Polynomial differential systems in \(\mathbb R^3\) having invariant weighted homogeneous surfaces
type: Item
instance of: scholarly article
title: Polynomial differential systems in \(\mathbb R^3\) having invariant weighted homogeneous surfaces
zbMATH Open document ID: 1388.34044
author: T. M. Dalbelo
author: Marcelo Messias
author: Alisson C. Reinol
published in: Bulletin of the Brazilian Mathematical Society. New Series
publication date: 2018-04-13T00:00:00Z
review text: Invariant algebraic surfaces are quite important in understanding global dynamics of many dynamical systems. Weighted homogeneous surfaces, which are defined as \((x, y, z) \in \mathbb{R}^3\) such that \(f (x, y, z) = 0\), where \(f: \mathbb{R}^3\rightarrow \mathbb{R}\) is a weighted homogeneous polynomial, are an important class of surfaces contained in the Euclidean space \(\mathbb{R}^3\). The authors give the normal form of all polynomial differential systems in \(\mathbb{R}^3\) having a weighted homogeneous surface \(f = 0\) as an invariant algebraic surface and characterize among these systems those having a Darboux invariant constructed uniquely using this invariant surface. The obtained results are applied to study the Vallis system with an invariant cone, and a dynamical analysis of the flow of this system restricted to the invariant cone is presented.
reviewed by: Feng Xie
Mathematics Subject Classification ID: 34C45
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34C14
zbMATH DE Number: 6859579
zbMATH Keywords: polynomial differential systems
zbMATH Keywords: Darboux theory of integrability
zbMATH Keywords: invariant algebraic surfaces
zbMATH Keywords: weighted homogeneous surfaces
zbMATH Keywords: singular varieties
zbMATH Keywords: stratified vector fields
zbMATH Keywords: Vallis system
MaRDI profile type: Publication
full work available at URL: s00574-017-0045-9
OpenAlex ID: W2727966868
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DOI: 10.1007/S00574-017-0045-9
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description: scientific article; zbMATH DE number 6859579