 1539-1545.CrossRefGoogle Scholar
  67. 67.
    P. Frederickson, J. L. Kaplan, E. D. Yorke, and J. A. Yorke, The Lyapunov dimension of strange attractors, J. Diff. Eqs. 49 (1983) 185-207.CrossRefGoogle Scholar

Authors and Affiliations

  • L.A. Bunimovich
    • 1
  1. 1.Georgia Institute of TechnologySchool of MathematicsAtlantaUSA

Personalised recommendations