Weakly structural stability of cubic homogeneous parameterized vector fields
type: Item
instance of: scholarly article
title: Weakly structural stability of cubic homogeneous parameterized vector fields
zbMATH Open document ID: 0967.34053
author: Yun Tang
author: Kamal H. Yasir
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-12-12T00:00:00Z
review text: The authors study the weak structural stability of the vector field \(X\) with the property \[ j_3(X(x, y))_{(0,0)}= X_3(x,y),\quad j_2(X(x, y))_{(0,0)}= 0, \] where \(J_k(X(x, y))_{(0,0)}\) denotes the \(k\)-jet of \(X\) near the origin, and where \(X_3\) is of the form \[ X_3(x,y)= (a_1 x^3+ a_2 x^2y+ a_3 xy^2, b_2x^2y+ b_3xy^2+ b_4 y^3). \] They apply the theorems and technique of Takens to find sufficient conditions on the parameter space such that \(X\) is weakly structurally stable. Specifically, they prove that \(X\) is weakly structurally stable under one of the following conditions: \[ b_4\neq a_3,\quad b_3\neq a_2,\quad b_2\neq a_1;\quad b_4\neq a_3,\quad b_3= a_2,\quad b_2\neq a_1. \]
Mathematics Subject Classification ID: 34D30
zbMATH DE Number: 1542473
zbMATH Keywords: weakly structural stability
zbMATH Keywords: cubic homogeneous parameterized vector fields
reviewed by: Anthony D. Osborne
MaRDI profile type: Publication
full work available at URL: jmaa.2000.7107
OpenAlex ID: W1971982750
cites work: The singularity \(x\partial /\partial y\)
cites work: Algebraic and topological classification of the homogeneous cubic vector fields in the plane
cites work: Singularities of vector fields on the plane
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: Q5336331
cites work: Q4067388
cites work: Bifurcations and trajectories joining critical points
cites work: Singularities of vector fields
cites work: Q4359110
DOI: 10.1006/JMAA.2000.7107
description: scientific article; zbMATH DE number 1542473