Boundedness of solutions for a class of oscillating potentials without the twist condition
type: Item
instance of: scholarly article
title: Boundedness of solutions for a class of oscillating potentials without the twist condition
zbMATH Open document ID: 1223.34053
author: Zhiguo Wang
author: Yiqian Wang
publication date: 2011-03-02T00:00:00Z
review text: The authors prove that the second order differential equation \[ x''+x^{2n+1}f(x)+p(t)=0 \] with \(p(t+1)=p(t)\), \(f(x+T)=f(x)\) smooth and \(f(x)>0\), possesses Lagrangian stability by using a variant of Moser's twist theorem. The idea to prove the boundedness for all solutions is as follows. By means of several transformations, the system is transformed into a small perturbation of an integrable system. The Poincaré map of this system is close to a twist map. Using a variant of Moser's twist theorem, the authors prove the existence of large invariant curves diffeomorphic to circles and surrounding the origin in the \((x, \dot{x})\) plane. Every such curve is the base of a time-periodic invariant flow cylinder, which confines the solutions in its interior and therefore leads to boundedness of solutions.
reviewed by: Zaihong Wang
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34C20
zbMATH DE Number: 5859375
zbMATH Keywords: boundedness of solutions
zbMATH Keywords: oscillating potential
zbMATH Keywords: twist condition
zbMATH Keywords: KAM theorem
MaRDI profile type: Publication
full work available at URL: s10114-010-9133-0
OpenAlex ID: W2039167903
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DOI: 10.1007/S10114-010-9133-0
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5859375