Abel-like differential equations with a unique limit cycle
type: Item
instance of: scholarly article
title: Abel-like differential equations with a unique limit cycle
zbMATH Open document ID: 1228.34060
author: Yong-Cai Geng
author: Sumit K. Garg
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2011-06-01T00:00:00Z
full work available at URL: http://ddd.uab.cat/record/150444
review text: Consider the scalar differential equation \[ {dx\over dt}= \Biggl(\sum^k_{l=1} a_l\sin^{i_l}(t)\cos^{j_l}(t)\Biggr)\, x^n+ b\sin^{i_b}(t)\cos^{j_b}(t) x^m,\tag{\(*\)} \] where \(n,m\geq 2\), \(i_l\), \(i_b\), \(j_l\), \(j_b\) are nonnegative integers. An isolated periodic solution of \((*)\) is called a limit cycle of \((*)\). The supremum of all limit cycles of \((*)\) over \((a_1,\dots, a_k, b)\in\mathbb{R}^{k+1}\) is called the Hilbert number \({\mathcal H}\) of \((*)\). The authors derive conditions which characterize the cases \({\mathcal H}= 0\) and \({\mathcal H}= 1\).
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 5903372
zbMATH Keywords: periodic solutions
zbMATH Keywords: Abel equation
MaRDI profile type: Publication
OpenAlex ID: W2026455125
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DOI: 10.1016/J.NA.2011.02.049
description: scientific article; zbMATH DE number 5903372