Roots of unity and unbounded motions of an asymmetric oscillator
type: Item
instance of: scholarly article
title: Roots of unity and unbounded motions of an asymmetric oscillator
zbMATH Open document ID: 0899.34023
author: Rafael Ortega
author: Jose Miguel-Alonso
published in: Journal of Differential Equations
publication date: 1998-11-15T00:00:00Z
review text: The piecewise linear equation of the form \[ \ddot x+ ax^+- bx^-= f(t)\tag{1} \] is considered, where \(a\) and \(b\) are positive constants with \(a\neq b\), \(x^+= \max\{x, 0\}\), \(x^-= \min\{x, 0\}\) and \(f(t)\) is a \(2\pi\)-periodic function. A resonance phenomenon for the asymmetric oscillator can be described by equation (1). In a previous paper, the authors [J. Math. Anal. Appl. 196, No. 3, 965-986 (1995; Zbl 0844.34047)] used a technique based on Lyapunov functions to study the existence of unbounded solutions to differential equations at resonance. The general questions that the authors have in mind are: 1) under what conditions on \(a\), \(b\) and \(f(t)\) one can say that equation (1) has unbounded solutions; 2) are there equations of the form (1), where unbounded and periodic solutions can coexist? To answer the questions, the authors use a class of Poincaré mappings in polar coordinates (the action-angle variables on the plane).
Mathematics Subject Classification ID: 34C15
zbMATH DE Number: 1130540
zbMATH Keywords: differential equations
zbMATH Keywords: resonance phenomenon
zbMATH Keywords: unbounded soultions
zbMATH Keywords: periodic solutions
reviewed by: Sergei Georgievich Zhuravlev
MaRDI profile type: Publication
full work available at URL: jdeq.1997.3367
OpenAlex ID: W1976474211
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DOI: 10.1006/JDEQ.1997.3367
description: scientific article; zbMATH DE number 1130540