The Structure of Chaos

  • G. Contopoulos
Part of the NATO ASI Series book series (NSSB, volume 331)


We describe the structure of chaotic regions in 2 and 3 degrees of freedom. The lobes formed by the asymptotic curves of unstable periodic orbits follow certain rules that allow predictions of their structure. When the energy becomes larger than the escape energy there are some “limiting asymptotic curves” defining the regions of escape. When there are no escapes the asymptotic curves become longer as the energy increases and produce an infinity of new tangencies, followed by the appearance of new stable periodic orbits.

In systems of 3 degrees of freedom the asymptotic curves of complex unstable orbits are spirals, described by a linear theory. In their vicinity there is an infinity of periodic orbits arranged along certain spirals. The asymptotic curves of doubly unstable orbits oscillate in a way different from that of unstable 2-D systems.


Black Hole Periodic Orbit Rotation Number Chaotic Region Unstable Periodic Orbit 
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  1. Barbanis, B., 1993, Celest. Mech. Dyn. Astron.55:87.CrossRefzbMATHMathSciNetADSGoogle Scholar
  2. Contopoulos, G., 1966, in. “Les Nouvelles Méthodes de la Dynamique Stellaire”, F. Nahon and M. Henon eds., CNRS, Paris, Bull. Astron. Ser. 3, 2: 223.Google Scholar
  3. Contopoulos, G., 1971, Astron. J. 76:147.CrossRefMathSciNetADSGoogle Scholar
  4. Contopoulos, G., 1990a, Astron. Astrophys. 231:41.MathSciNetADSGoogle Scholar
  5. Contopoulos, G., 1990b, Proc. R. Soc. London A 431:183.CrossRefzbMATHMathSciNetADSGoogle Scholar
  6. Contopoulos, G and Barbanis, B., 1989, Astron. Astrophys. 222:329.MathSciNetADSGoogle Scholar
  7. Contopoulos, G. and Magnenat, P., 1985, Celest. Mech. 37:387.CrossRefzbMATHMathSciNetADSGoogle Scholar
  8. Contopoulos, G. and Kaufmann, D., 1992, Astron. Astrophys. 253:379.zbMATHMathSciNetADSGoogle Scholar
  9. Contopoulos, G. and Polymilis, C., 1993, Phys. Rev. E 46:1546.CrossRefMathSciNetADSGoogle Scholar
  10. Contopoulos, G., Kandrup, H.E. and Kaufmann, D., 1993, Physica D 64:310.CrossRefzbMATHADSGoogle Scholar
  11. Chirikov, B.V., 1969, Phys. Rep. 52:263.CrossRefMathSciNetADSGoogle Scholar
  12. Churchill, R.C. Pecelli, G. and Rod, D.L., 1975, J. Diff. Equ. 17:329.CrossRefzbMATHMathSciNetADSGoogle Scholar
  13. Churchill, R.C., Pecelli, G. and Rod, D.L., 1979 in “Stochastic Behavior in Classical and Quantum Quantum Hamiltonian Systems”, G. Casati, and J. Ford, eds., Springer Verlag, Heidelberg, p. 76.CrossRefGoogle Scholar
  14. Heggie, D.C., 1985, Celest. Mech. 35:357.CrossRefzbMATHMathSciNetADSGoogle Scholar
  15. Henon, M., 1988, Physica D 33:132.CrossRefzbMATHMathSciNetADSGoogle Scholar
  16. Jung, C. and Scholz, H.Z., 1987, J. Phys. A 20:3607.CrossRefzbMATHMathSciNetADSGoogle Scholar
  17. Jung, C. and Scholz, H.Z., 1988, J. Phys. A 21:2301.CrossRefzbMATHMathSciNetADSGoogle Scholar
  18. Laskar, J., Froeschle, C. and Celetti, A., 1992, Physica D 56:253.CrossRefzbMATHMathSciNetADSGoogle Scholar
  19. Newhouse, S.E., 1977, Am. J. Math. 99:1061.CrossRefzbMATHMathSciNetGoogle Scholar
  20. Newhouse, S.E., 1983, in “Chaotic Behaviour of Deterministic Systems”, G. Iooss, R.H.G. Helleman and R. Stora, eds. North Holland, Amsterdam, p. 443.Google Scholar
  21. Papadaki, H., Contopoulos, G. and Polymilis, C., 1994, in Roy, A.E. (ed). “From Newton to Chaos: Modern Techniques for Understanding and Coping with Chaos in N-Body Dynamical Systems”, Plenum Press, N. York (in press).Google Scholar
  22. Rod, D.L., 1973, J. Diff. Equ. 14:129.CrossRefzbMATHMathSciNetADSGoogle Scholar
  23. Rosenbluth, M.N., Sagdeev, R.A., Taylor, J.B. and Zaslavskii, M., 1966, Nucl. Fusion 6:297.CrossRefGoogle Scholar
  24. Shirts, R.B. and Reinhardt, W.P., 1982, J. Chem. Phys. 77:5204.CrossRefMathSciNetADSGoogle Scholar
  25. Sohos, G., Bountis, T. and Polymilis, H., 1989, Nuovo Cim. 104B:339.CrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • G. Contopoulos
    • 1
    • 2
  1. 1.Department of AstronomyUniversity of Athens PanepistimiopolisAthensGreece
  2. 2.Department of AstronomyUniversity of FloridaGainesvilleUSA

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