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Lyapunov Exponents and Synchronization of Cellular Automata

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Complex Systems

Part of the book series: Nonlinear Phenomena and Complex Systems ((NOPH,volume 6))

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Abstract

In these notes we discuss the concept of Lyapunov exponents of cellular automata (CA). We also present a synchronization mechanism for CA. We begin with an introduction to CA, introduce the concept of Boolean derivative and show that any CA has a finite expansion in terms of the Boolean derivatives. The Lyapunov exponents are defined as the rate of exponential growth of the linear part of this expansion using a suitable norm. We then present a simple mechanism for the synchronization of CA and apply it to totalistic one-dimensional CA. The CA with a nonzero synchronization threshold exhibit complex nonperiodic space time patterns and vice versa. This synchronization transition is related to directed percolation. The synchronization threshold is strongly correlated to the maximum Lyapunov exponent and we propose approximate relations between these quantities.

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References

  1. Alves, A.S. ed.(1991) Discrete Models of Fluid Dynamics, World Scientific Publishing Co., Inc., River Edge, NJ.

    MATH  Google Scholar 

  2. Bagnoli, F. (1992) Boolean Derivatives and Computation of Cellular Automata, Int. J. Mod. Phys. C, 3, p. 307.

    Article  MathSciNet  MATH  Google Scholar 

  3. Bagnoli, F., Baroni, L. and Palmerini, P. (1999) Synchronization and Directed Percolation in Coupled Map Lattices, Phys. Rev. E, 59, pp. 409–416.

    Article  Google Scholar 

  4. Bagnoli, F., Palmerini, P. and Rechtman, R. (1997) Algorithmic Mapping from Criticality to Self-Organized Criticality, Phys. Rev. E, 55, p. 3970.

    Article  Google Scholar 

  5. Bagnoli, F., Rechtman, R. and Ruffo, S. (1992) Damage Spreading and Lyapunov Exponents in Cellular Automata, Phys. Lett. A, 172, pp. 34–38.

    Article  Google Scholar 

  6. Bagnoli, F., Rechtman, R. and Ruffo, S. Maximal Lyapunov Exponent for ID Cellular Automata, in [11].

    Google Scholar 

  7. Bagnoli, F., Rechtman, R. and Ruffo, S. (1994) Lyapunov Exponents for Cellular Automata, in M. López de Haro, C. Varea eds., Lectures on Thermodynamics and Statistical Mechanics, World Scientific Publishing Co., Singapore.

    Google Scholar 

  8. Bagnoli, F. and Rechtman, R. (1999) Synchronization and Maximum Lyapunov Exponents of Cellular Automata, Phys. Rev. E, 59, p. R1307–R1310.

    Article  Google Scholar 

  9. Benettin, G., Galgani, L., Giorgilli, A. and Strelcyn, J.-M. (1980) Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems: A method for Computing all of them, Part 1: Meccanica, p. 9; Part 2: Meccanica, p. 21.

    Google Scholar 

  10. Berlekamp, E., Conway, J. and Guy, R. eds, (1982) Winning Ways, Vol. 2, chap. 25, Academic Press, New York.

    MATH  Google Scholar 

  11. Boccara, N., Goles, E., Martinez, S. and Picco, P. eds. (1993) Cellular Automata and Cooperative Systems, Kluwer Academic Publishers Group, Dordrecht.

    MATH  Google Scholar 

  12. Bozoyan, S.E. (1978) Some Properties of Boolean Differentials and of Activities of Arguments of Boolean Functions, Prof. Pederachi Infor., 14(1), pp. 77–89.

    MATH  Google Scholar 

  13. Cohen, E.G.D. (1992) New Types of Diffusion in Lattice Gas Cellular Automata, in M. Mareschal, B. L. Holian eds., Microscopic Simulations of Complex Hydrodynamic Phenomena, Plenum Press, New York.

    Google Scholar 

  14. Devaney, R.L. (1989) An Introduction to Chaotic Dynamical Systems, 2nd. edition, Addison Wesley Publishing Company, Redwood City, CA.

    MATH  Google Scholar 

  15. Domany, E. (1984) Exact Results for Two — and Three — Dimensional Ising and Potts Models, Phys. Rev. Lett., 52, p. 871.

    Article  MathSciNet  Google Scholar 

  16. Doolen, G.D. ed. (1989) Lattice Gas Methods, Theory, Applications and Hardware, MIT Press, Cambridge, MA.

    Google Scholar 

  17. Frisch, U., d’Humières, D., Hasslacher, B., Lallemand, P., Pomeau, Y. and Rivet, J.P. (1987) Complex Systems, 1, p. 649.

    MathSciNet  MATH  Google Scholar 

  18. Frisch, U., Hasslacher, B. and Pomeau, Y. (1986) Lattice Gas Automata for the Navier Stokes Equation, Phys. Rev. Lett., 56, p. 1505.

    Article  Google Scholar 

  19. Fujisaka, H. (1983) Stability Theory of Synchronized Motions in Coupled-Oscillator Systems, Prog. Them. Phys., 70, p. 1264.

