Sunflower model: time-dependent coefficients and topology of the periodic solutions set
type: Item
instance of: scholarly article
title: Sunflower model: time-dependent coefficients and topology of the periodic solutions set
zbMATH Open document ID: 1333.34119
author: Luca Bisconti
author: Marco Spadini
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2015-11-09T00:00:00Z
review text: The authors investigate \(T\)-periodic solutions of parametrized delay differential equations of the form \[ \ddot y(t) = a(t)\dot y(t) + \lambda b(t) \phi(y(t),y(t-r)) , \quad \lambda \geq 0, \tag{1} \] where \(\phi\) is a continuous function, \(r>0\) is a finite time delay, and \(a\) and \(b\) are assumed to be continuous, \(T\)-periodic with \(a\) of nonzero average and \(b(t) \neq 0\) for all \(t\). These equations generalize in some sense the sunflower equation, i.e., \[ \ddot y(t) = -\frac{\alpha}{r} \dot y(t) - \frac{\beta}{r} \sin(y(t-r)), \tag{2} \] in which \(\alpha\) and \(\beta\) are experimental parameters. By a combination of topological methods, such as degree theory and fixed point index theory, along with a classical Liénard-plane analysis, the authors are able to obtain information on the structure of the set of \(T\)-periodic solutions of (1) as \(\lambda \geq 0\) varies. Roughly speaking, the main result of the paper gives a topological condition on the function \(p\mapsto\phi(p,p)\) which implies that there exists a connected set of ``nontrivial'' \(T\)-periodic solutions of (1) whose closure is not compact and intersects the set of constant solutions of (1) for \(\lambda =0\). The paper closes with an application to a variant of the original sunflower equation (2), namely, to an equation with time-periodic coefficients.
reviewed by: Alessandro Calamai
Mathematics Subject Classification ID: 34K60
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 6506036
zbMATH Keywords: sunflower equation
zbMATH Keywords: coupled differential equations
zbMATH Keywords: branches of periodic solutions
zbMATH Keywords: fixed point index
MaRDI profile type: Publication
full work available at URL: s00030-015-0336-z
OpenAlex ID: W891123534
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DOI: 10.1007/S00030-015-0336-Z
description: scientific article; zbMATH DE number 6506036