Bifurcations of certain family of planar vector fields tangent to axes
type: Item
instance of: scholarly article
title: Bifurcations of certain family of planar vector fields tangent to axes
zbMATH Open document ID: 0648.34068
DOI: 10.1016/0022-0396(87)90138-0
author: Henryk Żołądek
published in: Journal of Differential Equations
publication date: 1987-01-01T00:00:00Z
review text: This interesting paper deals with the families of planar vector fields of the following form: \(V_{\mu}(x,y)=[xW_ 1(x,y;\mu),yW_ 2(x,y;\mu)],\) where \(\mu =(\mu_ 1,\mu_ 2)\in {\mathbb{R}}^ 2\). If \(dV_ 0(0)=0\), the vector field \(V_ 0\) is called singular. The author proves that the space of germs of deformations of singular vector fields is divided into degenerate germs and nondegenerate germs such that the degenerate germs form a union of a finite number of submanifolds of codimension one. Every nondegenerate deformation is \(C^ 0\)-orbitally equivalent to the family \[ M_{\mu}(x,y)=[x(\mu_ 1+x+y),y(\mu_ 2-(\alpha +1)x/\beta -\alpha y/(\beta +1)+\nu R(x,y)], \] where R(x,y) is a specially chosen perturbation. The bifurcation diagram of the family \(M_{\mu}(x,y)\) is investigated in detail. Some new resuls about the function \(h\mapsto \iint_{H\leq h}Pdxdy\), where P(x,y) and H(x,y) are not polynomials, are obtained.
Mathematics Subject Classification ID: 34D30
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 4057999
zbMATH Keywords: versal deformation
zbMATH Keywords: equivariant vector field
zbMATH Keywords: nondegenerate deformation
zbMATH Keywords: perturbation
zbMATH Keywords: bifurcation diagram
reviewed by: A. Klíč
MaRDI profile type: Publication
full work available at URL: 0022-0396(87)90138-0
OpenAlex ID: W2043560105
cites work: Q5824493
cites work: On the bifurcation of separatrices in the problem of stability loss of auto-oscillations near 1:4 resonance
cites work: Q5842261
cites work: Q5789702
cites work: Abelian integrals and bifurcation theory
cites work: Q3948194
cites work: Q5650865
cites work: Q5672739
cites work: Singularities of vector fields on the plane
cites work: Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3
cites work: Q3905755
cites work: A note on: Abelian integrals and bifurcation theory
cites work: Q3940514
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: Q4743116
cites work: Q4167320
cites work: Q3923300
cites work: On reducible non-linear differential equations occurring in mechanics
cites work: Q3891465
cites work: Bifurcations and trajectories joining critical points
cites work: Bifurcations of the phase pattern of an equation system arising in the problem of stability loss of selfoscillations close to 1:4 resonance
cites work: Q3338368
cites work: A codimension two bifurcation with a third order Picard-Fuchs equation
cites work: Qualitative investigation of a system of differential equations of the theory of oscillations
cites work: Forced oscillations and bifurcations
cites work: Singularities of vector fields
cites work: Bifurcation into invariant tori at points of resonance
cites work: Q3334683
cites work: Q3664089
description: scientific article; zbMATH DE number 4057999