Spectra of stretching numbers and helicity angles in dynamical systems

  • G. Contopoulos
  • N. Voglis

Abstract

We define a “stretching number” (or “Lyapunov characteristic number for one period”) (or “stretching number”) a = In % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaada% Wcaaqaaiabe67a4jaadshacqGHRaWkcaaIXaaabaGaeqOVdGNaamiD% aaaaaiaawEa7caGLiWoaaaa!3F1E!\[\left| {\frac{{\xi t + 1}}{{\xi t}}} \right|\]as the logarithm of the ratio of deviations from a given orbit at times t and t + 1. Similarly we define a “helicity angle” as the angle between the deviation % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdGNaam% iDaaaa!3793!\[\xi t\]and a fixed direction. The distributions of the stretching numbers and helicity angles (spectra) are invariant with respect to initial conditions in a connected chaotic domain. We study such spectra in conservative and dissipative mappings of 2 degrees of freedom and in conservative mappings of 3-degrees of freedom. In 2-D conservative systems we found that the lines of constant stretching number have a fractal form.

Key words

Lyapunov characteristic numbers stretching numbers helicity angles conservative and dissipative mappings 

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References

  1. Ababarnel, H.D.I., Brown, R. and Kennel, M.B.: 1991, Int. J. Mod. Phys. B5, 134.Google Scholar
  2. Arnold, V.I.: 1964, Dokl. Akad. Nauk SSSR 156, 9.Google Scholar
  3. Arnold, V.I. and Avez, A.: 1967, Problèmes Ergodiques de la Mécanique Classique, Gauthier-Villars, Paris.Google Scholar
  4. Benzi, R., Paladin, G., Parisi, G. and Vulpiani, A.: 1985. J. Phys. A. 18, 2157.Google Scholar
  5. Chirikov, B.V.: 1979, Phys. Rep. 52, 263.Google Scholar
  6. Chirikov, B.V., Ford, J. and Vivaldi, F.: 1979, in Month, M. and Herreira, J.C. (Eds), “Nonlinear Dynamics and the Beam-Beam Interaction”, Amer. Inst. Phys., N. York.Google Scholar
  7. Contopoulos, G.: 1966, in Nahon, F. and Hénon, M. (eds) “Les Nouvelles Méthodes de la Dynamique Stellaire”, CNRS, Paris ≡ Bull. Astron. Ser. 3, 2, 223.Google Scholar
  8. Froeschle, C., Froeschle, Ch. and Lohinger, E.: 1993, Celest. Mech. Dyn. Astron. 56, 307.Google Scholar
  9. Fujisaka, H.: 1983, Prog. Theor. Phys. 70, 1264.Google Scholar
  10. Grassberger, P. and Procaccia, I.: 1984, Physica D13, 34.Google Scholar
  11. Laskar, J.: 1993, Physica D67, 257.Google Scholar
  12. Laskar, J., Froeschle, C. and Celletti, A.: 1992, Physica D56, 253.Google Scholar
  13. Lichtenberg, A.J. and Lieberman, M.A.: 1983, Regular and Stochastic Motion, Springer Verlag, N.York.Google Scholar
  14. Lohinger, E., Froeschle, C. and Dvorak, R.: 1993, Cel. Mech. Dyn. Astron. 56, 315Google Scholar
  15. Rosenbluth, M.N., Sagdeev, R.A., Taylor, J.B. and Zaslavskii, M.: 1966, Nucl. Fusion 6, 297.Google Scholar
  16. Sepulveda, M.A., Badii, R. and Pollak, E.: 1989, Phys. Rev. Lett. 63, 1226.Google Scholar
  17. Sinai, Ya.G.: 1989, Dynamical Systems II, Springer, Berlin, Heidelberg, N. York.Google Scholar
  18. Udry, S. and Pfenniger, D.: 1988, Astron. Astrophys. 198, 135.Google Scholar
  19. Voglis, N. and Contopoulos, G.: 1994, J. Phys. A27, 4899.Google Scholar
  20. Zaslavskii, G. M. and Chirikov, B.V.: 1972, Sov. Phys. Uspekhi 14, 549.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • G. Contopoulos
    • 1
  • N. Voglis
    • 1
  1. 1.Department of AstronomyUniversity of Athens PanepistimiopolisAthensGreece

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