    Article  MathSciNet  MATH  Google Scholar 

  20. Ghilezan, C. (1982) Les Dérivées Partielles des Fonctions Pseudo-Booléennes Généralisées, Discrete Appliesd Math., 4, pp. 37–45.

    Article  MathSciNet  MATH  Google Scholar 

  21. Grassberger, P. (1982) Z. Phys. B, 47, p. 365.

    Article  MathSciNet  Google Scholar 

  22. Grassberger, P. (1995) J. Stat. Phys., 79, p. 13.

    Article  MATH  Google Scholar 

  23. Grassberger, P. Synchronization of Coupled Systems with Spatiotempral chaos, unpublished, http://www.xxz.lanl.gov./cond-mat/9808199.

  24. Guckenheimer, J. (1979) Sensitive Dependence on Initial Conditions for One Dimensional Maps, Commun. Math. Phys., 70, p. 113.

    Article  MathSciNet  Google Scholar 

  25. Gutowitz H. ed. (1990) Cellular Automata: Theory and Experiment, North Holland.

    Google Scholar 

  26. Hardy, H., Pomeau, Y. and de Pazzis, O. (1973) J. Math. Phys., 14, p. 1736; H. Hardy, Y. Pomeau, O. de Pazzis (1976) Phys. Rev. A, 13, p. 1949.

    Article  Google Scholar 

  27. Herrmann, H.J. (1984) J. Stat. Phys., 32, p. 271.

    Google Scholar 

  28. Janssen, H.K. (1981) Z. Phys. B, 42, p. 151.

    Article  Google Scholar 

  29. Kapral, R. (1991) Discrete Models for Chemically Reacting Species, J. Math. Chem., 6, p. 113; R. Kapral, A. Lawniczak, P. Maziar, Complex Dynamics in Reactive Lattice-gas Models, in M. López de Haro, C. Varea eds. (1991) Lectures on Thermodynamics and Statistical Mechanics, World Scientific Publishing Co., Singapore.

    Article  MathSciNet  Google Scholar 

  30. Kinzel, W. (1983) in Percolation Structures and Processes, G. Deutsch, R. Zallen and J. Adler eds, Hilger, Bristol.

    Google Scholar 

  31. Kinzel, W. (1985) Phase Transitions of Cellular Automata, Z. Phys. B., 58, p. 229.

    Article  MathSciNet  MATH  Google Scholar 

  32. Kong, X.P. and Cohen, E.G.D. (1991) Diffusion and Propagation in a Triangular Lorentz Lattice Gas, J. Stat. Phys., 62, p. 737.

    Article  Google Scholar 

  33. Kong, X.P. and Cohen, E.G.D. (1991) A Kinetic Theorist’s look at Lattice Gas Cellular Automata, Physica D, 47, p. 9.

    Article  Google Scholar 

  34. Li, W., Packard, N.H. and Langton, CG. (1990) Transition Phenomena in Cellular Automata Rule Space, Physica, d 45, pp. 77–94.

    MathSciNet  Google Scholar 

  35. Lomnitz-Adler, J., Knopoff, L. and Martfnez-Mekler, G. (1992) Avalanches and Epidemic Models of Fracturing in Earthquakes, Phys. Rev. A, 45, p. 2211.

    Article  Google Scholar 

  36. Lübeck, S., Schreckenberg, M. and Usadel, K.D. (1998) Density Fluctuations and Phase Transition in the Nagel-Schreckenberg Traffic Flow Model, Phys. Rev E, 57, p. 1171.

    Article  Google Scholar 

  37. Manneville, P. Boccara, N., Vichniac, G.Y. and Bidaux, R. eds. (1989) Cellular Automata Modelling of Complex Physical Systems, Springer-Verlag, Berlin.

    Google Scholar 

  38. Morelli, L.G. and Zanette, D.H. (1998) Synchronization of Stochastically Coupled Cellular Automata, Phys. Rev E, 58, p. R8.

    Article  Google Scholar 

  39. Nagel, K. and Schreckenberg, M. (1992) J. Physique, 2, p. 2221.

    Article  Google Scholar 

  40. Noble, B. (1959) Applied Linear Algebra, Prentice Hall, Inc., Englewood Cliffs, N.J., chap. 13.

    Google Scholar 

  41. Oseledec, V.I. (1968) Trans. Moscow Math. Soc., 18, p. 21.

    Google Scholar 

  42. Ott, E. (1993) Chaos in Dynamical Systems, Cambridge University Press, New York.

    MATH  Google Scholar 

  43. Packard, N. (1986) Lattice Models for Solidification and Aggregation, in Y. Katoh et al. eds., Procceedings of the First International Symposium for Science on Form, KTK Scientific Publisher, Dordrecht.

    Google Scholar 

  44. Pikovsky, A.S. and Grassberger, P. (1991) J. Phys. A: Math. Gen., 24, p. 4587.

    Article  MathSciNet  Google Scholar 

  45. Pomeau, Y. (1984) Invariant in Cellular Automata, J. Phys., A17, L415.

    MathSciNet  Google Scholar 

  46. Rechtman, R., Salcido, A. and Calles, A. (1991) The Ehrenfest’s Wind-Tree Model and the Hypothesis of Molecular Chaos, Eur. J. Phys., 12, p. 27.

