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Persistence of invariant torus in Hamiltonian systems with two-degree of freedom. (English) Zbl 1134.37025
Consider the following Hamiltonian dynamical system: \(.{q}=H_p(p,q), \quad .{p}=-H_q(p,q)\) where the Hamiltonian function is \(H=h(p)+f(q,p)\). The classical KAM theorem asserts that if \(h\) is not degenerate i.e. det\((h_{pp})\neq 0\) then most of the invariant tori can persist when \(f\) is sufficiently small. In general, the nondegeneracy condition is necessary for KAM theorems. However, the Hamiltonian systems with two degrees of freedom have some special properties and so, the present paper is devoted to a KAM theorem for a class of 2D Hamiltonian systems without any nondegeneracy condition. The main tool is the so-called KAM iteration.

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
70H08 Nearly integrable Hamiltonian systems, KAM theory
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[1] Kolmogorov, A.N., On the persistence of quasi-periodic motions under small perturbations of the Hamiltonian, Dokl. akad. nauk USSR, 98, 527-530, (1954) · Zbl 0056.31502
[2] Arnol’d, V.I., Proof of A.N. Kolmogorov’s theorem on the preservation of quasi-periodic motions under small perturbations of the Hamiltonian, Russian math. surveys, 18, 5, 9-36, (1963) · Zbl 0129.16606
[3] Eliasson, L.H., Perturbations of stable invariant tori for Hamiltonian systems, Ann. sc. norm. super. Pisa cl. sci. (5), 15, 115-147, (1998) · Zbl 0685.58024
[4] J. Pöschel, A lecture on the classical KAM theorem, School on Dynamical Systems, May 1992
[5] Pöschel, J., On elliptic lower dimensional tori in Hamiltonian systems, Math. Z., 202, 559-608, (1989) · Zbl 0662.58037
[6] H. Rüssmann, On twist Hamiltonians, Talk on the Colloque International: Mécanique céleste et systémes hamiltonients, Marseille, 1990
[7] H. Rüssmann, Invariant tori in the perturbation theory of weakly non-degenerate integrable Hamiltonian systems, preprint, July 1998
[8] Rüssmann, H., Convergent transformations into a normal form in analytic Hamiltonian systems with two degree of freedom on the zero energy surface near degenerate elliptic singularities, Ergodic theory dynam. systems, 23, 1-44, (2003)
[9] Sevryuk, M.B., KAM-stable Hamiltonians, J. dyn. control syst., 1, 351-366, (1995) · Zbl 0951.37038
[10] Xu, J.; You, J.; Qiu, Q., Invariant tori of nearly integrable Hamiltonian systems with degeneracy, Math. Z., 226, 375-387, (1997) · Zbl 0899.34030
[11] Xu, J.; You, J., Persistence of lower dimensional tori under the first Melnikov’s non-resonance condition, J. math. pures appl., 80, 10, 1045-1067, (2001) · Zbl 1031.37053
[12] J. Xu, On reducibility for two dimensional linear differential equations with quasi-periodic coefficients close to constants, preprint, 2005
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