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The perturbative method for discrete processes and its physical application

  • Géza Györgyi
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1163)

Abstract

In this note chaotic iterations are considered which only slightly differ from processes of known properties. Averages are calculated in case of noisy perturbation of systems exhibiting originally a fixed point and where the iteration goes along a discrete one-dimensional chain or on the Cayley tree. As a physical example the disordered Ising model is considered on these lattices and results on spin fluctuations are presented. Finally, probabilistic properties of perturbed one-dimensional iterations are reviewed, where even the unperturbed systems show chaotic behaviour. For the latter ones in the examples the hat function and the logistic map in the fully developed chaotic state are taken, and we also discuss a noisy iteration describing a one-dimensional disordered quantum mechanical system.

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References

  1. [1]
    Haken, H., (ed.): Chaos and Order in Nature, Springer (1981).Google Scholar
  2. [2]
    Haken, H., (ed.): Evolution of Order and Chaos, Springer(1982)Google Scholar
  3. [3]
    Garrido, L. (ed.): Dynamical Systems and Chaos, Lecture Notes in Physics 179 Springer (1983).Google Scholar
  4. [4]
    Eilenberger, G., Müller-Krumbhaar, H. (ed.): Nichtlineare Dynamik in kondensierter Materie, KFA Institut für Festkörperforschung, Jülich (1983).Google Scholar
  5. [5]
    Cohen, E.G.D. (ed.): Proceedings of the NUFFIC Summer School on Statistical Mechanics, Trondheim (1984), to appear.Google Scholar
  6. [6]
    Kramers, H.A.: Physica 7, 284 (1940).MathSciNetCrossRefGoogle Scholar
  7. [7]
    Moyal, J.E.: J.R. Stat. Soc. 11, 151 (1949).MathSciNetGoogle Scholar
  8. [8]
    Bruinsma, R., Aeppli, G.: Phys. Rev. Lett. 50, 1494 (1983).MathSciNetCrossRefGoogle Scholar
  9. [9]
    Bruinsma, R., Aeppli, G.: Phys. Lett. 97A, 177 (1983).MathSciNetGoogle Scholar
  10. [10]
    Györgyi, G., Ruján, P.: J. Phys. C: Solid St. Phys. 17, 4207 (1984).CrossRefGoogle Scholar
  11. [11]
    Sütö, A., Zimányi, G.T.: Lecture Notes in Physics 206, Springer (1984).Google Scholar
  12. [12]
    Derrida, B., Pomeau Y., Vannimenus, J.: J. Phys. C: Solid St. Phys. 11, 4749 (1978).CrossRefGoogle Scholar
  13. [13]
    Cayley, A.: Coll. Math. Papers 3, 242 (1889).Google Scholar
  14. [14]
    Györgyi, G., Szépfalusy, P.: J. Stat. Phys. 34, 451 (1984).CrossRefGoogle Scholar
  15. [15]
    Misiurewicz, M.: Publ. Math. IHES 53, 17 (1981).MathSciNetCrossRefGoogle Scholar
  16. [16]
    Lasota, A., Yorke, J.A.: Trans. Am. Math. Soc. 183, 481 (1973).MathSciNetCrossRefGoogle Scholar
  17. [17]
    Grossmann, S., Thomae, S.: Z. Naturforsch. 32a, 1353 (1977).MathSciNetGoogle Scholar
  18. [18]
    Keller, G.: Analytic Perturbations of N-Fold Covering Interval Mappings, (1984) unpublished.Google Scholar
  19. [19]
    Geisel, T., Heldstab, J., Thomas, H.: Z. Phys. B55, 165 (1984).MathSciNetCrossRefGoogle Scholar
  20. [20]
    Grossmann, S.: Z. Phys. B57, (1984) to appear.Google Scholar
  21. [21]
    Derrida, B., Gardner, E.: J. de Phys. 45, 1283 (1984).MathSciNetCrossRefGoogle Scholar
  22. [22]
    Ishii, K.: Sup. Prog. Theor. Phys. 53, 77 (1973).CrossRefGoogle Scholar
  23. [23]
    Sarker, S.: Phys. Rev. B25, 4304 (1982).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Géza Györgyi
    • 1
  1. 1.Institute for Theoretical PhysicsEötvös UniversityBudapest

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