Regular and Chaotic Dynamics

, Volume 19, Issue 3, pp 363–373 | Cite as

Normal form and Nekhoroshev stability for nearly integrable hamiltonian systems with unconditionally slow aperiodic time dependence

  • Alessandro FortunatiEmail author
  • Stephen Wiggins


The aim of this paper is to extend the result of Giorgilli and Zehnder for aperiodic time dependent systems to a case of nearly integrable convex analytic Hamiltonians. The existence of a normal form and then a stability result are shown in the case of a slow aperiodic time dependence that, under some smallness conditions, is independent of the size of the perturbation.


Hamiltonian systems Nekhoroshev theorem aperiodic time dependence 

MSC2010 numbers

70H08 37J25 37J40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arnol’d, V. I., Proof of a Theorem of A. N. Kolmogorov on the Invariance of Quasi-periodic Motions under Small Perturbations of the Hamiltonian, Russian Math. Surveys, 1963, vol. 18, no. 5, pp. 9–36; see also: Uspekhi Mat. Nauk, 1963, vol. 18, no. 5, pp. 13–40.CrossRefzbMATHGoogle Scholar
  2. 2.
    Benettin, G., Galgani, L., and Giorgilli, A., A Proof of Nekhoroshev’s Theorem for the Stability Times in Nearly Integrable Hamiltonian Systems, Celestial Mech., 1985, vol. 37, no. 1, pp. 1–25.zbMATHMathSciNetGoogle Scholar
  3. 3.
    Bounemoura, A., Effective Stability for Slow Time-Dependent Near-Integrable Hamiltonians and Application, C. R. Math. Acad. Sci. Paris, 2013, vol. 351, nos. 17–18, pp. 673–676.CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Gallavotti, G., Quasi-Integrable Mechanical Systems, in Phénomènes critiques, systèmes aléatoires, théories de jauge (Les Houches, 1984): Part 1, 2, Amsterdam: North-Holland, 1986, pp. 539–624.Google Scholar
  5. 5.
    Giorgilli, A., Notes on Exponential Stability of Hamiltonian Systems, in Dynamical Systems: Part 1. Hamiltonian Systems and Celestial Mechanics, Pisa: Centro di Recerca Matematica Ennio De Giorgi, Scuola Normale Superiore, 2002.Google Scholar
  6. 6.
    Glimm, J., Formal Stability of Hamiltonian Systems, Comm. Pure Appl. Math., 1964, vol. 17, pp. 509–526.CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Giorgilli, A. and Zehnder, E., Exponential Stability for Time Dependent Potentials, Z. Angew. Math. Phys., 1992, vol. 43, no. 5, pp. 827–855.CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Kolmogorov, A.N., On the Conservation of Conditionally Periodic Motions under Small Perturbation of the Hamiltonian, Dokl. Akad. Nauk SSSR, 1954, vol. 98, pp. 527–530 (Russian). (See also: Stochastic Behaviour in Classical and Quantum Hamiltonian Systems (Volta Memorial Conference, Como, 1977), G.Casati, J.Ford (Eds.), Lect. Notes Phys. Monogr., vol. 93, Berlin: Springer, 1979, pp. 51–56).zbMATHMathSciNetGoogle Scholar
  9. 9.
    Koshy, Th., Catalan Numbers with Applications, Oxford: Oxford Univ. Press, 2009.zbMATHGoogle Scholar
  10. 10.
    Littlewood, J.E., The Lagrange Configuration in Celestial Mechanics, Proc. London Math. Soc. (3), 1959, vol. 9, pp. 525–543; addendum: 1959, vol. 10, p. 640.CrossRefzbMATHGoogle Scholar
  11. 11.
    Littlewood, J.E., On the Equilateral Configuration in the Restricted Problem of Three Bodies, Proc. London Math. Soc. (3), 1959, vol. 9, pp. 343–372.CrossRefMathSciNetGoogle Scholar
  12. 12.
    Lochak, P., Canonical Perturbation Theory: An Approach Based on Joint Approximations, Russian Math. Surveys, 1992, vol. 47, no. 6, pp. 57–133; see also: Uspekhi Mat. Nauk, 1992, vol. 47, no. 6(288), pp. 59–140.CrossRefMathSciNetGoogle Scholar
  13. 13.
    Morbidelli, A. and Giorgilli, A., Superexponential Stability of KAM Tori, J. Statist. Phys., 1995, vol. 78, nos. 5–6, pp. 1607–1617.CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Moser, J., Stabilitätsverhalten kanonischer Differentialgleichungssysteme, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. IIa, 1955, vol. 1955, pp. 87–120zbMATHGoogle Scholar
  15. 15.
    Nekhoroshev, N. N., An Exponential Estimate of the Stability Time of Near-Integrable Hamiltonian Systems, Russian Math. Surveys, 1977, vol. 32, no. 6, pp. 1–65; see also: Uspekhi Mat. Nauk, 1977, vol. 32, no. 6(198), pp. 5–66CrossRefzbMATHGoogle Scholar
  16. 16.
    Nekhoroshev, N. N., An Exponential Estimate of the Time of Stability of Nearly Integrable Hamiltonian Systems: 2, Trudy Sem. Petrovsk., 1979, no. 5, pp. 5–50 (Russian).Google Scholar
  17. 17.
    Poincaré, H., Les méthodes nouvelles de la mécanique céleste, Paris: Gauthier-Villars, 18Google Scholar
  18. 18.
    Pöschel, J., Nekhoroshev Estimates for Quasi-Convex Hamiltonian Systems, Math. Z., 1993, vol. 213, no. 2, pp. 187–216.CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Pöschel, J., A Lecture on the Classical KAM Theorem, in Smooth Ergodic Theory and Its Applications (Univ. of Washington, Seattle, July 26–August 13, 1999), A. B. Katok et al. (Eds.), Proc. Sympos. Pure Math., vol. 69, Providence, R.I.: AMS, 2001, 707–732.CrossRefGoogle Scholar
  20. 20.
    Perry, A.D. and Wiggins, S., KAM Tori Are Very Sticky: Rigorous Lower Bounds on the Time to Move Away from an Invariant Lagrangian Torus with Linear Flow, Phys. D, 1994, vol. 71, nos. 1–2, pp. 102–121.CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Wiggins, S. and Mancho, A., Barriers to Transport in Aperiodically Time-Dependent Two-Dimensional Velocity Fields: Nekhoroshev’s Theorem and “Nearly Invariant” Tori, Nonlin. Processes Geophys., 2014, vol. 21, pp. 165–185.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.School of MathematicsUniversity of BristolBristolUK

Personalised recommendations