A note on micro-instability for Hamiltonian systems close to integrable (Q2790263)

Let \(n \geq 2\) be an integer, \(B = B_1\subseteq \mathbb{R}^n\) be the unit open ball with respect to the supremum norm and \(\mathbb{T}^n := \mathbb{R}^n/\mathbb{Z}^n\). The smooth, at least \(C^2\), Hamiltonian function \(H\) defined on the domain \(\mathbb{T}^n \times B\) of the form NEWLINENEWLINE\[NEWLINE H(\theta, I) = h(I) +\varepsilon f(\theta, I), \varepsilon \geq 0, (\theta, I) \in \mathbb{T}^n \times B, \eqno{(H)}NEWLINE\]NEWLINE NEWLINEand its associated Hamiltonian system NEWLINENEWLINE\[NEWLINE\begin{aligned} \dot{\theta}(t) = \partial_IH(\theta(t), I(t)) = \partial_Ih(I(t)) + \varepsilon \partial_If(\theta(t), I(t)), \\NEWLINE\dot{I}(t) =-\partial_{\theta}H(\theta(t), I(t)) = -\varepsilon\partial_{\theta}f(\theta(t), I(t))\end{aligned}NEWLINE\]NEWLINE NEWLINEare considered.NEWLINENEWLINEFor \(\varepsilon=0\), the system is stable in the sense that the action variables \(I(t)\) of all solutions are constant, and these solutions are quasi-periodic.NEWLINENEWLINEThe aim of the article is to show, using the method of the authors [Mosc. Math. J. 14, No. 2, 181--203 (2014; Zbl 1347.37107)], that for a generic integrable Hamiltonian (this genericity condition being the existence of a resonant point) and for a generic perturbation (the associated averaged perturbation is nonconstant), a phenomenon of ``micro-instability'' arises: the existence of a solution whose action variables drift of order \(\sqrt{\varepsilon}\) after a time of order \((\sqrt{\varepsilon})^{-1}\).