Advertisement

Space–Time Dynamics

  • José María AmigóEmail author
Chapter
Part of the Springer Series in Synergetics book series (SSSYN)

Abstract

All applications of ordinal analysis hitherto had to do with time series analysis or abstract dynamical systems. A remaining challenge is to expand the applications to physical systems.

In order to tackle the viability of this program, we are going to study the permutation complexity of two simple models of spatially extended physical systems: cellular automata (CA) and coupled map lattices (CMLs). CA were presented in Sect. 1.5. CMLs can be considered as a generalization of the CA; they retain the space coarse graining of the CA, but the state variable take on real values. Despite their apparent simplicity, these are the preferred models when studying the emergence of collective phenomena (such as turbulence, space–time chaos, symmetry breaking, ordering) in systems of many particles interacting nonlinearly. Indeed, their ability to reproduce complex phenomena in, say, fluid dynamics and solid state physics, is impressive. For this reason, they are the ideal choice for our purpose.

Keywords

Cellular Automaton Cellular Automaton Topological Entropy Local Rule Ordinal Pattern 
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.

  1. 18.
    C. Anteneodo, A.M. Batista, and R.L. Viana, Synchronization threshold in coupled logistic map lattices, Physica D. Nonlinear Phenomena 223 (2006) 270–275.zbMATHCrossRefMathSciNetADSGoogle Scholar
  2. 19.
    N. Aoki and K. Hiraide, Topological theory of dynamical systems. North Holland, Amsterdam, 1994.Google Scholar
  3. 23.
    H. Atmanspacher and H. Scheingraber, Inherent global stabilization of unstable local behavior in coupled map lattices, International Journal of Bifurcation and Chaos 15 (2005) 1665–1676.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 42.
    L.A. Bunimovich and Y.G. Sinai, Space-time chaos in coupled map lattices, Nonlinearity 1 (1988) 491–518.zbMATHCrossRefMathSciNetADSGoogle Scholar
  5. 44.
    L.A. Bunimovich, Coupled map lattices: Some topological and ergodic properties, Physica D. Nonlinear Phenomena 103 (1997) 1–17.CrossRefMathSciNetADSzbMATHGoogle Scholar
  6. 46.
    R. Carretero-González, Low dimensional travelling interfaces in coupled map lattices, International Journal of Bifurcations and Chaos 7 (1997) 2745–2754.zbMATHCrossRefGoogle Scholar
  7. 47.
    R. Carretero-González, D.K. Arrowsmith, and F. Vivaldi, One-dimensional dynamics for traveling fronts in coupled map lattices, Physical Review E 61 (2000) 1329–1336.CrossRefMathSciNetADSGoogle Scholar
  8. 50.
    H. Chaté and P. Manneville, Coupled map lattices as cellular automata, Journal of Statistical Physics 56 (1989) 357–370.CrossRefMathSciNetADSGoogle Scholar
  9. 54.
    L.O. Chua, V.I. Sbitnev, and S. Yoon, A nonlinear dynamics perspective of Wolfram’s New Kind of Science –Part II: Universal neuron, International Journal of Bifurcation and Chaos 13 (2003) 2377–2491.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 55.
    L.O. Chua, V.I. Sbitnev, and S. Yoon, A nonlinear dynamics perspective of Wolfram’s new kind of science –Part IV: From Bernoulli shift to 1/f spectrum, International Journal of Bifurcation and Chaos 15 (2005) 1045–1183.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 58.
    M. Courbage, D. Mercier, and S. Yasmineh, Traveling waves and chaotic properties in cellular automata, Chaos 9 (1999) 893–901.zbMATHCrossRefMathSciNetADSGoogle Scholar
  12. 62.
    M. D’amico, G. Manzini, and L. Margara, On computing the entropy of cellular automata, Theoretical Computer Science 290 (2003) 1629–1646.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 71.
