Relevance of Exponentially Large Time Scales in Practical Applications: Effective Fractal Dimensions in Conservative Dynamical Systems

  • Antonio Giorgilli
Part of the NATO ASI Series book series (NSSB, volume 176)

Abstract

The problem of evaluating the fractal dimension of a chaotic orbit in a conservative dynamical system is revisited in the light of exponentially large time scales rigorously introduced by recent results of classical perturbation theory. The possible relevance for the problem of comparing theoretical previsions with experimental results in statistical models is pointed out.

Keywords

Manifold Peri 

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References

  1. [1]
    A.N. Kolmogorov, Dokl. Akad. Nauk SSSR, 98, 527 (1954).MathSciNetMATHGoogle Scholar
  2. V.I. Arnold, Usp. Mat. Nauk, 18, 13 (1963); Russ. Math. Surv., 18,9 (1963)Google Scholar
  3. Usp. Math. Nauk 18 N.6, 91 (1963); Russ. Math. Surv. 18 N.6, 85 (1963). J. Moser, Proceedings of the International Conference on Functional Analysis and Related Topics, p. 60 (Tokio, 1969).Google Scholar
  4. [2]
    N. N. Nekhoroshev, Usp. Mat. Nauk. 32, (1977) [Russ. Math. Surv. 32, 1 (1977)]; Trudy Sem. Petrows. No. 5, 5 (1979)Google Scholar
  5. G. Benettin, L. Galgani and A. Giorgilli, Cel. Mech. 37, 1–25 (1985).MathSciNetADSMATHCrossRefGoogle Scholar
  6. G. Benettin and G. Gallavotti, J. Stat. Phys. 44, 293–338 (1986)MathSciNetADSMATHCrossRefGoogle Scholar
  7. [3]
    A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and C. Simò, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem, preprint.Google Scholar
  8. [4]
    B. B. Mandelbrot,The fractal geometry of nature, Freeman, San Francisco (1982).MATHGoogle Scholar
  9. [5]
    D. K. Umberger and J. D. Farmer, Phys. Rev. Lett. 55 N. 7, 661–664 (1985)MathSciNetADSCrossRefGoogle Scholar
  10. C. Grebogi, S. W. McDonald, E. Ott and J. A. Yorke, Phys. Lett. 110A, 1, 1–4 (1985).MathSciNetADSGoogle Scholar
  11. [6]
    M. Pettini and A. Vulpiani, Phys. Lett. A, 106, 5,6, 207–211 (1984)MathSciNetADSCrossRefGoogle Scholar
  12. A. Malagoli, G. Paladin and A. Vulpiani, Phys. Rev. A, 34, 1550–1555 (1986).ADSCrossRefGoogle Scholar
  13. [7]
    E. Ott, Rev. Mod. Phys. 53, 4, 655–671 (1981).MathSciNetADSMATHCrossRefGoogle Scholar
  14. [8]
    S. E. Newhouse, Soc. Math, de France, Astérisque 51, 223–334 (1978).Google Scholar
  15. [9]
    A. Giorgilli, D. Casati, L. Sironi and L. Galgani, Phys. Lett. A, 115, 5, 202–206 (1986).MathSciNetADSCrossRefGoogle Scholar
  16. [10]
    G. Benettin, D. Casati, L. Galgani, A. Giorgilli and L. Sironi, Phys. Lett. A, 118, 325–330 (1986).MathSciNetADSCrossRefGoogle Scholar
  17. [11]
    A.I. Neishtadt, PMM U.S.S.R, 48, N. 2, 133–139 (1984).MathSciNetGoogle Scholar
  18. [12]
    G. Benettin, C. Cercignani, L. Galgani and A. Giorgilli, Lett. Nuovo Cim., 28 N. 1, 1–4 (1980); 29 N. 6, 163–166 (1980).CrossRefGoogle Scholar
  19. [13]
    L. Pontrjagin and L. Schnirelmann, Ann. Math. 33, 156 (1932)MathSciNetCrossRefGoogle Scholar
  20. A.N. Kolmogorov and V.M. Tihomirov, Usp. Mat. Nauk 14, 3 (1959) (english trans.: Am. Math. Soc. Transi., series 2, 17, 277 (1961)).MathSciNetMATHGoogle Scholar
  21. [14]
    P. Grassberger, Phys. Lett. A, 97 N. 6, 224–226 (1983).MathSciNetADSCrossRefGoogle Scholar
  22. [15]
    G. Benettin, L. Galgani and A. Giorgilli, Phys. Lett. A, 120, 23–27 (1987).ADSCrossRefGoogle Scholar
  23. [16]
    G. Benettin, L. Galgani and A. Giorgilli, Realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory, Comm. Math. Phys., to appear.Google Scholar
  24. [17]
    L. Boltzmann, Nature 51, 413 (1885).ADSCrossRefGoogle Scholar
  25. [18]
    H. Jeans, Phil. Mag. 6, 279 (1903).Google Scholar
  26. Jeans, Phil. Mag. 10, 91 (1905).Google Scholar

Copyright information

© Plenum Press, New York 1988

Authors and Affiliations

  • Antonio Giorgilli
    • 1
  1. 1.Dipartimento di Fisica dell’UniversitàMilanoItalia

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