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Shil’nikov Chaos: How to Characterize Homoclinic and Heteroclinic Behaviour

  • F. T. Arecchi
  • A. Lapucci
  • R. Meucci
Part of the NATO ASI Series book series (NSSB, volume 208)

Abstract

We introduce the concepts of Shil’nikov chaos and competing instabilities in a nonlinear dynamics including at least a saddle focus and a saddle point, a parameter change induces a smooth transition from a homoclinic to a heteroclinic trajectory.

In terms of the return map to a given Poincaré section, the two trajectories have the following characterization. In the homoclinic case, the global behavior is recovered from the local linear dynamics within a unit box around the saddle focus. The heteroclinic case requires the composition of two linearized maps around the two unstable points.

By an exponential transformation the geometrical map yields the return map of the orbital times. This new map represents the most appropriate indicator for experimental situations whenever a symbolic dynamics built on geometric position does not offer a sensitive test. Furthermore the time maps display a large sensitivity to noise. This offers a criterion to discriminate between a simulation (either analog or digital) with a few variables and experiment dealing with the physical variables embedded in the real world and thus acted upon by noise.

Keywords

Homoclinic Orbit Saturable Absorber Population Inversion Symbolic Dynamic Unstable Point 
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.

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References

  1. 1.
    L.P. Shil’nikov, Dokl. Akad. Nauk SSSR 160, 558 (1965);MathSciNetGoogle Scholar
  2. L.P. Shil’nikov, Mat. Sb. 77, 119, 461 (1968);MathSciNetGoogle Scholar
  3. L.P. Shil’nikov, Mat. Sb. 81, 92, 123 (1970).Google Scholar
  4. 2.
    A. Arneodo, P.H. Coullet, E.A. Spiegel, and C. Tresser, Physica 140, 327 (1985).MathSciNetGoogle Scholar
  5. 3.
    F.T. Arecchi, R. Meucci and W. Gadomski, Phys. Rev. Lett. 58, 2205 (1987).CrossRefGoogle Scholar
  6. 4.
    F. Argoul, A. Arneodo, and P. Richetti, Phys. Lett. Al20, 269 (1987).CrossRefGoogle Scholar
  7. 5.
    P. Glendinning and C. Sparrow, J. Stat. Phys. 35, 645 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  8. P. Gaspard, R. Kapral and G. Nicolis, J. Stat. Phys. 35, 697 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 6.
    F.T. Arecchi, A. Lapucci, R. Meucci, J.A. Roversi, and P. Coullet, i) ELITE Meeting, Torino, Italy, march 1987.Google Scholar
  10. F.T. Arecchi, A. Lapucci, R. Meucci, J.A. Roversi, and P. Coullet, ii) International Workshop on Instabilities and Chaos in Nonlinear Optical systems, Il Ciocco, Italy, July 1987 paper WC 13–2.Google Scholar
  11. F.T. Arecchi, A. Lapucci, R. Meucci, J.A. Roversi, and P. Coullet, iii) Europhys. Lett. 6, 677 (1988).CrossRefGoogle Scholar
  12. 7.
    F.T. Arecchi, W. Gadomski, A. Lapucci, R. Meucci, H. Mancini, and J.A. Roversi, JOSA B5, 1153 (1988).CrossRefGoogle Scholar
  13. 8.
    J.P. Crutchfield, D. Farmer, and B.A. Hubermann, Phys. Rev. 92, 45 (1982).Google Scholar
  14. 9.
    F.T. Arecchi, W. Gadomski, and R. Meucci, Phys. Rev. A 34, 1617 (1986).CrossRefGoogle Scholar
  15. 10.
    F.T. Arecchi, V. Degiorgio, and B. Querzola, Phys. Rev. Lett. 19, 168 (1967);CrossRefGoogle Scholar
  16. F.T. Arecchi, and A. Politi, and L. Ulivi, Nuovo Cimento 71B, 119, (1982).Google Scholar
  17. 11.
    D. Hennequin, F. De Tomasi, B. Zambon, and E. Arimondo, Phys. Rev. A 37, 243 (1988).CrossRefGoogle Scholar
  18. 12.
    D. Dangoisse, A. Bekkali, F. Papoff, and P. Glorieux, Europhys. Lett. 6, 335 (1988).CrossRefGoogle Scholar
  19. 13.
    M. Tachikawa, K. Tanii, and T. Shimizu, J.O.S.A. B 5, 1077 (1988).CrossRefGoogle Scholar
  20. 14.
    C.0. Weiss, N.B. Abraham, and U. Hubner, Phys. Rev. Lett. 61, 1587, (1988).CrossRefGoogle Scholar
  21. 15.
    F. Papoff, A. Fioretti, E. Arimondo, and N.B. Abraham, “Return time and distribution in the laser with saturable absorber”, this same ConferenceGoogle Scholar
  22. 16.
    F.T. Arecchi, A. Lapucci, and R. Meucci, to be published.Google Scholar

Copyright information

© Plenum Press, New York 1989

Authors and Affiliations

  • F. T. Arecchi
    • 1
  • A. Lapucci
    • 1
  • R. Meucci
    • 1
  1. 1.Istituto Nazionale di OtticaFirenzeItaly

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