Invariant tori of Duffing-type equations
type: Item
instance of: scholarly article
title: Invariant tori of Duffing-type equations
zbMATH Open document ID: 0911.34042
published in: Journal of Differential Equations
publication date: 1999-05-04T00:00:00Z
review text: A class of Duffing-type equations \[ {d^2x\over dt^2}+x^{2n+1} +\sum^{2 n}_{j=0} p_j(t)x^j= 0,\;n\geq 1,\tag{1} \] where \(p_j(t)\) \((j=0,1,\dots,2n)\) are 1-periodic functions, is considered. It is known that equation (1) possesses invariant tori in the \((x,\dot x,t)\)-space arbitrary far from the \(t\)-axis when \(p_j(t)\) are sufficiently smooth. The existence of invariant tori means that all solutions to (1) are bounded and there exist quasiperiodic solutions to (1). The problem of differentiability of the functions \(p_j(t)\) is discussed. Recently it was proved that the functions \(p_j(t)\) are 1-periodic \(C^{(5+\varepsilon)}\) functions \((\varepsilon >0)\). Here, the required degree of differentiability of the \(p_j(t)\)'s decreased to \(C^{(1)}\) for \(p_j(t)\), \(j=0,1,\dots,n\), and to \(C^{(2)}\) for \(p_j(t)\), \(j=n+1,\dots,2n\). The following theorem is proved. Assume that \(p_j(t)\), \(j=0,1,\dots,n\), are 1-periodic \(C^{(1)}\) functions and \(p_j(t)\), \(j=n+1,\dots,2n\), are 1-periodic \(C^{(2)}\) functions, then the Poincaré map \(P:(x,\dot x)_{t=0} \mapsto(x,\dot x)_{t= 1}\) of equation (1) possesses countable many invariant curves whose radii tend to infinity. The boundedness and the existence of quasiperiodic solutions to (1) are obtained as simple corollaries of the theorem.
Mathematics Subject Classification ID: 34C27
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 1212872
zbMATH Keywords: Duffing type equations
zbMATH Keywords: invariant tori
zbMATH Keywords: bounded solutions
zbMATH Keywords: quasiperiodic solutions
author: Xiaoping Yuan
reviewed by: Sergei Georgievich Zhuravlev
MaRDI profile type: Publication
full work available at URL: jdeq.1997.3356
OpenAlex ID: W1988376998
cites work: Q3803454
cites work: Stable and Random Motions in Dynamical Systems
cites work: Boundedness for solutions of nonlinear periodic differential equations via Moser's twist theorem
cites work: Invariant curves and time-dependent potentials
cites work: Quasiperiodic motions in superquadratic time-periodic potentials
cites work: Q3294369
DOI: 10.1006/JDEQ.1997.3356
description: scientific article; zbMATH DE number 1212872