Chaotic Dynamical Systems as Machines

  • J. L. McCauley
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 38)

Abstract

The subject of this discussion belongs to no single well-established field but overlaps physics, mathematics and computer science. It can begin with a rather simple question, “when can a chaotic dynamical system be regarded as a machine, or as a model of a machine?” Also, “when is the output (behaviour) of a machine chaotic?” The two questions are related and one is led to them by asking, “to what extent is the orbit of a deterministic but chaotic dynamical system computable?” and, “in what sense is such a computed orbit chaotic?” In the past, appeals have been made to symbolic dynamics and to algorithmic complexity, but those efforts have not resolved these questions. The systems that we shall concentrate upon are purely deterministic, so that the chaos must be generated entirely by the dynamical system during the computation, and not by the effect of any external noise.

Keywords

Entropy Peri Alan Wolfram Carus 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J.L. McCauley, J.I. Palmore: Phys. Lett. A115, 433 (1986).CrossRefMathSciNetGoogle Scholar
  2. 2.
    J. von Neumann: Theory of Self-Reproducing Automata ( Univ. of Illinois Pr., Urbana 1966 ).Google Scholar
  3. 3.
    J.E. Hopcroft, J.D. Ullman: Introduction to Automata Theory, Language, and Computation ( Addison-Wesley, Reading, MA 1979 ).Google Scholar
  4. 4.
    D.E. Knuth: The Art of computer Programming II. Semi-Numerical Algorithms ( Addison-Wesley, Reading, MA 1981 ).Google Scholar
  5. 5.
    R. Shaw: Z. Naturforsch. 36a, 80 (1980).ADSGoogle Scholar
  6. 6.
    J.I. Palmore, J.L. McCauley: Shadowing by Computable Chaotic Orbits, to be published, Phys. Let. A (1987).Google Scholar
  7. 7.
    H.G. Schuster: Deterministic Chaos ( Physik verlag, Mosbach 1984 ).MATHGoogle Scholar
  8. 8.
    S. Grossmann, S, Thomas: Z. Naturforsch. 32a, 1353 (1977).ADSMathSciNetGoogle Scholar
  9. 9.
    C. Beck, G. Roepstorff: Univ. of Aachen preprint (1986)Google Scholar
  10. 10.
    G. Benettin, G.M. Casartelli, L. Galgani, A. Giorgilli, J.M. Strelcyn: Nuovo Cimento B44 183 (1978).CrossRefMathSciNetGoogle Scholar
  11. 11.
    J. Guckenheimer, P. Holmes: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer Verlag, New York, Berlin, Heidelberg, Tokyo 1983 ).Google Scholar
  12. 12.
    O.E. Lanford III: in Chaotic Behaviour of Deterministic Systems, ed by Ioos, Helleman, Stora ( North Holland, Amsterdam 1983 )Google Scholar
  13. 13.
    J.L. McCauley, J.I. Palmore: in Scaling of Disordered Systems,eds. R. Pynn, A. Skjeltorp ( Plenum Pr., New York 1985 ).Google Scholar
  14. 14.
    J.I. Palmore, J.L. McCauley: Statistics of Computable Hyperbolic Systems, preprint (1987).Google Scholar
  15. 15.
    A. Hodges: Alan Turing: The Enigma ( Simon and Schuster, New York 1980 ) pp. 91–110.Google Scholar
  16. 16.
    A.M. Turing: Proc. London Math. Sa (2) 42, 230 (1937).Google Scholar
  17. 17.
    M.L. Minsky: Computation, Finite and Infinite Machines ( Prentice-Hall, London 1967 ).MATHGoogle Scholar
  18. 18.
    Niven: Irrational Numbers, The Carus Mathematical Monographs No. 11 (1956).Google Scholar
  19. 19.
    M. Kac: Statistical Independence in Probability Analysis and Number Theory. The Carus Mathematical Monographs No. 12 (1959).Google Scholar
  20. 20.
    J.L. McCauley: Z. Naturforsh. 42a (1987), in press.Google Scholar
  21. 21.
    S. Waggoner: Math. Intelligencer 7, 65 (1985)MathSciNetGoogle Scholar
  22. 22.
    P. Martin-Löf: Inf. Control 9, 602 (1966).CrossRefGoogle Scholar
  23. 23.
    J. Ford: Physics Today 36, 40 (1983).CrossRefGoogle Scholar
  24. 24.
    K. Preston, Jr., M.J.B. Duff: Modern Cellular Automata ( Plenum Pr., London 1984 ).Google Scholar
  25. 25.
    S. Wolfram: Physica 10D, 1 (1984).MathSciNetGoogle Scholar
  26. 26.
    U. Frisch, B. Hasslacher, Y. Pomeau: Phys. Rev. Lett. 56, 1505 (1986); N. Margolus, T. Toffoli, G. Vichniac: Phys. Rev. Lett. 56, 1694 (1986).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • J. L. McCauley
    • 1
  1. 1.Institute for Energy TechnologyKjellerNorway

Personalised recommendations