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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
##### Keywords:
KAM theory; hyperbolic tori; fast-slow systems
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##### References:
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