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Self-similar transport in incomplete chaos

G. M. Zaslavsky

D. Stevens and H. Weitzner

  • Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012
  • Physics Department, New York University, 4 Washington Place, New York, New York 10003

  • Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012

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Phys. Rev. E 48, 1683 – Published 1 September, 1993

DOI: https://doi.org/10.1103/PhysRevE.48.1683

Abstract

Particle chaotic dynamics along a stochastic web is studied for three-dimensional Hamiltonian flow with hexagonal symmetry in a plane. Two different classes of dynamical motion, obtained by different values of a control parameter, and corresponding to normal and anomalous diffusion, have been considered and compared. It is shown that the anomalous transport can be characterized by powerlike wings of the distribution function of displacement, flights which are similar to Lévy flights, approximate trappings of orbits near the boundary layer of islands, and anomalous behavior of the moments of a distribution function considered as a function of the number of the moment. The main result is related to the self-similar properties of different topological and dynamical characteristics of the particle motion. This self-similarity appears in the Weierstrass-like random-walk process that is responsible for the anomalous transport exponent in the mean-moment dependence on t. This exponent can be expressed as a ratio of fractal dimensions of space and time sets in the Weierstrass-like process. An explicit form for the expression of the anomalous transport exponent through the local topological properties of orbits has been given.

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References (45)

  1. G. M. Zaslavsky and B. V. Chirikov, Usp. Fiz. Nauk. 101, 3 (1972) [Sov. Phys. Usp. 14, 549 (1972)].
  2. B. V. Chirikov, Phys. Rep. 52, 264 (1979).
  3. A. B. Rechester and R. B. White, Phys. Rev. Lett. 44, 1586 (1980).
  4. J. R. Cary and J. D. Meiss, Phys. Rev. A 24, 2664 (1981).
  5. C. F. F. Karney, Physica D 8, 360 (1983).
  6. I. Dana, N. W. Murray and I. C. Percival, Phys. Rev. Lett. 62, 233 (1989).
  7. A. Lichtenberg and M. A. Liberman, Regular and Stochastic Motion (Springer, New York, 1992).
  8. I. C. Percival, Nonlinear Dynamics and Beam-Beam Interaction, in Proceedings of the Symposium on Nonlinear Dynamics and Beam-Beam Interaction, AIP Conf. Proc. No. 57, edited by M. Month and J. C. Herrera (AIP, New York, 1979).
  9. S. Aubry, Physica D 7, 240 (1983).
  10. V. I. Arnold, Dokl. Akad. Nauk SSSR 156, 9 (1964) [Sov. Math. Dokl. 5, 581 (1964)].
  11. G. M. Zaslavsky, M. Yu. Zakharov, R. Z. Sagdeev, D. A. Usikov and A. A. Chernikov, Zh. Eksp. Teor. Fiz. 91, 500 (1986) [Sov. Phys. JETP 64, 294 (1986)].
  12. G. M. Zaslavsky, R. Z. Sagdeev and A. A. Chernikov, Zh. Eksp. Teor. Fiz. 94, 102 (1988) [Sov. Phys. JETP 67, 270 (1988)].
  13. G. M. Zaslavsky, R. Z. Sagdeev, D. A. Usikov, and A. A. Chernikov, Weak Chaos and Quasiregular Patterns (Cambridge University Press, Cambridge, England, 1991).
  14. V. V. Beloshapkin, A. A. Chernikov, M. Ya. Natenzon, B. A. Petrovichev, R. Z. Sagdeev and G. M. Zaslavsky, Nature (London) 337, 113 (1989).
  15. A. A. Chernikov, B. A. Petrovichev, A. V. Rogalsky, R. Z. Sagdeev and G. M. Zaslavsky, Phys. Lett. A 144, 127 (1990).
  16. V. V. Afanas'ev, R. Z. Sagdeev and G. M. Zaslavsky, Chaos 1, 143 (1991).
  17. D. K. Chaikovsky and G. M. Zaslavsky, Chaos 1, 463 (1991).
  18. M. Schwägerl and J. Krug, Physica D 52, 143 (1991).
  19. B. A. Petrovichev, A. V. Rogalsky, R. Z. Sagdeev and G. M. Zaslavsky, Phys. Lett. A 150, 391 (1990).
  20. M. Ding, C. Grebogi, E. Ott and J. A. York, Phys. Rev. A 42, 7025 (1990).
  21. B. V. Chirikov and D. L. Shepelyansky, in Renormalization Group, edited by D. V. Shirkov, D. I. Kazakov, and A. A. Vladimirov (World Scientific, Singapore, 1988), p. 221.
  22. B. V. Chirikov, Chaos, Solitons and Fractals 1, 79 (1991).
  23. G. M. Zaslavsky and M. Tippett, Phys. Rev. Lett. 67, 3251 (1991).
  24. P. Lévy, Theiaaorie de l'Addition des Variables Aleiaaatoires (Gauthier-Vilars, Paris, 1937).
  25. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1982).
  26. M. F. Schlesinger, B. J. West and J. Klafter, Phys. Rev. Lett. 58, 1100 (1987).
  27. E. Montroll and M. F. Shlesinger, in Studies in Statistical Mechanics, edited by J. Leibowitz and E. Montroll (North-Holland, Amsterdam, 1984), Vol. II, pp. 1–121.
  28. G. M. Zaslavsky, Fluid Dyn. Res. 8, 127 (1991).
  29. M. F. Schlesinger, Annu. Rev. Phys. Chem. 39, 269 (1988).
  30. H. Scher, M. F. Shlesinger, and J. T. Bendler, Phys. Today 44, (1), 26 (1991).
  31. G. M. Zaslavsky, in Topological Aspects of the Dynamics of Fluids and Plasmas, edited by H. K. Moffatt, G. M. Zaslavsky, P. Comte, and M. Tabor (Kluwer, Boston, 1992).
  32. V. I. Arnold, C. R. Acad. Sci. Paris 261, 17 (1965).
  33. D. Stevens, New York University, CIMS Report No. MF-104, 1985 (unpublished).
  34. G. D. Birkhoff, Collected Mathematical Papers (American Mathematical Society, New York, 1950), Vol. 2.
  35. V. I. Arnold, Sov. Math. Dokl. 2, 247 (1961).
  36. J. D. Meiss, Phys. Rev. A 34, 2375 (1986).
  37. D. F. Escande, Phys. Rep. 121, 165 (1985).
  38. J. Meiss and E. Ott, Phys. Rev. Lett. 55, 2741 (1985).
  39. J. Meiss and E. Ott, Physica D 20, 387 (1986).
  40. B. D. Hughes, M. F. Shlesinger and E. W. Montroll, Proc. Natl. Acad. Sci. U.S.A. 78, 3287 (1981).
  41. G. M. Zaslavsky, Chaos in Dynamic Systems (Harwood Academic, New York, 1985).
  42. G. H. Hardy, Trans. Am. Math. Soc. 17, 301 (1980).
  43. M. V. Berry and Z. V. Lewis, Proc. R. Soc. London, Ser. A 370, 459 (1980).
  44. B. D. Hughes, E. W. Montroll and M. F. Shlesinger, J. Stat. Phys. 28, 111 (1982).
  45. B. V. Chirikov and D. L. Shepelyansky, Physica D 13, 395 (1984).

Outline

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Phys. Rev. E 48, 1683– Published 1 September, 1993
  • Received 3 May 1993

© 1993 American Physical Society

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