Invariant tori of a class of point transformations: Preservation of an invariant torus under perturbations
type: Item
instance of: scholarly article
title: Invariant tori of a class of point transformations: Preservation of an invariant torus under perturbations
zbMATH Open document ID: 1071.37016
author: A. Yu. Kolesov
author: N. Kh. Rozov
author: A. N. Kulikov
published in: Differential Equations
publication date: 2005-06-17T00:00:00Z
review text: A perturbed mapping is defined as a mapping of the form \(\Pi:(v,\varphi)\to(Av+f_0(\varphi)+f(v,\varphi),\, \varphi+g_0(\varphi)+g(v,\varphi))\), where \(A: V\to V\) is a linear bounded \(\|A\|<1\) operator, \(f_0(\varphi)\) and \(g_0(\varphi)\) are \(2\pi\)-periodic functions of the vector argument \(\varphi\in \mathbb{R}^p\) of the class \(C^1\) ranging in \(V\) and \(\mathbb{R}^p\) respectively, the equation \(\widetilde{\varphi}=\varphi+g_0(\varphi)\) has a unique solution \(\varphi=S_0(\tilde{\varphi})\in\mathbb{R}^p\) and \(\text{det}(I+g_0'(\varphi))\neq 0\) for all \(\varphi\in\mathbb{R}^p\). The perturbation \((f(v,\varphi), g(v,\varphi))\) belongs to the class \(\Sigma(\mu)\) of vector functions jointly continuous in \((v,\varphi)\) and \(2\pi\)-periodic with respect to the vector argument \(\varphi\), \(f(0,\varphi)\equiv 0\), \(g(0,\varphi)\equiv 0\) and \(\|f(v_1,\varphi_1)-f(v_2,\varphi_2)\|\leq\mu(\|v_1-v_2\|+\|\varphi_1-\varphi_2\|)\), \(\|g(v_1,\varphi_1)-g(v_2,\varphi_2)\|\leq\mu(\|v_1-v_2\|+\|\varphi_1-\varphi_2\|)\), for arbitrary \((v_k,\varphi_k)\), \(k=1,2\). Under some additional conditions, the authors prove that the perturbed mapping \(\Pi\) has an invariant torus close to the torus of the unperturbed mapping if the parameter \(\mu\) satisfying the perturbation \((f,g)\) is sufficiently small.
Mathematics Subject Classification ID: 37C29
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 2176565
zbMATH Keywords: quasilinear mapping
zbMATH Keywords: invariant tori
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
Wikidata QID: Q59448662
full work available at URL: b:dieq.0000008405.33370.ff
OpenAlex ID: W2042042849
DOI: 10.1023/B:DIEQ.0000008405.33370.FF
description: scientific article; zbMATH DE number 2176565