Limit cycles of Abel equations of the first kind
type: Item
instance of: scholarly article
title: Limit cycles of Abel equations of the first kind
zbMATH Open document ID: 1326.34027
published in: Journal of Mathematical Analysis and Applications
publication date: 2014-10-31T00:00:00Z
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 6363023
zbMATH Keywords: periodic solutions
zbMATH Keywords: limit cycles
zbMATH Keywords: Abel equation
describes a project that uses: Mathematica
MaRDI profile type: Publication
full work available at URL: j.jmaa.2014.10.019
OpenAlex ID: W1992538858
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DOI: 10.1016/J.JMAA.2014.10.019
author: A. Álvarez
author: J. L. Bravo
author: M. Fernández
review text: Consider the scalar differential equation NEWLINE\[NEWLINE{dx\over dt}= \sum^m_{i=0} a_i(t) x^{n_i},\tag{\(*\)}NEWLINE\]NEWLINE where all \(a_i\) are \(T\)-periodic analytic functions, and \(1\leq n_i\leq n\). For any (given) polynomial \(Q(x)= x^{n_0}- \sum^m_{i=1} \alpha_i x^{n_i}\), \((*)\) can be written in the form NEWLINE\[NEWLINE{dx\over dt}= a_0(t) Q(x)+ R(t,x).\tag{\(**\)}NEWLINE\]NEWLINE Let \(W\) be the Wronskian of \(Q\) and \(R\), that is NEWLINE\[NEWLINEW(t,x)= Q(x) R_x(t,x)- R(t,x) Q'(x).NEWLINE\]NEWLINE The authors prove that if \(V\) is a connected component of NEWLINE\[NEWLINEU:= \{(t,x)\in \mathbb{R}^2: W(t,x) Q(x)\neq 0\},NEWLINE\]NEWLINE then there is at most one closed orbit of \((**)\) in \(V\). Under additional conditions which imply that closed orbits are either contained in \(U\) or disjoint with \(U\), it is proved that the maximum number of connected components of \(U\) is \(3n-1\) which implies an upper bound for the closed orbits of \((**)\).
description: scientific article; zbMATH DE number 6363023