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Hyperbolic tori in Hamiltonian systems with slowly varying parameter. (English. Russian original) Zbl 1305.37027
Sb. Math. 204, No. 5, 661-682 (2013); translation from Mat. Sb. 204, No. 5, 45-66 (2013).
The author studies the persistence of 2-dimensional hyperbolic invariant tori of a Hamiltonian system having one of their frequencies of the same order as the perturbation, while the other varies is a non-degenerate fashion.
The setting is the following. Let \(H(v,\tau)\) be a real analytic Hamiltonian, with \(v\in M\), a \(2n\)-dimensional real analytic symplectic manifold, \(2\pi\)-periodic in \(\tau\). For each value of \(\tau\), \(H(\cdot,\tau)\) defines a time-independent Hamiltonian on \(M\). It is assumed that for all \(\tau\) and \(h\in(h_1,h_2)\), the Hamiltonian \(H(\cdot,\tau)\) has a hyperbolic periodic orbit \(L_{h,\tau}\) in the energy shell \(\{ v;H(v,\tau) = h\}\), analytic with respect to \(h\) and \(\tau\). Let \(g(h,\tau)\) be the frequency of the periodic orbit \(L_{h,\tau}\) and let \[ \langle g(h) \rangle = \frac{1}{2\pi} \int_0^{2\pi} g(h,\tau)\,d\tau \] be the averaged frequency.
In the extended phase space, with \(\dot \tau = 0\), each of these periodic orbits can be seen as a hyperbolic \(2\)-torus with frequency \((g(h,\tau),1)\). The paper deals with the persistence of these \(2\)-tori when \(\dot \tau = \varepsilon\), for small \(\varepsilon\).
Of course, one can only guarantee the persistence of the tori whose frequencies satisfy a Diophantine condition. In this case, a vector \((g,\varepsilon)\in \mathbb{R}^2\) is said to be Diophantine if \[ |g \ell_1 + \varepsilon \ell_2 |\geq \frac{\varepsilon}{(|\ell_1|+ |\ell_2|)^2} \quad \forall (\ell_1,\ell_2) \in \mathbb{Z}^2, \; (\ell_1,\ell_2) \neq (0,0), \] where \(\varepsilon \in (0,\varepsilon_0)\) is a small parameter. The measure of the values of \(g\in [0,1]\) which do not satisfy the above condition is bounded by \(10\varepsilon\).
The result proven by the author is the following. If, for some \(h_0\in (h_1,h_2)\), \[ \int_0^{2\pi} \frac{\partial g}{\partial h}(h_0,\tau)\, d\tau \neq 0, \] then there exists \(\varepsilon_0\) such that for any \(\varepsilon \in (0,\varepsilon_0)\) for which \((\langle g(h_0) \rangle,\varepsilon)\) satisfies the Diophantine condition above, the non-autonomous Hamiltonian system \(H(v,\varepsilon t)\) has a hyperbolic \(2\)-torus with frequencies \((\langle g(h_0) \rangle,\varepsilon)\).
The claim is proven by means of a KAM scheme. First of all, appropriate coordinates are found around the torus in question, and the Hamiltonian is written as a sum of a system with a hyperbolic torus and a perturbation. Then the author constructs the iterative lemma through a sequence of changes (up to 10) of coordinates defined by generating functions, which reduce the size of the perturbative terms.
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
70H08 Nearly integrable Hamiltonian systems, KAM theory
Full Text: DOI
[1] В. И. Арнольд, “О поведении адиабатического инварианта при медленном периодическом изменении функции Гамильтона”, Докл. АН СССР, 142:4 (1962), 758 – 761 · Zbl 0119.43701
[2] V. I. Arnol/d, “On the behavior of an adiabatic invariant under slow periodic variation of the Hamiltonian”, Soviet Math. Dokl., 3 (1962), 136 – 140 · Zbl 0119.43701
[3] S. M. Graff, “On the conservation of hyperbolic invariant tori for Hamiltonian systems”, J. Differential Equations, 15 (1974), 1 – 69 · Zbl 0257.34048 · doi:10.1016/0022-0396(74)90086-2
[4] Ю. Н. Бибиков, “Усиление одной теоремы Мозера”, Докл. АН СССР, 213:4 (1973), 766 – 769 · Zbl 0301.34041
[5] Yu. N. Bibikov, “A sharpening of a theorem of Moser”, Soviet Math. Dokl., 14 (1973), 1769 – 1773 · Zbl 0301.34041
[6] Г. Н. Пифтанкин, “Скорость диффузии в задаче Мезера”, Докл. РАН, 408:6 (2006), 736 – 737 · mi.mathnet.ru
[7] S. Bolotin, D. Treschev, “Unbounded growth of energy in nonautonomous Hamiltonian systems”, Nonlinearity, 12:2 (1999), 365 – 388 · Zbl 0989.37050 · doi:10.1088/0951-7715/12/2/013 · adsabs.harvard.edu
[8] A. Delshams, R. de la Llave, T. M. Seara, “A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of \(\mathbb{T}^2\)”, Comm. Math. Phys., 209:2 (2000), 353 – 392 · Zbl 0952.70015 · doi:10.1007/PL00020961 · adsabs.harvard.edu
[9] N. Bra\"nnstro\"m, V. Gelfreich, “Drift of slow variables in slow-fast Hamiltonian systems”, Phys. D, 237:22 (2008), 2913 – 2921 · Zbl 1153.37338 · doi:10.1016/j.physd.2008.05.001 · arxiv:0710.1576
[10] V. Gelfreich, D. Turaev, “Unbounded energy growth in hamiltonian systems with a slowly varying parameter”, Comm. Math. Phys., 283:3 (2008), 769 – 794 · Zbl 1152.37023 · doi:10.1007/s00220-008-0518-1 · adsabs.harvard.edu
[11] S. V. Bolotin, D. V. Treschev, “Remarks on the definition of hyperbolic tori of Hamiltonian systems”, Regul. Chaotic Dyn., 5:4 (2000), 401 – 412 · Zbl 1005.70015 · doi:10.1070/rd2000v005n04ABEH000156 · web.uni.udm.ru
[12] D. Treschev, O. Zubelevich, Introduction to the perturbation theory of Hamiltonian systems, Springer Monogr. Math., Springer-Verlag, Berlin, 2010 · Zbl 1181.37001 · doi:10.1007/978-3-642-03028-4 · arxiv:1206.1496
[13] В. И. Арнольд, А. Б. Гивенталь, Симплектическая геометрия, РХД, Ижевск, 2000
[14] Лекции по симплектической геометрии и топологии, ред. Я. Элиашберга, Л. Трейнор, МЦНМО, М., 2008
[15] Symplectic geometry and topology. Lecture notes from the graduate summer school program (Park City, UT, USA, 1997) Ya. Eliashberg, L. Traynor, Amer. Math. Soc., Providence, RI, 1999 · Zbl 0921.00023
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