Advertisement

Self-generated diffusion and universal critical properties in chaotic systems

  • T. Geisel
  • J. Nierwetberg
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 179)

Abstract

This paper reviews the universal critical properties exhibited by some 1d discrete dynamical systems: period-doubling systems and systems generating diffusive motion. While the period-doubling bifurcations have the universal asymptotic bifurcation rate δ=4.6692..., the tangent bifurcations present within the chaotic region do not follow this rate. We show that the tangent bifurcations giving rise to a fine structure of periodic windows have bifurcation rates γk which can be calculated analytically. They converge to a universal constant γ=2.94805... We have found that a class of dynamical systems show the onset of a diffusive motion in addition to period-doubling. The diffusion is self-generated and does not rely on the presence of random external forces. The onset of diffusion has strong analogies with a phase-transition. The diffusion coefficient is the order parameter and has a universal critical exponent. The dependence on random external fluctuations is also universal and can be expressed in terms of a universal scaling function which is calculated analytically.

Keywords

Periodic Point Universality Class Fundamental Period Discrete Dynamical System Periodic Window 
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.
    M.J. Feigenbaum, J. Stat. Phys. 19, 25 (1978) and 21, 669 (1979).CrossRefGoogle Scholar
  2. 2.
    P. Coullet and C. Tresser, J. Phys. Coll. (Paris) 39, C5–25 (1978).Google Scholar
  3. 3.
    S. Grossmann and S. Thomae, Z. Naturforsch. 32a, 1353 (1977).Google Scholar
  4. 4.
    P. Manneville and Y. Pomeau, Phys. Lett. 75A, 1 (1979) and Physica 1D, 219 (1980).Google Scholar
  5. 5.
    J.E. Hirsch, B.A. Huberman and D.J. Scalapino, Phys. Rev. 25A, 519 (1982)Google Scholar
  6. 6.
    T. Geisel and J. Nierwetberg, Phys. Rev. Lett. 48, 7 (1982).Google Scholar
  7. 7.
    B.A. Huberman and J.P. Crutchfield, Phys. Rev. Lett. 43, 1743 (1979).Google Scholar
  8. 8.
    A. Libchaber and J. Maurer, J. Phys. Lett. (Paris) 40, 419 (1979).Google Scholar
  9. 9.
    R.M. May, Nature 261, 459 (1976).PubMedGoogle Scholar
  10. 10.
    T. Geisel and J. Nierwetberg, Phys. Rev. Lett. 47, 975 (1981).Google Scholar
  11. 11.
    N. Metropolis, M.L. Stein and P.R. Stein, J. Comb. Theory A15, 25 (1973).Google Scholar
  12. 12.
    M. Misiurewicz, Publ. Math. IHES 53, 17 (1981).Google Scholar
  13. 13.
    H. Haken and G. Mayer-Kress, Z. Phys. B43, 185 (1981).Google Scholar
  14. 14.
    A. Lasota and J.A. Yorke, Trans. Am. Math. Soc. 186, 481 (1973).Google Scholar
  15. 15.
    S. Grossmann and H. Fujisaka, preprint.Google Scholar
  16. 16.
    M. Schell, S. Fraser and R. Kapral, preprint. *** DIRECT SUPPORT *** A3418136 00003Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • T. Geisel
    • 1
  • J. Nierwetberg
    • 1
  1. 1.Institut für Theoretische PhysikUniversität RegensburgRegensburgW.Germany

Personalised recommendations