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Invariant tori with prescribed frequency for nearly integrable Hamiltonian systems. (English) Zbl 1345.70027

The celebrated KAM Theorem was founded by Kolmogorov, Arnold and Moser to conquer the problem of the persistance of quasi-periodic solutions and invariant tori for nearly integrable Hamiltonian systems.
In this paper an improved KAM iterative scheme is used to obtain that not only the frequency of perturbed invariant tori keeps the original direction (an analogous result to iso-energetic KAM Theorem), but also the unperturbed and perturbed invariant torus are bridged by an analytic conjugation transformation. Moreover, the entire tori form a one-parameter analytic family.

MSC:

70H08 Nearly integrable Hamiltonian systems, KAM theory
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[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, Russ. Math. Surv., 18, 5, 9-36 (1963) · Zbl 0129.16606 · doi:10.1070/RM1963v018n05ABEH004130
[2] Arnold, V. I., Mathematical Methods of Classical Mechanics (1980)
[3] Bourgain, J., On Melnikov”s persistency problem, Math. Res. Lett., 4, 445-458 (1997) · Zbl 0897.58020 · doi:10.4310/MRL.1997.v4.n4.a1
[4] Broer, H. W.; Huitema, G. B., A proof of the isoenergetic KAM-theorem from the “Ordinary” one, J. Differ. Equ., 90, 52-60 (1991) · Zbl 0721.58020 · doi:10.1016/0022-0396(91)90160-B
[5] Bruno, A. D., Analytic form of differential equations, Trans. Moscow Math. Soc., 25, 131-288 (1971)
[6] Cheng, C.; Sun, Y., Existence of KAM tori in degenerate Hamiltonian systems, J. Differ. Equ., 114, 288-335 (1994) · Zbl 0813.58050 · doi:10.1006/jdeq.1994.1152
[7] Chow, S.; Li, Y.; Yi, Y., Persistence of invariant tori on submanifolds in Hamiltonian systems, J. Nonlinear Sci., 12, 585-617 (2003) · Zbl 1012.37043 · doi:10.1007/s00332-002-0509-x
[8] Eliasson, L. H., Perturbations of stable invariant tori for Hamiltonian systems, Ann. Sci. Norm. Superior. Pisa, 15, 115-147 (1988) · Zbl 0685.58024
[9] Huitema, G. B., Unfolding of quasi-periodic tori (1988)
[10] Kolmogorov, A. N., On conservation of conditionally periodic motions for a small change in Hamilton”s function, Dokl. Akad. Nauk SSSR (N.S.), 98, 527-530 (1954) · Zbl 0056.31502
[11] Moser, J., On invariant curves of area preserving mappings of an annulus, Nachr. Akad. Wiss. Gött. Math. Phys. Kl., II, 1-20 (1962) · Zbl 0107.29301
[12] Pöschel, J., A lecture on the classical KAM theorem, Proc. Symp. Pure Math., 69, 707-732 (2001) · Zbl 0999.37053 · doi:10.1090/pspum/069/1858551
[13] Rüssmann, H., On twist Hamiltonian, Talk on the Colloque International: Mécanique Céleste et Systèmes Hamiltoniens (1990)
[14] Rüssmann, H., Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regular Chaotic Dynam., 6, 119-204 (2001) · Zbl 0992.37050 · doi:10.1070/RD2001v006n02ABEH000169
[15] Rüssmann, H., Addendum to ‘Invariant tori in non-degenerate nearly integrable Hamiltonian systems, “ Regular Chaotic Dynam., 10, 21-31 (2005) · Zbl 1078.30508 · doi:10.1070/RD2005v010n01ABEH000297
[16] Sevryuk, M. B., KAM-stable Hamiltonians, J. Dyn. Control Syst., 1, 351-366 (1995) · Zbl 0951.37038 · doi:10.1007/BF02269374
[17] Sevryuk, M. B., Partial preservation of frequencies in KAM theory, Nonlinearity, 19, 5, 1099-1140 (2006) · Zbl 1100.37037 · doi:10.1088/0951-7715/19/5/005
[18] Xu, J.; You, J.; Qiu, Q., Invariant tori for nearly integrable Hamiltonian systems with degeneracy, Math. Z., 226, 375-387 (1997) · Zbl 0899.34030 · doi:10.1007/PL00004344
[19] Xu, J.; You, J., Persistence of the non-twist torus in nearly integrable hamiltonian systems, Proc. Am. Math. Soc., 138, 7, 2385-2395 (2010) · Zbl 1202.37079 · doi:10.1090/S0002-9939-10-10151-8
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