Melnikov-type theorem for time reversible system
Applying the Craig-Wayne-Bourgain method (without the coordinate transformation), the authors solve the problem of the persistence of quasi-periodic solutions for non-linear perturbations of autonomous ODEs. They establish a Melnikov type theorem for finitely dimensional time reversible systems of differential equations in the form \[ \mathrm{i} \, \frac{\mathrm{d}z}{\mathrm{d}t} + \Lambda z + \varepsilon g \left(z, \overline{z} \right) = 0, \qquad z \in \mathbb{C}^n, \tag{1} \] where \(\Lambda\) is a real diagonal matrix and \(g\) is analytic in \(z\) and \(\overline{z}\) and satisfies \[ \overline{ g \left(z, \overline{z}\right)} = g \left(\overline{z}, z\right). \] Note that \(\mathrm{i}\) is the imaginary unit and the overline denotes the complex conjugation. In brief, the authors construct lower dimensional invariant tori for Eq. (1) (without using the second Melnikov condition). Their main result reads as follows. \textbf{Theorem.} Let \(U = (0, 1)^d\) and \(a = (a_1, \dots, a_d) \in \mathbb{C}^d\) be small. There exists \(\vartheta > 0\) with the property that, for any \(\varepsilon \in (0, \vartheta)\), there exist \(U_\infty (\varepsilon, a) \subset U\) satisfying \[ \mathrm{mes}\left(U \smallsetminus U_\infty (\varepsilon, a)\right) \to 0 \qquad \text{as } \varepsilon \to 0^+. \] For any \(\omega \in U_\infty (\varepsilon, a)\), \[ z (t) = \sum\limits_{j=1}^n \sum\limits_{k \in \mathbb{Z}^d} \widehat{z}_j (k) \mathrm{e}^{\mathrm{i} \langle k, \widetilde{\omega} t \rangle } e_j \] is an analytic quasi-periodic solution of Eq. (1), where \(\{e_j\}_{j=1}^n\) is the standard basis of \(\mathbb{R}^n\), \(\widehat{z}_j (k)\) is the \(k\)-th Fourier coefficients of \(z_j\), it holds \[ \widehat{z}_j \left(e_j \right) = a_j , \qquad 1 \leq j \leq d, \] and \[ \sum\limits_{j=1}^n \sum\limits_{k \in \mathbb{Z}^d, k \ne e_j} \left| \widehat{z}_j (k) \right| \mathrm{e}^{- \varrho \left|k\right| } < \sqrt{\varepsilon} \] for some \(\varrho > 0\), and the perturbed frequency \(\widetilde{\omega} \in \mathbb{R}^d\) satisfies \[ \left| \widetilde{\omega} - \omega \right| < C \varepsilon \] for some \(C > 0\).
scientific article; zbMATH DE number 7489698
scientific article