Hierarchical structures in the phase space and fractional kinetics: I. Classical systems

Chaos

Volumn 10, Issue 1, 2000, Pages 135-146

  Zaslavsky, G M ^a,b   Edelman, M ^a  

^a NEW YORK UNIVERSITY   (United States)

^b POLYTECHNIC UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034147028     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.166481     Document Type: Article

References (72)

1. 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), p. 481.
2. G. M. Zaslavsky, Chaos 4, 25 (1994); Physica D 76, 110 (1994).
3. G. M. Zaslavsky, Chaos 4, 25 (1994); Physica D 76, 110 (1994).
4. M. F. Shlesinger, G. M. Zaslavsky, and J. Klafter, Nature (London) 363, 31 (1993).
5. E. W. Montroll and M. F. Shlesinger, in Studies in Statistical Mechanics, edited by J. Lebowitz and E. Montroll (North-Holland, Amsterdam, 1984), Vol. 11, p. 1.
6. J. P. Bouchaud and A. Georges, Phys. Rep. 95, 127 (1990).
7. Lévy Flights and Related Topics in Physics, edited by M. Shlesinger, G. M. Zaslavsky, and U. Frisch (Springer-Verlag, Heidelberg, 1995).
8. T. Geisel, J. Nierwetberg, and A. Zacherl, Phys. Rev. Lett. 54, 616 (1984); T. Geisel, A. Zacherl, and G. Radons, ibid. 59, 2503 (1987); Z. Phys. B: Condens. Matter 71, 117 (1988).
9. T. Geisel, J. Nierwetberg, and A. Zacherl, Phys. Rev. Lett. 54, 616 (1984); T. Geisel, A. Zacherl, and G. Radons, ibid. 59, 2503 (1987); Z. Phys. B: Condens. Matter 71, 117 (1988).
10. T. Geisel, J. Nierwetberg, and A. Zacherl, Phys. Rev. Lett. 54, 616 (1984); T. Geisel, A. Zacherl, and G. Radons, ibid. 59, 2503 (1987); Z. Phys. B: Condens. Matter 71, 117 (1988).
11. M. F. Shlesinger, Annu. Rev. Phys. Chem. 39, 269 (1988).
12. V. V. Afanas'ev, R. Z. Sagdeev, and G. M. Zaslavsky, Chaos 1, 143 (1991).
13. J. Klafter, G. Zumofen, and M. F. Shlesinger, in Lévy Flights and Related Topics in Physics, edited by M. F. Shlesinger, G. M. Zaslavsky, and U. Frisch (Springer-Verlag, New York, 1995), p. 196.
14. G. M. Zaslavsky and B. V. Chirikov, Sov. Phys. Dokl. 9, 989 (1965).
15. J. D. Meiss, Phys. Rev. A 34, 2375 (1986).
16. J. D. Meiss, Rev. Mod. Phys. 64, 795 (1992).
17. B. V. Chirikov, Chaos, Solitons and Fractals 1, 79 (1991).
18. V. I. Melnikov, in Transport, Chaos, and Plasma Physics 2, edited by S. Benkadda, F. Doveil, and Y. Elskens (World Scientific, Singapore, 1994), p. 142.
19. C. F. F. Karney, Physica D 8, 360 (1983).
20. R. S. MacKay and J. D. Meiss, Hamiltonian Dynamical Systems (Adam Hilger, Bristol, 1987).
21. J. D. Hanson, J. R. Cary, and J. D. Meiss, J. Stat. Phys. 39, 327 (1985).
22. J. D. Meiss and E. Ott, Phys. Rev. Lett. 55, 2741 (1984); M. Ding, T. Bountis, and E. Ott, Phys. Lett. A 151, 395 (1990).
23. J. D. Meiss and E. Ott, Phys. Rev. Lett. 55, 2741 (1984); M. Ding, T. Bountis, and E. Ott, Phys. Lett. A 151, 395 (1990).
24. G. M. Zaslavsky, M. Edelman, and B. A. Niyazov, Chaos 7, 159 (1997).
25. G. M. Zaslavsky and B. A. Niyazov, Phys. Rep. 283, 73 (1997).
26. Y. H. Ichikawa, T. Kamimura, and T. Hatori, Physica D 29, 247 (1987); R. Ishizaki, T. Horito, T. Kobayashi, and M. Mori, Prog. Theor. Phys. 85, 1013 (1991).
27. Y. H. Ichikawa, T. Kamimura, and T. Hatori, Physica D 29, 247 (1987); R. Ishizaki, T. Horito, T. Kobayashi, and M. Mori, Prog. Theor. Phys. 85, 1013 (1991).
28. V. Rom-Kedar and G. M. Zaslavsky, Chaos 9, 697 (1999).
29. G. M. Zaslavsky and N. N. Filonenko, Sov. Phys. JETP 25, 851 (1968).
30. V. Rom-Kedar, Nonlinearity 7, 441 (1994).
31. A. Iomin, D. Gangardt, and S. Fishman, Phys. Rev. E 57, 4054 (1998).
