On Melnikov's persistency problem
Let \(H_\varepsilon(I,\theta,y)=\langle\lambda_0, I\rangle + \sum_{s=1}^r\mu_s| y_s| ^2+| I|^2+\varepsilon H_1(I,\theta, y)\) be a family of real analytic Hamiltonians defined on \(\mathbb{R}^b\times\mathbb{R}^b\times\mathbb{R}^r\), depending upon a real parameter \(\varepsilon\). Here \(\lambda_0\) is an integer vector satisfying \(\langle\lambda_0, k\rangle \neq \mu_s\) for all \(s\) and any integer vector \(k\). Melnikov's persistency problem is whether an invariant torus \(T^b\times 0\times 0\) for \(H_0\) persists as a nearby invariant torus for \(H_\varepsilon\), \(\varepsilon \neq 0\). The main result is that for small \(\varepsilon >0\) and \(t\) taken in a set of positive measure there is a perturbed torus with frequency vector \(\lambda = t\lambda_0\) and \(t \approx 1\). A similar result had been established by \textit{L. H. Eliasson} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 15, No. 1, 115-147 (1988; Zbl 0685.58024)], but with a stronger nonresonance condition on \(\lambda_0\).
scientific article; zbMATH DE number 1082972