On the configurations of the singular points and their topological indices for the spatial quadratic polynomial differential systems
type: Item
instance of: scholarly article
title: On the configurations of the singular points and their topological indices for the spatial quadratic polynomial differential systems
zbMATH Open document ID: 1454.34055
author: Jaume Llibre
author: Claudia Valls
published in: Journal of Differential Equations
publication date: 2020-10-13T00:00:00Z
review text: Consider a real planar quadratic polynomial differential system \[ \dot x = P(x,y), \ \ \dot y = Q(x,y) \] that has exactly 4 real singular points. The well-known Berlinskii Theorem states that if the quadrilater formed by these points is convex, then two opposite singular points are anti-saddles (i.e., nodes, foci or centers) and the other two are saddles. If this quadrilater is not convex, then either the three exterior vertices are saddles and the interior vertex is an anti-saddle or the exterior vertices are anti-saddles and the interior vertex is a saddle. The authors extend the Berlinskii Theorem for quadratic polynomial differential systems in the plane to the space. They deal with differential systems \[ \dot x = P(x,y,z), \ \ \dot y = Q(x,y,z), \ \ \dot z = R(x,y,z), \] where \(P,Q,R\) are real polynomials of degrees 2. Assuming that the above system has 8 real singular points, the authors obtain all the configurations of the singular points and their topological indices.
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 37C25
zbMATH DE Number: 7260117
zbMATH Keywords: singular point
zbMATH Keywords: topological index
zbMATH Keywords: polynomial differential systems
zbMATH Keywords: Berlinskii theorem
zbMATH Keywords: Euler-Jacobi formula
reviewed by: Alexey O. Remizov
MaRDI profile type: Publication
full work available at URL: j.jde.2020.07.022
OpenAlex ID: W3087388776
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DOI: 10.1016/J.JDE.2020.07.022
description: scientific article; zbMATH DE number 7260117