    Article  Google Scholar 

  47. Robert, F. (1986) Discrete Iterations, Springer-Verlag, Berlin.

    Book  MATH  Google Scholar 

  48. Schadschneider, A. and Schreckenberg, M. (1993) Cellular Automaton Models of Traffic Flow, J. Phys A, 26, p. L679.

    Article  Google Scholar 

  49. Seiden, P.E. and Schulman, L.S. (1990) Percolation Model of Galactic Structure, Adv. Phys., 39, p. 1.

    Article  MathSciNet  Google Scholar 

  50. Shereshevsky, M.A. (1992) Lyapunov Exponents for One Dimensional Cellular Automata, J. Nonlinear Sci., 2, p. 1.

    Article  MathSciNet  MATH  Google Scholar 

  51. Signorini, J. Complex Computing with Cellular Automata, in Ref. [37].

    Google Scholar 

  52. Thayse, A. and Davio, A. (1973) Boolean Differential Calculus and its Application to Switching Theory, IEEE Transactions on Computers, C22(4), pp. 409–420.

    Article  MathSciNet  Google Scholar 

  53. Ulam, S. (1962) On Some Mathematical Problems Connected with Patterns of Growth and Figures, Proc. of Symposia in Applied Mathematics, 14, pp. 215–224, (American Mathematical Society, Rhode Island; R. Schrandt, S. Ulam, On Recursively Defined Geometrical Objects and Patterns of Growth, in A. Burks ed., Essays in Cellular Automata, University of Illinois Press, 1970.

    Google Scholar 

  54. Urías, J., Enciso, A. and Rechtman, R. (1997) Sensitive Dependence on Initial Conditions for Cellular Automata, Chaos, 7, p. 688.

    Article  MathSciNet  MATH  Google Scholar 

  55. Urías, J., Salazar, G. and Ugalde, E. (1998) Synchronization of Cellular Automaton Pairs, Chaos, 8, pp. 1–5.

    Article  Google Scholar 

  56. Vichniac, G. (1990) Boolean Derivatives on Cellular Automata, Physica D, 45, p. 63. Reprinted in Ref. [25].

    Article  MathSciNet  MATH  Google Scholar 

  57. von Neumann, J. (1996) Theory of Self-Reproducing Automata, University of Illinois Press. Completed and edited by A. W. Burks.

    Google Scholar 

  58. Wolfram, S. (1983) Statistical Mechanics of Cellullar Automata, Rev. Mod. Phys., 55, 601. Reprinted in Ref. [61], pp. 7–50.

    Article  MathSciNet  Google Scholar 

  59. Wolfram, S. (1984) Universality and Complexity in Cellular Automata, Physica, 10D, p. 1. Reprinted in [61] p. 91.

    MathSciNet  Google Scholar 

  60. Wolfram, S. (1986) Cellular Automata Fluids 1: Basic Theory, J. Stat. Phys., 45, p. 471.

    Article  MathSciNet  MATH  Google Scholar 

  61. Wolfram, S. ed. (1986) Theory and Applications of Cellular Automata, World Scientific Publishing Co., Singapore.

    MATH  Google Scholar 

  62. Zabolitzky, J.G., Herrmann, H.J. (1988) J. Comp. Phys., 76, p. 426.

    Article  MATH  Google Scholar 

  63. Ziff, R.M., Kong, X.P. and Cohen, E.G.D. (1991) A Lorentz Lattice Gas and Kinetic Walk Model, Phys. Rev. A, 44, p. 2410.

    Article  Google Scholar 

  64. A guide to recent literature can be found in http: //alife.santafe.edu/alife/topics/cas/ca-faq/biblio/biblio.html

    Google Scholar 

  65. The Contributions in the Column on Mathematical Games in Scientific American are reprinted in M. Gardner, Wheels, Life and Other Mathematical Amusements, W. H. Freeman and Co., San Francisco (1983).

    Google Scholar 

  66. See the Appendix of Ref. [61].

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica Applicata, Università di Firenze, Via S. Marta 3, I-50139, Firenze, Italy

    Franco Bagnoli

  2. Centro de Investigatión en Energía, Universidad Nacional Autónoma de México, Apdo. Postal No. 34, Temixco, 62580, Morelos, Mexico

    Raul Rechtman

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  1. Franco Bagnoli
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  2. Raul Rechtman
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Editors and Affiliations

  1. Department of Mathematical Engineering, Faculty of Physical and Mathematical Sciences, University of Chile, Santiago, Chile

    Eric Goles  & Servet Martínez  & 

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Bagnoli, F., Rechtman, R. (2001). Lyapunov Exponents and Synchronization of Cellular Automata. In: Goles, E., Martínez, S. (eds) Complex Systems. Nonlinear Phenomena and Complex Systems, vol 6. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0920-1_2

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  • DOI: https://doi.org/10.1007/978-94-010-0920-1_2

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