Predictability, Stability and Chaos in Dynamical Systems

  • Christian Marchal
Part of the NATO ASI Series book series (NSSB, volume 272)


Progress in the theory of stability is presented in the particular example of the Lagrangian motions of the three-body problem and is then generalized.

The notion of predictability is discussed and, surprisingly, in some cases the chaotic motions allow better predictions than the regular motions.

The Arnold diffusion conjecture gives a general picture of Hamiltonian dynamical systems, a picture in which chaotic motions have a major part.

All these recent advances have renewed the picture of Physics.


Chaotic Motion Strange Attractor Effective Stability Chaotic Solution Observable Universe 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A Giorgilli — A. Delshams — E. Fontich — L. Galgani — C. Simo — “ Effective Stability for a Hamiltonian System near an Elliptic Equilibrium Point, with an Application to the Restricted Three Body Problem”. Journal of Differential Equations - 77 p.l67–198 (1989).MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. Celletti - A. Giorgilli, “On the stability of the Lagrangian points in the restricted problem of three bodies”. CARR reports in Mathematical Physics - nl9/90 - July 1990.Google Scholar
  3. 3.
    A Bennett - “Characteristics exponents of the Five Equilibrium Solutions in the Elliptically Restricted Problem” Icarus 4, p.177 (1965).ADSCrossRefGoogle Scholar
  4. 4.
    J.M.A. Danby - “Stability of Triangular Points in the Elliptic Restricted Problem of Three Bodies” Astronomical Journal, p.165 (1964).Google Scholar
  5. 5.
    A.B. Leontovitch - “On the stability of the Lagrange periodic solution of the restricted three-body problem” Dokl-Akad. Nauk.SSSR. Vol.143 No.3-p.525–528 (1962).MathSciNetGoogle Scholar
  6. 6.
    A. Deprit - A. Deprit-Bartholome “Stability of the triangular Lagrangian point” Astronomical Journal, 72, No.2, p.173–179 (1967).ADSCrossRefGoogle Scholar
  7. 7.
    C. Marchal - “Etude de la stabilite des solutions de Lagrange du problém des trois corps. Cas où 1’excentricité et les trois masses sont quelconques”. Séminaire du Bureau des Longitudes (7 mars 1968).Google Scholar
  8. 8.
    J. Tschauner - “Die Bewegung in der Nähe der Dreieckspunkte des elliptischen eingeschränkten Dreikörperproblems”. Celestial Mechanics 3, p.189–196 (1971).ADSCrossRefMATHGoogle Scholar
  9. 9.
    A. Alothman-Alragheb - “Influence des perturbations d’ordre élevé sur la stabilité des systémes hamiltoniens (cas à deux degrés de liberté)”. Thése de Doctorat d’Etat. Observatoire de Paris - ler juillet 1986.Google Scholar
  10. 10.
    L. El Bakkali - “Voisinage et stabilité des solutions periodiques des systémes hamiltoniens. Application aux solutions de Lagrange du Problemé des trois corps”. Thése de Doctorat d’Etat. Observatoire de Paris. 27 juin 1990.Google Scholar
  11. 11.
    C. Marchai - “The three-body problem”. Elsevier Science Publishers B.V. Amsterdam (1990).Google Scholar
  12. 12.
    V.I. Arnold - A. Avez - “Problémes ergodiques de la Mécanique classique” Gauthier-Villars, Paris (1967).Google Scholar
  13. 13.
    E. N. Lorenz - “Deterministic non-Periodic Flow”. J. Atmos. Sci, Vol.20 p.130–141 (1963).ADSCrossRefGoogle Scholar
  14. 14.
    M. Hénon - “Numerical study of quadratic area-preserving mappings” Quarterly of Applied Mathematics - Vol.27 - No.3 p.291 (1969).MathSciNetMATHGoogle Scholar
  15. 15.
    M. Henon - “A two-dimensional mapping with a strange attractor” Communications in Mathematical Physics p.69 (1976).Google Scholar
  16. 16.
    D. Ruelie - F. Takens - “On the nature of turbulence”. (Commun. Math. Physics - Vol.20, p.167–192 (1971).ADSCrossRefGoogle Scholar
  17. 17.
    P. Collet - J.P. Eckmann- “Iterated maps on the interval as dynamical systems” Progress in Physics 1 - A. Jaffe - D. Ruelle editors - Birkhäuser (1983).Google Scholar
  18. 18.
    P. Bérge - Y. Pomeau - C. Vidal - “L’ordre dans le chaos”. Hermann (1984).MATHGoogle Scholar
  19. 19.
    J. Laskar - “Numerical experiment on the chaotic behaviour of the Solar System” Nature 338 - p.237–238 (1989).ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • Christian Marchal
    • 1
  1. 1.D.E.S. — OneraChatillonFrance

Personalised recommendations