Planar vector fields with a given set of orbits
type: Item
instance of: scholarly article
title: Planar vector fields with a given set of orbits
zbMATH Open document ID: 1247.34079
author: Jaume Llibre
author: Natalia Sadovskaia
author: R. Ramirez
published in: Journal of Dynamics and Differential Equations
publication date: 2012-01-10T00:00:00Z
full work available at URL: http://ddd.uab.cat/record/150414
review text: The authors first present a general result on determining all \(C^1\) planar vector fields with a given set of orbits of the form \(y-y_i(x)=0\) (\(i=1,\dots,s\)), where \( y_i(x)\) are \(C^1\) functions satisfying some assumptions. Then they study in detail few cases when the orbits are branches of an algebraic curve. In particular, they consider the quadratic vector field associated to the system \[ \dot x=p_0 y^2+p_1 y+p_2, \quad \dot y=q_0 y^2+q_1 y+q_2, \tag{1} \] where \(p_j=\sum_{n=0}^j p_{jn} x^n, \;q_j=\sum_{n=0}^j q_{jn} x^n, \;j=1, 2,\) for which \[ g=\sum_{j=0} ^S a_j (x) y^{S-j}=0 \tag{2} \] is the unique irreducible algebraic invariant curve. It is proven that if the curve \(g=0\) has \(S>1\) branches with respect to the variable \(y\), then deg\,\(g\leq 4S\). The authors also present the following two conjectures: - if system (1) admits a unique invariant irreducible algebraic curve (2), then deg\,\(g\leq 3 S\); - if system (1) with a unique invariant irreducible algebraic curve (2) does not admit a rational first integral, then deg\,\(g\leq 12\).
Mathematics Subject Classification ID: 34C45
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 5996176
zbMATH Keywords: invariant curve
zbMATH Keywords: singular algebraic curve
zbMATH Keywords: branches
zbMATH Keywords: quadratic vector fields
zbMATH Keywords: orthogonal polynomial
zbMATH Keywords: orbits
MaRDI profile type: Publication
OpenAlex ID: W20199705
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DOI: 10.1007/S10884-011-9219-0
description: scientific article; zbMATH DE number 5996176