32. G. M. Zaslavsky and B. V. Chirikov, Sov. Phys. Usp. 14, 549 (1972).
33. B. V. Chirikov, Phys. Rep. 52, 264 (1979).
34. R. Mackay, J. Meiss, and I. Percival, Physica D 13, 55 (1984).
35. S. Newhouse, Am. J. Math. 99, 1061 (1977).
36. S. Benkadda, S. Kassibrakis, R. White, and G. Zaslavsky, Phys. Rev. E 55, 4909 (1997).
37. G. M. Zaslavsky, D. Stevens, and H. Weitzner, Phys. Rev. E 48, 1683 (1993).
38. M. Kac, Probability and Related Topics in Physical Sciences, Boulder, Colorado. 1957 (Wiley Interscience, New York, 1958).
39. Y. G. Sinai, Topics in Ergodic Theory (Princeton University Press, Princeton, NJ, 1994).
40. J. D. Meiss, Chaos 7, 39 (1997).
41. S. Benkadda, S. Kassibrakis, R. White, and G. M. Zaslavsky, Phys. Rev. E 59, 3761 (1999).
42. A. N. Kolmogorov, Usp. Mat. Nauk 5, 5 (1938).
43. A. Saichev and G. M. Zaslavsky, Chaos 7, 753 (1997).
44. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives and Their Applications (Nauka i Tekhnika, Minsk, 1987) (translation by Harwood Academic Publishers).
45. G. M. Zaslavsky, in Applications of Fractional Calculus in Physics, edited by R. Hilfer (World Scientific, Singapore, 1999).
46. A. A. Chernikov, B. A. Petrovichev, A. V. Rogalsky, R. Z. Sagdeev, and G. M. Zaslavsky, Phys. Lett. A 144, 127 (1990).
47. G. Zumofen and J. Klafter, Europhys. Lett. 25, 565 (1994).
48. J. Klafter, G. Zumofen, and M. F. Shlesinger, in Ref. 10, p. 196.
49. G. M. Zaslavsky and M. Edelman, Phys. Rev. E 56, 5310 (1997).
50. P. Castiglione, A. Mazzino, P. Muratore-Ginanneschi, and A. Vulpiani (not published).
51. L. A. Bunimovich and Y. G. Sinai, Math. USSR-Sb. 90, 407 (1973); Commun. Math. Phys. 78, 479 (1981);
52. L. A. Bunimovich and Y. G. Sinai, Math. USSR-Sb. 90, 407 (1973); Commun. Math. Phys. 78, 479 (1981);
53. J. Machta, J. Stat. Phys. 32, 555 (1983); J. Machta and B. Reinhold, ibid. 42, 949 (1986); J. Machta and R. Zwanzig, Phys. Rev. Lett. 50 1959 (1983).
54. J. Machta, J. Stat. Phys. 32, 555 (1983); J. Machta and B. Reinhold, ibid. 42, 949 (1986); J. Machta and R. Zwanzig, Phys. Rev. Lett. 50 1959 (1983).
55. J. Machta, J. Stat. Phys. 32, 555 (1983); J. Machta and B. Reinhold, ibid. 42, 949 (1986); J. Machta and R. Zwanzig, Phys. Rev. Lett. 50 1959 (1983).
56. T. Niemeijer and J. van Leeuwen, Phase Transitions and Critical Phenomena, edited by C. Domb and M. Green (Academic Press, London, 1976), Vol. 6, p. 425.
57. M. F. Shlesinger and B. West, Phys. Rev. E 67, 2106 (1991).
58. B. D. Hughes, M. F. Shlesinger, and E. W. Montroll, Proc. Natl. Acad. Sci. USA 78, 3287 (1981).
59. D. Sornette, Phys. Rep. 297, 239 (1998).
60. B. V. Chirikov and D. L. Shepeliansky, Physica D 13, 395 (1984).
61. J. Klafter, A. Blumen, and M. F. Shlesinger, Phys. Rev. A 35, 3081 (1987).
62. S. C. Venkataramani, T. M. Antonsen, Jr., and E. Ott, Phys. Rev. Lett. 78, 3864 (1997).
63. R. White, S. Benkadda, S. Kassibrakis, and G. M. Zaslavsky, Chaos 8, 757 (1998).
64. L. Kuznetsov and G. M. Zaslavsky (submitted).
65. V. V. Beloshapkin and G. M. Zaslavsky, Phys. Lett. A 97, 121 (1983).
66. I. C. Percival, Nonlinear Dynamics and Beam-Beam Interaction, AIP Conf. Proc. 57, edited by M. Month and J. C. Herrera (American Institute of Physics, New York, 1979), p. 302.
67. J. R. Cary and J. D. Meiss, Phys. Rev. A 24, 2664 (1981).
68. R. S. MacKay, Physica D 7, 283 (1983).
69. G. M. Zaslavsky and S. S. Abdullaev, Chaos 7, 182 (1997).
70. G. M. Zaslavsky, Chaos 5, 653 (1995).
71. V. Afraimovich, Chaos 7, 12 (1997).
72. V. Afraimovich and G. M. Zaslavsky, Phys. Rev. E 55, 5418 (1997).