Sudden Jumps in the Logistic Map with Periodic Modulation: Theory and Experiment

  • M. Bucher
  • S. Zhu
  • Y. Pan
Conference paper
Part of the Woodward Conference book series (WOODWARD)


When the logistic map is augmented with a periodic modulation, bistability as well as sudden jumps occur in response to changes of the control parameters λ and μ. Using the inverse-curve method for graphical iterations of one-dimensional maps, we are able to specify the conditions for bistability and identify three basic mechanisms for the bifurcation discontinuities. These mechanisms are (1) basin crossing, (2) curve separation, and (3) transfer crisis. In order to verify experimentally the findings from computer iterations, we use an anharmonic electrical oscillator circuit with a junction diode as the nonlinear element, driven with a superposition of sinusoidal and square-wave signals of amplitudes λ and μ, respectively. The experiments confirm, on a qualitative level, our theoretical predictions.


Bifurcation Diagram Shift Parameter Nonlinear Element Sudden Jump Oscilloscope Trace 
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  1. 1.
    E. Ott, Rev. Mod. Phys. 53 655 (1981).MathSciNetADSCrossRefMATHGoogle Scholar
  2. 2.
    P. Collet and J. P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, (Birkhäuser, Boston, 1980).MATHGoogle Scholar
  3. 3.
    J. Testa, J. Perez, and C. Jeffries, Phys. Rev. Lett. 48 714 (1982); 49 1054 (1982); 49 1055 (1982).Google Scholar
  4. 4.
    C. Jeffries and J. Perez, Phys. Rev. A 26 2117 (1982).MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    J. Perez and C. Jeffries, Phys. Lett. 92A 82 (1982).MathSciNetADSGoogle Scholar
  6. 6.
    P. S. Linsay, Phys. Rev. 47 1349 (1981).ADSGoogle Scholar
  7. 7.
    H. Ikezi, J. S. deGrassie, and T. H. Jensen, Phys. Rev. A 28 1207 (1983).MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    R. C. Hilborn, Phys. Rev. A 31 378 (1985).ADSCrossRefGoogle Scholar
  9. 9.
    C. Jeffries and K. Wiesenfeld, Phys. Rev. A 31 1077 (1985).ADSCrossRefGoogle Scholar
  10. 10.
    J. M. Perez, Phys. Rev. A 32 2513 (1985).ADSCrossRefGoogle Scholar
  11. 11.
    J. Mevissen, R. Seal, and L Waters, Phys. Rev. A 32 2990, (1985).ADSCrossRefGoogle Scholar
  12. 12.
    N. Metropolis, M. L. Stein, and P. R. Stein, J. Comb. Theory, Ser. A 15 25 (1973).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    M. J. Feigenbaum, J. Stat. Phys. 19 25 (1978); Physica 7D 16 (1983).Google Scholar
  14. 14.
    P. Manneville and Y. Pomeau, Phys. Lett. 75A 1 (1979).MathSciNetADSGoogle Scholar
  15. 15.
    C. Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. Lett. 48 1507 (1982).MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    Y. Yamaguchi and K. Sakai, Phys. Rev. A 27 2755 (1983).ADSCrossRefGoogle Scholar
  17. 17.
    D. P. Siemens and M. Bucher, Physica 20D 363 (1986).MathSciNetADSGoogle Scholar
  18. 18.
    Y. Pan, M. S. Thesis, California State University, Fresno, 1985.Google Scholar
  19. 19.
    S. Zhu, M. S. Thesis, California State University, Fresno, 1986.Google Scholar
  20. 20.
    J. Testa and G. A. Held, Phys. Rev. A 28 3085 (1983).ADSCrossRefGoogle Scholar
  21. 21.
    H. Nakatsuka, S. Asaka, H. Itho, K. Ikeda, and M. Matsuoka, Phys. Rev. Lett. 50 109 (1983).ADSCrossRefGoogle Scholar
  22. 22.
    W. Lange, F. Mitschke, R. Deserno, and J. Mlynek, Phys. Rev. A 32 1271 (1985).ADSCrossRefGoogle Scholar
  23. 23.
    B. Ritchie and C. M. Bowden, Phys. Rev. A 32 2293 (1985).ADSCrossRefGoogle Scholar
  24. 24.
    J.M. Sancho, R. Mannella, P. V. E. McClintock, F. Moss, Phys. Rev. A 32 3639 (1985).ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1990

Authors and Affiliations

  • M. Bucher
    • 1
  • S. Zhu
    • 1
  • Y. Pan
    • 1
  1. 1.Physics DepartmentCalifornia State UniversityFresnoUSA

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