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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.
MSC:
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
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