Existence of quasiperiodic solutions and Littlewood's boundedness problem of super-linear impact oscillators
type: Item
instance of: scholarly article
title: Existence of quasiperiodic solutions and Littlewood's boundedness problem of super-linear impact oscillators
zbMATH Open document ID: 1247.37048
author: Zhiguo Wang
author: Yiqian Wang
published in: Applied Mathematics and Computation
publication date: 2011-03-28T00:00:00Z
review text: Most applications of KAM theory have been restricted to smooth dynamical systems. This paper applies KAM theory to analyze the dynamical behavior of Duffing-type equations with impact. The authors consider a special but typical case of suplinear impact oscillators as follows: \[ \begin{cases} \ddot{x} + x^{2n+1} = p(t),\quad & \text{for }x(t)>0, \\ x(t)\geqslant 0\\ \dot{x}(t_0^+)=-\dot{x}(t_0^-), & \text{if } x(t_0) = 0, \end{cases}\eqno{(0.1)} \] where \(n\in {\mathbb N}^+\), and \(p(t)\) is a \(C^5\) periodic function with period \(1\). This paper proves that all solutions of the above equation are bounded, and that there are infinitely many periodic and quasiperiodic solutions. The idea of proving the boundedness is as follows. First, the equation can be described by a Hamiltonian function \(H(\phi, \rho, t)\) in action-angle variables defined on the whole space \({\mathbb T}\times {\mathbb R}^+ \times {\mathbb T}\) by means of transformation theory. The Hamiltonian \(H(\phi, \rho, t)\) is continuous in \(\phi\) but is \(C^5\) smooth in \(\rho\) and \(t\). The authors introduce an equivalent Hamiltonian \({\mathcal H}(\theta, I, \tau)\) through a transformation by exchanging the roles of the variables \(t\) and \(\phi\) outside of a large disk. The new Hamiltonian \({\mathcal H}\) is sufficiently smooth in \(\theta\) and \(I\), and therefore Moser's theorem is applicable and guarantees the existence of arbitrarily large invariant curves. Going back to the original system, every such curve is a base of a time-periodic and flow-invariant cylinder in the extended phase space \((x,\dot{x}, t)\in {\mathbb R}^+\times {\mathbb R} \times {\mathbb R}\), which leads to a bound of these solutions if the uniqueness of the initial value problem holds.
reviewed by: Jinzhi Lei
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 34C27
zbMATH DE Number: 5870907
zbMATH Keywords: impact oscillators
zbMATH Keywords: boundedness of solutions
zbMATH Keywords: KAM theory
zbMATH Keywords: quasiperiodic solutions
zbMATH Keywords: action-angle variables
MaRDI profile type: Publication
full work available at URL: j.amc.2011.01.037
OpenAlex ID: W2084879312
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DOI: 10.1016/J.AMC.2011.01.037
description: scientific article; zbMATH DE number 5870907