A geometric approach to periodically forced dynamical systems in presence of a separatrix
type: Item
instance of: scholarly article
title: A geometric approach to periodically forced dynamical systems in presence of a separatrix
zbMATH Open document ID: 1080.34023
author: Gabriele Villari
author: Fabio Zanolin
published in: Journal of Differential Equations
publication date: 2005-02-22T00:00:00Z
review text: The main result of the article gives some sufficient conditions under which the periodic boundary problem for the nonlinear nonautonomous second-order ifferential equation \[ x''+F(x,x')=e(t) \] has a \(T\)-periodic solution. Here, \(e(t)\) is a continuous \(T\)-periodic forcing term and the function \(F(x,y)\) satisfies a standard sign condition and has properties guaranteeing the uniqueness of the solution. The authors propose three alternative conditions for the existence, that are related to properties of the trajectories of the associated autonomous equation. The main tool is a continuation theorem. A particular interest is addressed to the case when the nonautonomous equation and the autonomous one posses an unbounded separatrix, whose behavior provides a partial bound for the \(T\)-periodic solutions of the homotopic equation \[ x''+F(x,x')=\lambda e(t),\quad \lambda \in [0,1]. \] The last section of the paper contains some applications.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34A26
Mathematics Subject Classification ID: 34C37
zbMATH DE Number: 2136536
zbMATH Keywords: nonautonomous equation
zbMATH Keywords: periodic solutions
zbMATH Keywords: continuation theorem
reviewed by: Gabriela Liţcanu
MaRDI profile type: Publication
full work available at URL: j.jde.2003.09.005
OpenAlex ID: W1980861032
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DOI: 10.1016/J.JDE.2003.09.005
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description: scientific article; zbMATH DE number 2136536