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The Lyapunov Characteristic Exponents — Applications to Celestial Mechanics

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The Stability of Planetary Systems

Abstract

After a presentation of Lyapunov characteristic exponents (LCE) we recall their basic properties and numerical methods of computation. We review some numerical computations which are concerned with LCEs and mainly related to celestial mechanics problems.

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References

  1. Chaotic behaviour of deterministic systems, Cours des Houches XXXVI, North Holland, 1981.

    Google Scholar 

  2. Arnold V.I,, Mathematical methods of classical Mechanics. Translation Springer Verlag, Heidelberg, 1978.

    MATH  Google Scholar 

  3. Benettin G., Galgani L. and Strelcyn J.M., ‘Kolmogo’rov Entropy and numerical experiments’, Phys. Rev. A 14, p. 2338 – 2345, 1976.

    Article  ADS  Google Scholar 

  4. Benettin G., Strelcyn J.M., ‘Numerical experiments on the free motion of a point mass moving in a plane convex region; stochastic transition and Entropy’, Phys. Rev. 17, p. 773 – 785, 1978.

    Article  ADS  Google Scholar 

  5. Benettin G., Froeschlé C., Scheidecker J.P., ‘Kolmogorov Entropy of Dynamical systems with increasing number of degrees of freedom’, Phys. Rev. A 19, p. 2454 – 2460, 1979.

    Article  Google Scholar 

  6. Benettin G., Galgani L., Giorgilli A., Strelcyn J.M, Strelcyn J.M., ‘Lyapunov characteristic exponents for smooth Dynamical systems; a method for computing all of them’. Part 1: Theory; Part 2: Numerical application, Meccanica March 1980.

    Google Scholar 

  7. Cesari L., Asymptotic Behaviour and Stability Problems in Ordinary differential equations, Springer Verlag Berlin, 1959.

    Google Scholar 

  8. Chirikov B.V., ‘An universal instability of many dimensional oscillator systems’, Phys. Rep.52, p. 263, 1979.

    Article  MathSciNet  ADS  Google Scholar 

  9. Farmer J.D., Otte and Yorke J., ‘The dimension of chaotic attractors’ Physica 7D, p. 153–180, 1983.

    Google Scholar 

  10. Ford J., Fundamental problems in Statistical Mechanics, ed. E.G.D. Cohen, Vol III (North-Holland, Amsterdam) p. 215–255, 1975.

    Google Scholar 

  11. Froeschlé C., ‘Numerical Study of Dynamical systems with three degrees of Freedom, ’ I Numerical displays of four- dimensional sections’, Astron. Astrophys. 4, p. 115–128, 1970.

    ADS  Google Scholar 

  12. Froeschlé C., ‘Numerical Study of Dynamical systems with three degrees of Freedom, ’ II Numerical displays of four- dimensional sections’, Astron. Astrophys. 5, p. 177–183, 1970.

    ADS  Google Scholar 

  13. Froeschlé C., ‘A numerical Study of the Stochasticity of Dynamical Systems with two degrees of freedom’, Astron. Astrophys. 9, p. 15–23, 1970.

    ADS  MATH  Google Scholar 

  14. Froeschlé C. and Scheidecker J.P., ‘Numerical Study of the Stochasticity of Dynamical Systemswith more than two degrees of freedom’, Astron. Astrophys.22, p. 431, 1973.

    ADS  MATH  Google Scholar 

  15. Froeschlé C. and Scheidecker J.P., ‘On the disappearance of isolating integrals in systems with more than two degrees of freedom’, Astrophys. and Space Sc.25, p. 273, 1973.

    Google Scholar 

  16. Froeschlé C. and Scheidecker J.P., ‘Numerical study of the stochasticity of Dynamical System with more than two degrees of freedom’, J. Comp. Phys. Vol. 11, n° 3, 1973

    Article  Google Scholar 

  17. Froeschlé C., Scheidecker J.P., ‘Stochasticity of dynamical systems with increasing number of degrees of freedom’, Phys. Rev. A 12, p. 2137 – 2143, 1975.

    Article  ADS  Google Scholar 

  18. Froeschlé C. and Scholl H., ‘On the dynamical topology of the Kirkwood gaps’, Astron. Astrophys., 48, p. 389–393, 1976.

    ADS  Google Scholar 

  19. Froeschlé C. and Scholl H., ‘The stochasticity of pecular orbits in the 2/1 Kirkwood gap’, Astron.Astrophs.93, 62–66, 1981.

