Lower bounds for the Hilbert number of polynomial systems
type: Item
instance of: scholarly article
title: Lower bounds for the Hilbert number of polynomial systems
zbMATH Open document ID: 1247.34047
published in: Journal of Differential Equations
publication date: 2012-03-07T00:00:00Z
review text: Consider the planar polynomial systems \[ {dx\over dt}= P_n(x,y),\qquad{dy\over dt}= Q_n(x,y)\tag{E\(_n\)} \] and \[ {dx\over dt}= H_y(x,y)+\varepsilon P_n(x,y),\qquad {dy\over dt}=- H_x(x,y)+ \varepsilon Q_n(x,y),\tag{E\(^*_n\)} \] where \(P_n\) and \(Q_n\) are polynomials in \(x\) and \(y\) of degree \(n\), \(H\) is a polynomial in \(x\) and \(y\) of degree \(m\) with \(m\leq n+1\), \(\varepsilon\) is a small parameter. Let \(H(n)\) and \(H^*(n)\) denote the maximal number of limit cycles of system \((\text{E}_n)\) and \((\text{E}^*_n)\), respectively. The authors give a survey on improved lower bounds for \(H(n)\) and \(H^*(n)\). In particular, they prove \[ \begin{aligned} H(m)\geq m^2\quad &\text{for }m\geq 3,\;m\neq 6,\;m\neq 14,\\ H^*(m)\geq m^2\quad &\text{for }m\geq 23,\\ H^*(2k+1)\geq (2k+1)^2\quad &\text{for }k\geq 1,\end{aligned} \] \(H(m)\) grows at least as \({1\over 2\ln 2}(m+ 2)^2\ln(m+ 2)\).
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 6012787
zbMATH Keywords: polynomial system
zbMATH Keywords: Hilbert number
zbMATH Keywords: limit cycle bifurcation
zbMATH Keywords: lower bound
author: Maoan Han
author: Jibin Li
MaRDI profile type: Publication
full work available at URL: j.jde.2011.11.024
OpenAlex ID: W2041526529
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DOI: 10.1016/J.JDE.2011.11.024
description: scientific article; zbMATH DE number 6012787