Melnikov-type theorem for time reversible system
type: Item
instance of: scholarly article
title: Melnikov-type theorem for time reversible system
zbMATH Open document ID: 1495.34065
author: Yanyan Li
published in: Qualitative Theory of Dynamical Systems
publication date: 2022-03-15T00:00:00Z
review text: 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 freq…
reviewed by: Michal Veselý
Mathematics Subject Classification ID: 34C27
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 34C45
zbMATH DE Number: 7489698
zbMATH Keywords: Craig-Wayne-Bourgain method
zbMATH Keywords: quasi-periodic solutions
zbMATH Keywords: Melnikov-type theorem
zbMATH Keywords: time reversible systems
zbMATH Keywords: non-linear perturbations
MaRDI profile type: Publication
full work available at URL: s12346-021-00553-y
OpenAlex ID: W4211238798
Wikidata QID: Q113899538
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DOI: 10.1007/S12346-021-00553-Y
description: scientific article; zbMATH DE number 7489698