On the saddle order of polynomial differential systems at a resonant singular point
type: Item
instance of: scholarly article
title: On the saddle order of polynomial differential systems at a resonant singular point
zbMATH Open document ID: 1318.34047
author: Guangfeng Dong
author: Jiazhong Yang
published in: Journal of Mathematical Analysis and Applications
publication date: 2014-11-19T00:00:00Z
review text: For any system \[ \dot x = px +P(x,y), \quad \dot y= -q y + Q(x,y), \eqno{(1)} \] where \(P\) and \(Q\) are assumed to be complex polynomials of degree \(n\) without linear and constant terms, and \(p\) and \(q\) are positive integers with \(GCD(p,q)=1\), there exists a formal power series \(H(x,y) \) satisfying \(\dot H=\sum_{k\geq 1} L_k (x^q y^p)^{k+1}\). The quantity \(L_k\) is called the \(k\)-th saddle value of system (1); the saddle order of the system is the minimal number \(k\) such that \(L_k\neq 0\). It is proved that for system (1) with \(p=1\) and \(q=2\) for any \(n\geq 2\) such that either \(n+2\) or \(2 n+1\) is a prime number, there always exist polynomials \(P\) and \(Q\) of degree \(n\) such that the saddle order of the system is at least \(n^2-1\).
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34A25
zbMATH DE Number: 6370953
zbMATH Keywords: saddle order
zbMATH Keywords: focus order
zbMATH Keywords: integrability
zbMATH Keywords: p:-q resonance
MaRDI profile type: Publication
full work available at URL: j.jmaa.2014.10.085
OpenAlex ID: W2086469993
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DOI: 10.1016/J.JMAA.2014.10.085
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description: scientific article; zbMATH DE number 6370953