    ADS  Google Scholar 

  20. Froeschlé C., Gonczi P.., ‘Lyapunov characteristic numbers and Kolmogorov entropy of a four dimensional Mapping’, Nuovo Cimento, vol 55 B, n° 1, p. 59 – 69, 1980.

    MathSciNet  Google Scholar 

  21. Froeschlé C. and Scholl H., ‘The dynamical structure of the asteroidal belt (review paper)’ Proceeding of Uppsala meeting, p. 115–125, 1983.

    Google Scholar 

  22. Giffen R., ‘A study of commensurable motion in the asteroid belt’, Astron. Astrophys. 23, p. 387–403, 1973.

    ADS  Google Scholar 

  23. Gonczi R., Froeschlé C., ‘The Lyapunov characteristic exponents as indicators of stochasticity in the restricted Three-body Problem’, Cel. Mech. 25, p. 271 – 280, 1981.

    Article  ADS  MATH  Google Scholar 

  24. Guikenheimer J., Moser J., Newhouse S., Dynamical Systems, CIME lectures Birkhauser, 1978.

    Google Scholar 

  25. Hénon M. and Heiles C., ‘The applicatibility of the third integral of motion, some numerical experiments’, Astron. Journal 69, p. 73–79, 1964.

    Article  ADS  Google Scholar 

  26. Hénon M., ‘Exploration numerique du probleme restreint’, I Ann. Astr. 28, p. 499, 1965.

    ADS  Google Scholar 

  27. Hénon M., ‘Exploration numerique du probleme restreint’, II Ann. Astr. 28, p. 992, 1965.

    ADS  Google Scholar 

  28. Hénon M., ‘Exploration numerique du probleme restreint’, III Bull. Astr. Paris, 1, p. 57, 1966.

    Google Scholar 

  29. Hénon M., ‘Exploration numerique du probleme restreint’, IV Bull. Astr. Paris, 1, fasc. 2, p. 49, 1966.

    Google Scholar 

  30. Hénon M. and Wisdom J., ‘The Benettin-Strelcyn oval Billiard revisited’ Physica 8D, p. 157–169, 1983.

    Google Scholar 

  31. Jefferys W.H., ‘An atlas of surfaces of section for the restricted problem of the three bodies’, Publications of the department of Astronomy of the University of Texas at Austin, ser. II., 3, 6, 1971.

    ADS  Google Scholar 

  32. Jefferys W.H., and Zhao-Hua Yi, ‘Stability in the restricted problem of three bodies with Lyapunov characteristic numbers’, Cel. Mech. 30, p. 85–95, 1983.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. MacKay R.S., Meiss J.D. and Percival I.C., “Transport in Hamiltonian system”, preprint.

    Google Scholar 

  34. Oseledec V.I., ‘A multiplicative ergodic theorem. The Lyapunov characteristic numbers of Dynamical systems (in Russian)’, Trudy Mosk. Mat. Obsc. 19p. 179–210, 1968.

    MathSciNet  Google Scholar 

  35. Oseledec V.I., ‘A multiplicative ergodic theorem. The Lyapunov characteristic numbers of Dynamical systems (in Russian)’, English translation in Trans. Mosc. Math. Soc. 19p. 197, 1968.

    MathSciNet  Google Scholar 

  36. Schubart J., ‘Long-period effects in nearly commensurable cases of the restricted three-body problem in Smithsonian Astrophys’. Obs. Spec. Rep. n° 149, 1964.

    Google Scholar 

  37. Wisdom J. ‘Chaotic behaviour and the origin of 3/1 Kirkwood gap’, Icarus 56, p. 51–74, 1983.

    Article  ADS  Google Scholar 

  38. Szebehely V. Theory of orbits, Academic Press, 1977.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Observatoire de Nice, B.P. 139, 06003, Nice Cedex, France

    Cl. Froeschlé

Authors
  1. Cl. Froeschlé
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Editor information

Editors and Affiliations

  1. Department of Aerospace Engineering, University of Texas, Austin, Texas, USA

    R. L. Duncombe

  2. Institut für Astronomie, Universität Wien, Wien, Österreich

    R. Dvorak

  3. Department of Applied Mathematics and Theoretical Physics, Liverpool University, England

    P. J. Message

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© 1984 D. Reidel Publishing Company

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Froeschlé, C. (1984). The Lyapunov Characteristic Exponents — Applications to Celestial Mechanics. In: Duncombe, R.L., Dvorak, R., Message, P.J. (eds) The Stability of Planetary Systems. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5331-4_9

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  • DOI: https://doi.org/10.1007/978-94-009-5331-4_9

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-8854-1

  • Online ISBN: 978-94-009-5331-4

  • eBook Packages: Springer Book Archive

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