Abstract.
We study the persistence of invariant tori on resonant surfaces of a nearly integrable Hamiltonian system under the usual Kolmogorov non-degenerate condition. By introducing a quasi-linear iterative scheme to deal with small divisors, we generalize the Poincaré theorem on the maximal resonance case (i.e., the periodic case) to the general resonance case (i.e., the quasi-periodic case) by showing the persistence of majority of invariant tori associated to non-degenerate relative equilibria on any resonant surface.
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Arnold, V.I.: Proof of a theorem by A. N. Kolmogorov on the preservation of quasi-periodic motions under small perturbations of the Hamiltonian. Usp. Mat. Nauk. 18, 13–40 (1963)
Bolotin, S.V.: Homoclinic orbits to invariant tori of Hamiltonian systems. Dynamical Systems in Classical Mechanics. Amer. Math. Soc. Transl. 168 (2), 21–90 (1995)
Broer, H., Huitema, G., Sevryuk, M.: Quasi-Periodic Motions in Families of Dynamical Systems. Lect. Notes Math. 1645, Springer-Verlag, 1996
Cheng, C.-Q.: Birkhoff-Kolmogorov-Arnold-Moser tori in convex Hamiltonian systems. Comm. Math. Phys. 177, 529–559 (1996)
Chierchia, L., Gallavotti, G.: Drift and diffusion in phase space. Ann. Inst. H. Poincaré Phy. Th. 69, 1–144 (1994)
Cong, F.Z., Küpper, T., Li, Y., You, J.G.: KAM-type theorem on resonant surfaces for nearly integrable Hamiltonian systems. J. Nonl. Sci. 10, 49–68 (2000)
Eliasson, L.H.: Biasymptotic solutions of perturbed integrable Hamiltonian systems. Bol. Soc. Mat. 25, 57–76 (1994)
Kolmogorov, A.N.: On the conservation of conditionally periodic motions for a small change in Hamilton's function. Dokl. Akad. Nauk. SSSR 98, 525–530 (1954)
Moser, J.: On invariant curves of area preserving mappings of an annulus. Nachr. Akad. Wiss. Gött. Math. Phys. K1, 1–20 (1962)
Poincaré, H.: Les Méthodes Nouvelles de la Mécaniques Céleste, I–III, Gauthier-Villars, 1892, 1893, 1899 (The English translation: New Methods of Celestial Mechanics, AIP Press, Williston, 1992)
Pöschel, J.: Integrability of Hamiltonian systems on Cantor sets. Comm. Pure Appl. Math. 35, 653–696 (1982)
Rudnev, M., Wiggins, S.: KAM theory near multiplicity one resonant surfaces in perturbations of A-priori stable Hamiltonian systems. J. Nonl. Sci. 7, 177–209 (1997)
Siegel, C.L., Moser, J.K.: Lectures on Celestial Mechanics. Grundlenhren 187, Springer, Berlin, 1971
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970
Treshchev, D.V.: The mechanism of destruction of resonance tori of Hamiltonian systems. Math. USSR Sb. 68, 181–203 (1991)
Xu, J.X., You, J.G., Qiu, Q.J.: Invariant tori for nearly integrable Hamiltonian systems with degeneracy. Math. Z. 226, 375–387 (1997)
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The first author was partially supported by NSFC grant 19971042, the National 973 Project of China: Nonlinearity, and the outstanding young's project of the Ministry of Education of China.
The second author was partially supported by NSF grant DMS9803581.
Mathematics Subject Classification (2000): Primary 58F05, 58F27, 58F30
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Li, Y., Yi, Y. A quasi-periodic Poincaré's theorem. Math. Ann. 326, 649–690 (2003). https://doi.org/10.1007/s00208-002-0399-0
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DOI: https://doi.org/10.1007/s00208-002-0399-0