Branches of forced oscillations for periodically perturbed second-order autonomous ODEs on manifolds
type: Item
instance of: scholarly article
title: Branches of forced oscillations for periodically perturbed second-order autonomous ODEs on manifolds
zbMATH Open document ID: 0929.34032
author: Massimo Furi
author: Marco Spadini
published in: Journal of Differential Equations
publication date: 2000-01-16T00:00:00Z
review text: The authors examine the structure of the set of forced oscillations of periodic perturbations of second-order autonomous ODEs of the type \[ x''_{\pi} = h(x,x') + \lambda f(t,x,x') \] where \(h: TM \rightarrow \mathbb{R}^{k}\) and \(f:\mathbb{R}\times TM \rightarrow \mathbb{R}^{k}\) are continuous vector fields, tangent to a smooth submanifold \(M\) of \(\mathbb{R}^{k}\), \(f\) is \(\omega\)-periodic in \(t\) and \(\lambda \geq 0.\) Here, \(TM\) denotes the tangent bundle of \(M.\) In particular, the authors provide sufficient conditions to prove the existence of a noncompact connected component of solutions \((\lambda,x) \), with \(x\) an \(\omega\)-periodic function, emanating from the set of equilibria for \(\lambda = 0.\) Moreover, this component is unbounded if \(M\) is complete. An application to a perturbed pendulum equation is given. It is very important, in the proofs, the use of the notion of degree of a second-order vector field.
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 37C55
zbMATH DE Number: 1288031
zbMATH Keywords: parametrized second-order differential equations
zbMATH Keywords: forced oscillations
zbMATH Keywords: bifurcation
zbMATH Keywords: periodic perturbations
zbMATH Keywords: differentiable manifolds
zbMATH Keywords: topological degree
reviewed by: Antonio Cañada
MaRDI profile type: Publication
full work available at URL: jdeq.1998.3557
OpenAlex ID: W2057764274
cites work: A continuation principle for forced oscillations on differentiable manifolds
cites work: Q3976076
cites work: A continuation principle for periodic solutions of forced motion equations on manifolds and applications to bifurcation theory
cites work: On the set of harmonic solutions of periodically perturbed autonomous differential equations on manifolds
cites work: Multiplicity of forced oscillations for the spherical pendulum
cites work: Q4134602
cites work: Differential Topology
cites work: Q5510566
DOI: 10.1006/JDEQ.1998.3557
description: scientific article; zbMATH DE number 1288031