The global dynamics of a class of vector fields in \(\mathbb R^3\)
type: Item
instance of: scholarly article
title: The global dynamics of a class of vector fields in \(\mathbb R^3\)
zbMATH Open document ID: 1275.34063
author: Xinan Zhang
author: Zhaojun Liang
author: Lansun Chen
publication date: 2013-03-18T00:00:00Z
review text: This paper is devoted to the investigation of the two vector fields \[ X(x)=(x_1+f_1(x))\frac{\partial}{\partial x_1}(x) + (x_2+f_2(x))\frac{\partial}{\partial x_2}(x) + (x_3+f_3(x))\frac{\partial}{\partial x_3}(x) \] and \[ Y(x)=x+f(x)= (x_1+f_1(x), x_2+f_2(x), x_3+f_3(x)), \] where \(x=(x_1, x_2, x_3)\in\mathbb R^3,\) and \(f_i (x)\,(i=1,2,3)\) are homogeneous polynomials of degree \(n\) in \(\mathbb R^3\). The authors establish conditions such that all the possible nonwandering sets of the vector fields fall into three classes, namely singularities; closed orbits; graphs of unions of singularities and the trajectories connecting them. Moreover, criteria of the existence of closed orbits, heteroclinic orbits and homoclinic orbits of the vector fields and some applications of these results are given.
Mathematics Subject Classification ID: 34C45
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 6145917
zbMATH Keywords: heteroclinic orbit
zbMATH Keywords: tangent vector field
zbMATH Keywords: invariant cone
MaRDI profile type: Publication
full work available at URL: s10114-011-9053-7
OpenAlex ID: W2012002054
cites work: Q4768040
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: Q3994920
cites work: Q5580482
cites work: On hereditarily indecomposable Banach spaces
cites work: A certain class of integral curves in 3-space
cites work: Global topological properties of homogeneous vector fields in \(R^3\)
cites work: Evolutionary Games and Population Dynamics
cites work: Trajectories tending to a critical point in 3-space
cites work: Q5330830
cites work: Q5848416
cites work: The global dynamic behavior of the competition model of three species
DOI: 10.1007/S10114-011-9053-7
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6145917