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Adiabatic invariance and separatrix: Single separatrix crossing

  • Nonlinear Physics
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Abstract

Detailed numerical experiments on the dynamics and statistics of a single crossing of the separatrix of a nonlinear resonance with a time-varying amplitude are described. The results are compared with a simple approximate theory first developed by Timofeev and further improved and generalized by Tennyson and coworkers. The main attention is paid to a new, ballistic, regime of separatrix crossing in which the violation of adiabaticity is maximal. Some unsolved problems and open questions are also discussed.

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References

  1. L. I. Mandelshtam, Complete Works (Akad. Nauk SSSR, Moscow, 1948), Vol. 1, p. 297.

    Google Scholar 

  2. B. V. Chirikov, Reviews of Plasma Physics, Ed. by B. B. Kadomtsev (Énergoatomizdat, Moscow, 1984; Consultants Bureau, New York, 1987), Vol. 13.

    Google Scholar 

  3. A. Lichtenberg and M. Lieberman, Regular and Chaotic Dynamics (Springer, New York, 1992).

    Google Scholar 

  4. V. I. Arnold, Dokl. Akad. Nauk SSSR 156, 9 (1961).

    Google Scholar 

  5. B. V. Chirikov, Phys. Rep. 52, 263 (1979).

    Article  ADS  MathSciNet  Google Scholar 

  6. B. V. Chirikov and V. V. Vecheslavov, J. Stat. Phys. 71, 243 (1993); Zh. Éksp. Teor. Fiz. 112, 1132 (1997) [JETP 85, 616 (1997)].

    Article  Google Scholar 

  7. N. N. Nekhoroshev, Usp. Mat. Nauk 32(6), 5 (1977); P. Lochak, Phys. Lett. A 143, 39 (1990); Usp. Mat. Nauk 47 (6), 59 (1992); P. Lochak and A. Neishtadt, Chaos 2, 495 (1992).

    MATH  Google Scholar 

  8. B. V. Chirikov, Dokl. Akad. Nauk SSSR 125, 1015 (1959) [Sov. Phys. Dokl. 4, 390 (1959)]; B. V. Chirikov and D. L. Shepelyanskii, Zh. Tekh. Fiz. 52, 238 (1982) [Sov. Phys. Tech. Phys. 27, 156 (1982)].

    MATH  MathSciNet  Google Scholar 

  9. J. Tennyson et al., Phys. Rev. Lett. 56, 2117 (1986); Phys. Rev. A 34, 4256 (1986).

    Article  ADS  MathSciNet  Google Scholar 

  10. R. Best, Physica 40, 182 (1968).

    Article  Google Scholar 

  11. R. Aamodt and E. Jaeger, Phys. Fluids 17, 1386 (1974).

    Google Scholar 

  12. A. V. Timofeev, Zh. Éksp. Teor. Fiz. 48, 1303 (1978) [Sov. Phys. JETP 48, 656 (1978)].

    Google Scholar 

  13. V. Ya. Davydovskii and A. I. Matveev, Zh. Tekh. Fiz. 53, 2125 (1983) [Sov. Phys. Tech. Phys. 28, 1302 (1983)]; A. I. Neishtadt, Fiz. Plazmy 12, 992 (1986) [Sov. J. Plasma Phys. 12, 568 (1986)].

    Google Scholar 

  14. C. Eckart, Phys. Rev. 35, 1303 (1930); A. S. Bakav and Yu. P. Stepanovsky, Adiabatic Invariants (Naukova Dumka, Kiev, 1981).

    ADS  MATH  Google Scholar 

  15. B. Mirbach and G. Casati, Phys. Rev. Lett. 83, 1327 (1999).

    Article  ADS  Google Scholar 

  16. C. Menyuk, Phys. Rev. A 31, 3282 (1985).

    Article  ADS  Google Scholar 

  17. Y. Elskens and F. Escande, Physica D 62, 66 (1993).

    Article  ADS  MathSciNet  Google Scholar 

  18. L. Casetti, Phys. Scr. 51, 29 (1995).

    ADS  Google Scholar 

  19. E. Forest and R. Ruth, Physica D 43, 105 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  20. J. Cary and R. Skodje, Physica D 36, 287 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  21. D. Bruhwiler and J. Cary, Physica D 40, 265 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  22. B. Mirbach, private communication.

  23. J. Cary, F. Escande, and J. Tennyson, Report No. IFSR-155 (Institute for Fusion Studies Report, 1984).

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Authors and Affiliations

  1. Budker Institute of Nuclear Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090, Russia

    B. V. Chirikov & V. V. Vecheslavov

Authors
  1. B. V. Chirikov
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  2. V. V. Vecheslavov
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From Zhurnal Éksperimental’no\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} \) i Teoretichesko\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} \) Fiziki, Vol. 117, No. 3, 2000, pp. 644–652.

Original English Text Copyright © 2000 by Chirikov, Vecheslavov.

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Chirikov, B.V., Vecheslavov, V.V. Adiabatic invariance and separatrix: Single separatrix crossing. J. Exp. Theor. Phys. 90, 562–569 (2000). https://doi.org/10.1134/1.559139

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  • DOI: https://doi.org/10.1134/1.559139

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