Fractal Structure of Subharmonic Steps in Dissipative Systems Described by a Driven Damped Pendulum Equation

  • Preben Alstrøm
  • Mogens T. Levinsen


Since it became clear that the return map for the driven damped pendulum equation
with I(ø) = sinø at and below the transition to chaos is a circle map for every irrational rotation number W,1 the connection between the scaling laws for the pendulum system and the structures obtained by iterations of circle maps has been studied extensively. In particular, the universal scaling behavior of the complete devil’s staircase obtained by iterations of circle maps2 like the sine map
with a zero slope third order inflection point has been conjectured also to be found at the transition to chaos in every dissipative dynamical system with two competing frequencies.3


Rotation Number Dissipative System Critical Line Pendulum System Dissipative Dynamical System 


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  1. 1.
    P. Bak, T. Bohr, M. H. Jensen, and P. V. Christiansen, Josephson junctions and circle maps, Solid State Commun. 51: 231 (1984).CrossRefGoogle Scholar
  2. 2.
    M. H. Jensen, P. Bak, and T. Bohr, Complete devil’s staircase, fractal dimension, and universality of mode-locking structure in the circle map, Phys. Rev. Lett. 50: 1637 (1983).CrossRefGoogle Scholar
  3. 3.
    M. H. Jensen, p. Bak, and T. Bohr, Transition to chaos by interaction of resonances in dissipative systems. I. Circle maps, Phys. Rev. A 30:1960 (1984); II. Josephson junctions, charge-density waves, and standard maps, ibid. 30: 1970 (1984).MathSciNetGoogle Scholar
  4. 4.
    S. J. Shenker, Scaling behavior in a map of a circle onto itself: Empirical results, Physica D 5: 405 (1982).MathSciNetCrossRefGoogle Scholar
  5. 5.
    S. E. Brown, G. Mozurkewich, and G. Grüner, Subharmonic Shapiro steps and devil’s-staircase behavior in driven charge-density-wave systems, Phys. Rev. Lett. 52: 2277 (1984).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Preben Alstrøm
    • 1
  • Mogens T. Levinsen
    • 1
  1. 1.H.C. Ørsted InstituteCopenhagen ØDenmark

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