    Y. Dobyns and H. Atmanspacher, Characterizing spontaneous irregular behavior in coupled map lattices, Chaos, Solitons & Fractals 24 (2005) 313–327.zbMATHADSGoogle Scholar
  14. 92.
    G.A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Mathematical Systems Theory 3 (1969) 320–375.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 103.
    N. Israeli and N. Goldenfeld, Coarse-graining of cellular automata, emergence, and the predictability of complex systems, Physical Review E 73 (2006) 1–17.CrossRefMathSciNetGoogle Scholar
  16. 104.
    S. Jalan, J. Jost, and F.M. Atay, Symbolic synchronization and the detection of global properties of coupled dynamics from local information, Chaos 16 (2006) 033124.CrossRefADSGoogle Scholar
  17. 106.
    K. Kaneko, Transition from torus to chaos accompanied by frequency lockings with symmetry breaking, Progress in Theoretical Physics 69 (1983) 1427–1442.zbMATHCrossRefADSGoogle Scholar
  18. 107.
    K. Kaneko, Period-doubling of kink-antikink patterns, quasiperiodicity in anti-ferro-like structures and spatial intermittency in coupled logistic lattice, Progress in Theoretical Physics 72 (1984) 480–486.zbMATHCrossRefADSGoogle Scholar
  19. 108.
    K. Kaneko, Pattern dynamics in spatiotemporal chaos, Physica D. Nonlinear Phenomena 34 (1989) 1–41.zbMATHCrossRefMathSciNetADSGoogle Scholar
  20. 110.
    K. Kaneko, Chaotic traveling waves in a coupled map lattice, Physica D. Nonlinear Phenomena 68 (1993) 299–317.zbMATHCrossRefMathSciNetADSGoogle Scholar
  21. 123.
    B.P. Kitchens, Symbolic Dynamics. Springer Verlag, Berlin, 1998.Google Scholar
  22. 132.
    J. Kurths, D. Maraun, C.S. Zhou, G. Zamora-López, and Y. Zou, Dynamics in complex systems, European Review 17 (2009), 357–370.CrossRefGoogle Scholar
  23. 141.
    G. Manzini and L. Margara, A complete and efficiently computable topological classification of linear cellular automata over \(\mathbb{Z}_{m}\), Theoretical Computer Science 221 (1999) 157–177.zbMATHCrossRefMathSciNetGoogle Scholar
  24. 145.
    C. Masoller and A.C. Martí, Random delays and the synchronization of chaotic maps, Physical Review Letters 94 (2005) 134102.CrossRefADSGoogle Scholar
  25. 162.
    M. Newman, A.L. Barabási, and D.J. Watts, The Structure and Dynamics of Networks. Princeton University Press, Princeton, 2006.Google Scholar
  26. 170.
    S.D. Pethel, N.J. Corron, and E. Bollt, Symbolic dynamics of coupled map lattices, Physical Review Letters 96 (2006) 034105.CrossRefADSGoogle Scholar
  27. 171.
    S.D. Pethel, N.J. Corron, and E. Bollt, Deconstructing spatiotemporal chaos using local symbolic dynamics, Physical Review Letters 99 (2007) 214101.CrossRefADSGoogle Scholar
  28. 188.
    M.A. Shereshevsky, Expansiveness, entropy and polynomial growth for groups acting on subshifts by automorphisms. Indagationes Mathematicae 4 (1993) 203–210.zbMATHCrossRefMathSciNetGoogle Scholar
  29. 192.
    D. Sotelo Herrera and J. San Martín, Analytical solutions of weakly coupled map lattices using recurrence relations, Physics Letters A 373 (2009) 2704–2709.CrossRefMathSciNetADSGoogle Scholar
  30. 204.
    A. Wolf, J.B. Swift, H.L. Swinney, and J.A. Vastano, Determining Lyapunov exponents from a time series, Physica D. Nonlinear Phenomena 16 (1985) 285–317.zbMATHCrossRefMathSciNetADSGoogle Scholar
  31. 206.
    S. Wolfram, Universality and complexity in cellular automata, Physica 10D (1984) 1–35.MathSciNetADSGoogle Scholar
  32. 207.
    S. Wolfram, A New Kind of Science. Wolfram Media, Champaign, 2002.Google Scholar
  33. 210.
    G.C. Zhuang, J. Wang, Y. Shi, and W. Wang, Phase synchronization and its cluster feature in two-dimensional coupled map lattices, Physical Review E 66 (2002) 046201.CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Centro de Investigacion OperativaUniversidad Miguel HernandezElcheSpain

Personalised recommendations