On the isoclines of polynomial vector fields
type: Item
instance of: scholarly article
title: On the isoclines of polynomial vector fields
zbMATH Open document ID: 0855.34029
DOI: 10.1007/BF02104723
published in: Siberian Mathematical Journal
publication date: 1996-08-14T00:00:00Z
review text: The author considers autonomous systems (1) \(\dot x = P(x,y)\), \(\dot y = Q(x,y)\) with \(\{P(x,y), Q(x,y)\}\) a polynomial vector field. He proves the following results. Theorem 1. Assume \(P(x,y) = P_m (x,y) + P_n (x,y)\), \(Q(x,y) = Q_m (x,y) + Q_n (x,y)\) with \(m,n > 0\) and \(P_m\), \(Q_m\) and \(P_n\), \(Q_n\) homogeneous polynomials of degree \(m\) and degree \(n\) respectively. If \(m\) and \(n\) are of different parity, then at least one isocline passes through the singular point \((0,0)\). Moreover, either each straight line passing through \((0,0)\) is an isocline or there are at most \(m + n\) isoclines. Theorem 2. Let the vector field in (1) be polynomial with \((0,0)\) a singular point. Every straight line passing through this point is an isocline if and only if the field is collinear with a homogeneous vector field, i.e., there exist homogeneous polynomials \(P_k (x,y)\) and \(Q_k (x,y)\) \((k \geq 0)\) and a polynomial \(R(x,y)\) such that the representation \(P(x,y) \equiv P_k (x,y) R(x,y)\), \(Q(x,y) \equiv Q_k (x,y) R(x,y)\) holds.
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 917145
zbMATH Keywords: polynomial vector field
zbMATH Keywords: isocline
author: Vladimir Mikhaĭlovich Cheresiz
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf02104723
OpenAlex ID: W2007883650
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description: scientific article; zbMATH DE number 917145