Persistence of elliptic lower dimensional invariant tori for small perturbation of degenerate integrable Hamiltonian systems
type: Item
instance of: scholarly article
title: Persistence of elliptic lower dimensional invariant tori for small perturbation of degenerate integrable Hamiltonian systems
zbMATH Open document ID: 0889.58045
author: Junxiang Xu
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-02-18T00:00:00Z
review text: The following nearly integrable Hamiltonian differential system \[ \dot x=H_y=\omega+P_y,\quad\dot y=-H_x=-P_x, \] \[ \dot u=H_v=\Omega v+P_v,\quad\dot v=-H_u=-\Omega u-P_u \] is considered. Here the Hamiltonian \[ H=N+P,\quad(x,y,u,v)\in T^n\times\mathbb{R}^n\times\mathbb{R}^m\times\mathbb{R}^m\quad(1\leq m,n<+\infty), \] \(T^n\) the usual \(n\)-torus, \(N=\sum_{1\leq j\leq n} \omega_jy_j+(1/2)\sum_{j=1}^m\Omega_j(\omega)(u_j^2+v_j^2)\) is a normal form, and \(P=P(x,y,u,v;\epsilon)\) is a small perturbation term, where \(\epsilon\) is a small perturbation parameter. The frequency vectors \(\omega=(\omega_1,\omega_2,\dots,\omega_n)\in O\subset\mathbb{R}^n\) are also regarded as parameters, where \(O\) is a bounded closed subset of \(\mathbb{R}^n\) with nonempty interior. The corresponding nonperturbed system admits a family of invariant tori \(\{T_\omega\mid\omega\in O\}\). Some of the invariant tori can be destroyed by an arbitrary small perturbation. The problem whether some invariant tory can persist a under small perturbation is considered. The persistence of elliptic invariant tori for a class of small perturbations of degenerate integrable Hamiltonian systems is proved.
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34D10
Mathematics Subject Classification ID: 34C40
zbMATH DE Number: 1031577
zbMATH Keywords: degenerate integrable Hamiltonian system
zbMATH Keywords: perturbation
zbMATH Keywords: invariant tori
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5313
OpenAlex ID: W2045971694
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DOI: 10.1006/JMAA.1997.5313
description: scientific article; zbMATH DE number 1031577