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Integrability of Hamiltonian systems and Birkhoff normal forms in the simple resonance case

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References

  1. Arnol'd, V.I.: Mathematical methods of classical mechanics, 2nd ed. (Grad Texts Math., vol. 60) Berlin Heidelberg New York: Springer 1989

    Google Scholar 

  2. Arnol'd, V.I.: Geometrical methods in the theory of ordinary differential equations, 2nd ed. Berlin Heidelberg New York: Springer 1988

    Google Scholar 

  3. Arnol'd, V.I., Kozlov, V.V., Neishtadt, A.I.: Dynamical systems III: Mathematical aspects of classical and celestial mechanics. (Encycl. Math. Sci., vol. 3) Berlin Heidelberg New York: Springer 1988

    Google Scholar 

  4. Churchill, R.C., Lee, D.: Harmonic oscillators at low energies. In: Chudnovsky, D.V., Chudnovsky, G.V. (eds.) Classical and quantum models and arithmetic problems. (Lect. Notes Pure Appl. Math., vol. 92, pp. 239–286). New York Basel: Marcel Dekker 1984

    Google Scholar 

  5. Duistermaat, J.J.: Non-integrability of the 1∶1∶2-resonance. Ergodic Theory Dyn. Sys.4, 553–568 (1984)

    Google Scholar 

  6. Eliasson, H.: Normal forms for Hamiltonian systems with Poisson commuting integralselliptic case. Comment. Math. Helv.65, 4–35 (1990)

    Google Scholar 

  7. Hoveijn, I., Verhulst, F.: Chaos in the 1∶2∶3 Hamiltonian normal form. Physica D44, 397–406 (1990)

    Google Scholar 

  8. Humphreys, J.E.: Introduction to Lie algebras and representation theory, 2nd printing. (Grad. Texts Math., vol. 9) Berlin Heidelberg New York: Springer 1972

    Google Scholar 

  9. Ito, H.: Convergence of Birkhoff normal forms for integrable systems. Comment. Math. Helv.64, 412–461 (1989)

    Google Scholar 

  10. Ito, H.: Action-angle coordinates at singularities for analytic integrable systems. Math. Z.206, 363–407 (1991)

    Google Scholar 

  11. Markus, L., Meyer, K.R.: Generic Hamiltonian dynamical systems are neither integrable nor ergodic. Mem. Am. Math. Soc.144 (1974)

  12. Moser, J.: On the generalization of a theorem of A. Liapounoff. Commun. Pure Appl. Math.11, 257–271 (1958)

    Google Scholar 

  13. Moser, J.: Lectures on Hamiltonian systems. Mem. Am. Math. Soc.81 (1968)

  14. Moser, J.: Stable and random motions in dynamical systems, with special emphasis on celestial mechanics. (Ann. Math. Stud., Vol. 77) Princeton: Princeton University Press 1973

    Google Scholar 

  15. Siegel, C.L.: Über die Existenz einer Normalform analytischer Hamiltonischer Differentialgleichungen in der Nähe eine Gleichgewichtslösung. Math. Ann.128, 144–170 (1954)

    Google Scholar 

  16. van der Meer, J.-C.: The Hamiltonian Hopf Bifurcation. (Lect. Notes Math. vol. 1160) Berlin Heidelberg New York: Springer 1985

    Google Scholar 

  17. Vey, J.: Sur certain systèmes dynamiques séparables. Am. J. Math.100, 591–614 (1978)

    Google Scholar 

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Ito, H. Integrability of Hamiltonian systems and Birkhoff normal forms in the simple resonance case. Math. Ann. 292, 411–444 (1992). https://doi.org/10.1007/BF01444629

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