Hyperbolic tori in Hamiltonian systems with slowly varying parameter (Q2848645)

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.NEWLINENEWLINEThe 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 NEWLINE\[NEWLINE \langle g(h) \rangle = \frac{1}{2\pi} \int_0^{2\pi} g(h,\tau)\,d\tau NEWLINE\]NEWLINE be the \textit{averaged frequency}.NEWLINENEWLINEIn 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 \(\varepsilon\).NEWLINENEWLINEOf 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 NEWLINE\[NEWLINE |g \ell_1 + \varepsilon \ell_2 |\geq \frac{\varepsilon}{(|\ell_1|+ |\ell_2|)^2} \quad \forall (\ell_1,\ell_2) \in \mathbb{Z}^2, \; (\ell_1,\ell_2) \neq (0,0), NEWLINE\]NEWLINE 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 \(10\varepsilon\).NEWLINENEWLINEThe result proven by the author is the following. If, for some \(h_0\in (h_1,h_2)\), NEWLINE\[NEWLINE \int_0^{2\pi} \frac{\partial g}{\partial h}(h_0,\tau)\, d\tau \neq 0, NEWLINE\]NEWLINE 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 \((\langle g(h_0) \rangle,\varepsilon)\).NEWLINENEWLINEThe 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.