A note on a result of G. S. Petrov about the weakened 16th Hilbert problem
type: Item
instance of: scholarly article
title: A note on a result of G. S. Petrov about the weakened 16th Hilbert problem
zbMATH Open document ID: 0829.34022
author: Baoyi Li
author: Zhifen Zhang
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-01-17T00:00:00Z
review text: \textit{G. S. Petrov} [Funct. Anal. Appl. 22, 72-73 (1988); translation from Funkts. Anal. Prilozh. 22, 83-84 (1988; Zbl 0645.33003)] and \textit{P. Mardesic} [Ergodic Theory Dyn. Syst. 10, 523-529 (1990; Zbl 0706.58054)] have proved that for system \(\dot x = y + \varepsilon P(x,y)\), \(\dot y = 1 - 3x^2 + \varepsilon Q(x,y)\), where \(P(x,y)\) and \(Q(x,y)\) are polynomials of \(x,y\) with degree \(\leq N\), if the first order Melnikov function \(M_1 (h) \not \equiv 0\), then the lowest upper bound \(B(N)\) of the number of limit cycles of the above system is \(N - 1\). We prove that if \(M_1(h) \equiv 0\) and the second order Melnikov function \(M_2(h) \not \equiv 0\), then \[ B(N) = \begin{cases} 2N - 2, \quad & N \text{- even},\\ 2N - 3, \quad & N \text{-odd}. \end{cases} . \]
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 765136
zbMATH Keywords: Hilbert 16th problem
zbMATH Keywords: upper bound
zbMATH Keywords: number of limit cycles
zbMATH Keywords: second order Melnikov function
reviewed by: C. Chicone
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1088
OpenAlex ID: W2061545715
DOI: 10.1006/JMAA.1995.1088
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description: scientific article; zbMATH DE number 765136