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)


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.


Periodic Point Universality Class Fundamental Period Discrete Dynamical System Periodic Window 
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Copyright information

© Springer-Verlag 1983

Authors and Affiliations

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

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