Generalized Lyapunov exponents indicators in Hamiltonian dynamics: An application to a double star system

  • E. Lohinger
  • C. Froeschlé
  • R. Dvorak
Session On Chaos And Stability


The Lyapunov characteristic numbers (LCNs) which are defined as the mean value of the distribution of the local variations of the tangent vectors to the flow (=ln α k i ) (see Froeschlé, 1984) have been found to be sensitive indicators of stochasticity. So we computed the distribution of these local variations and determined the moments of higher order for the integrable and stochastic regions in a binary star system with μ=0.5.

Key words

Double star systems Lyapunov exponents chaotic motion 


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  1. Benettin G., Galgani L., Giorgilli A., Strelcyn J. M.: 1980, “Lyapunov characteristic exponents for smooth dynamical systems; a method for computing all of them”.Meccanica vol.15, part I+II, p. 9–30Google Scholar
  2. Bulirsch R., Stoer J.: 1966, “Numerical Treatment of Ordinary Differential Equations by Extrapolation Methods”.Numerische Mathematik 8, p. 1–13.Google Scholar
  3. Chirikov B. V.: 1979, “An universal instability of many dimensional oscillator systems”. Phys. Rep.52, p. 263–379.Google Scholar
  4. Ford J.: 1975, “The statistical mechanics of classical analytic dynamics, fundamental problems in statistical mechanics”. ed. E.G.D. Cohen, Vol. III (North-Holland, Amsterdam), p. 215–255.Google Scholar
  5. Froeschlé C.: 1970, “A numerical study of the stochasticity of dynamical systems with two degrees of freedom”.Astron. Astrophys. 9, p. 15–23.Google Scholar
  6. Froeschlé C., Scheidecker J. P.: 1973, “On the disappearance of isolating integrals in systems with more than two degrees of freedom”.Astrophys. and Space Sc., vol.25, p. 373–386.Google Scholar
  7. Froeschlé C.: 1984, “The Lyapunov characteristic exponents and applications”.Jounal de Mécanique théorique et appliquée, Numéro spécial, p. 101–132.Google Scholar
  8. Froeschlé C., Froeschlé Ch., Lohinger E.: 1992, “Generalized Lyapunov characteristic indicators and corresponding Kolmogorov like entropy of the standard mapping”, this volume.Google Scholar
  9. Gonczi R., Froeschlé C.: 1981, “The Lyapunov characteristic exponents as indicators of stochasticity in the restricted three-body problem”.Celest. Mech. 25, p. 271–280.Google Scholar
  10. Hénon M., Heiles C.: 1964, “The applicability of the third integral of motion, some numerical experiments”.Astron. Journal 69, p. 73–79.Google Scholar
  11. Lyapunov A. M.: 1907, “Problème Général de la Stabilité du Mouvement” (transl. from Russian),Ann. Fac. Sci. Univ. Toulouse 9, p. 203–475. Reproduce<d inAnn. Math. Study, vol.17, Princeton 1947.Google Scholar
  12. Oseldec V. I.: 1968, “The Multiplicative Ergodic”. Theorem. The Lyapunov Characteristic Numbers of Dynamical Systems (in Russian).Trudy Mosk. Mat. Obsch. 19, p. 179–210. English translation inTrans. Mosc. Math. Soc. 19, p. 197, 1968.Google Scholar
  13. Szebehely V.: 1967, “The Theory of Orbits”. Academic Press, New York and LondonGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • E. Lohinger
    • 1
  • C. Froeschlé
    • 2
  • R. Dvorak
    • 3
  1. 1.Institute of AstronomyUniversity of ViennaWienAustria
  2. 2.Observatoire de la Côte d'AzurNice Cedex 4France
  3. 3.Institute of AstronomyUniversity of ViennaWienAustria

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