Bifurcation diagrams for Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields
type: Item
instance of: scholarly article
title: Bifurcation diagrams for Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields
zbMATH Open document ID: 1309.34045
author: Ilker E. Colak
author: Jaume Llibre
author: Claudia Valls
published in: Journal of Differential Equations
publication date: 2014-11-21T00:00:00Z
review text: In a previous work the authors of this article demonstrated that there are six classes of Hamiltonian planar polynomial vector fields (identified with systems of polynomial differential equations on the plane) that have the form \(X = L + C\), where \(L\) and \(C\) are homogeneous polynomials of degrees 1 and 3, respectively, and where the eigenvalues of \(L\) are nonzero pure imaginary numbers, so that \(X\) has a ``linear center'' at the origin. That is, for any such vector field there exist a linear change of coordinates and a time rescaling that expresses it in one of six forms. They also found, up to topological equivalence and time reversal, all possible phase portraits on the Poincaré sphere of such systems. In this work they give an exhaustive description of conditions on the parameters in each class that determine the phase portrait of members of the class. For example, one class corresponds to the systems of differential equations of the form \[ \dot x = a x + b y, \] \[ \dot y = -((a^2+1)/b) x - a y + x^3. \] All such systems with \(b < 0\) have one particular phase portrait, all those with \(b > 0\) have another.
reviewed by: Douglas S. Shafer
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 6371762
zbMATH Keywords: Hamiltonian linear type center
zbMATH Keywords: vector field
zbMATH Keywords: phase portrait
MaRDI profile type: Publication
full work available at URL: j.jde.2014.10.006
OpenAlex ID: W1973474583
cites work: Q5832271
cites work: Integrability of a linear center perturbed by a fourth degree homogeneous polynomial
cites work: Integrability of a linear center perturbed by a fifth degree homogeneous polynomial
cites work: Local analytic integrability for nilpotent centers
cites work: Q5839236
cites work: Algebraic and topological classification of the homogeneous cubic vector fields in the plane
cites work: Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields
cites work: Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields
cites work: Phase portrait of Hamiltonian systems with homogeneous nonlinearities
cites work: A method for characterizing nilpotent centers
cites work: Probleme General de la Stabilite du Mouvement. (AM-17)
cites work: Q3035865
cites work: Algebraic Particular Integrals, Integrability and the Problem of the Center
cites work: Cubic vector fields symmetric with respect to a center
cites work: Q3660122
cites work: Q3828314
cites work: Recent advances on determining the number of real roots of parametric polynomials
cites work: Quadratic systems with center and their perturbations
cites work: The classification of reversible cubic systems with center
cites work: Remarks on ``The classification of reversible cubic systems with center''
DOI: 10.1016/J.JDE.2014.10.006
Recommended article: Bifurcation diagrams for Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields
Recommended article: Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields
Recommended article: Q4687885
Recommended article: Bifurcation of Limit Cycles from the Center of a Family of Cubic Polynomial Vector Fields
Recommended article: Limit cycle bifurcations from centers of symmetric Hamiltonian systems perturbed by cubic polynomials
Recommended article: On the bifurcation of limit cycles due to polynomial perturbations of Hamiltonian centers
Recommended article: The Bifurcation Diagram of Cubic Polynomial Vector Fields on CP1
Recommended article: Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields
Recommended article: Q5715780
Recommended article: Q4979294
description: scientific article; zbMATH DE number 6371762