Periodic solutions to two-dimensional forced systems: The Massera theorem and its extension
type: Item
instance of: scholarly article
title: Periodic solutions to two-dimensional forced systems: The Massera theorem and its extension
zbMATH Open document ID: 0914.34042
DOI: 10.1023/A:1022618000699
author: Paresh Murthy
published in: Journal of Dynamics and Differential Equations
publication date: 1998-12-08T00:00:00Z
review text: The author deals with two-dimensional systems of the form \[ \dot x= F(t,x),\tag{1} \] where \(F:\mathbb{R}^3\to \mathbb{R}^2\) is \(T\)-periodic in \(t\), \(T>0\). After a short survey of results concerning the existence of \(qT\)-periodic solutions to (1) (or higher-dimensional systems) with an integer \(q\geq 1\), the author presents a proof of the Massera theorem which is based on the Brouwer asymptotic fixed point theorem. Further, he extends the Massera theorem in the following way. He assumes that \(F\) is continuous and locally Lipschitz in the second variable and that all solutions to (1) can be continued from \(t= 0\) to \(t= T\). If there exists a bounded solution to (1) that is bounded away from a given \(T\)-periodic solution \(\zeta\), then there exists a \(qT\)-periodic solution (other than \(\zeta\)) for some integer \(q\geq 1\) or all solutions to (1) that are bounded and bounded away from \(\zeta\) have a well-defined irrational rotation number about \(\zeta\).
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 1210117
zbMATH Keywords: Massera theorem
zbMATH Keywords: embeddings
zbMATH Keywords: rotation numbers
zbMATH Keywords: subharmonics
reviewed by: Irena Rachůnková
MaRDI profile type: Publication
full work available at URL: a:1022618000699
OpenAlex ID: W52872908
description: scientific article; zbMATH DE number 1210117