Phase portraits of linear type centers of polynomial Hamiltonian systems with Hamiltonian function of degree 5 of the form \(H = H_1(x)+H_2(y)\)
type: Item
instance of: scholarly article
title: Phase portraits of linear type centers of polynomial Hamiltonian systems with Hamiltonian function of degree 5 of the form \(H = H_1(x)+H_2(y)\)
zbMATH Open document ID: 1412.37059
author: Jaume Llibre
author: Y. P. Martínez
author: Claudio Vidal
published in: Discrete and Continuous Dynamical Systems
publication date: 2018-12-19T00:00:00Z
review text: The phase portraits in the Poincaré disc of the linear-type centers of the polynomial Hamiltonian system \[ \dot{x}=-y(1+b_3y+b_4y^2+b_5y^3)~,~~\dot{y}=x(1+a_3x+a_4x^2+a_5x^3), \] are analyzed. \par The main result gives topological equivalences of the phase portraits in the Poincaré disc of the linear-type centers of the considered system. This result provides the global phase portraits in the Poincaré disc as well as some local phase portraits. The results depend on the roots of the polynomials $\hat p(y)=1+b_3y+b_4y^2+b_5y^3$ and $\hat q(x)=1+a_3x+a_4x^2+a_5x^3.$ A characterization of the polynomials $\hat p$ and $\hat q$ according to their roots is given. The local phase portraits of all the possible finite equilibrium points in relation to their eigenvalues and linear matrix are analyzed. Moreover the infinite equilibrium points and their stability are characterized.
reviewed by: Cristian Lăzureanu
Mathematics Subject Classification ID: 37J25
Mathematics Subject Classification ID: 37C15
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 70K05
zbMATH DE Number: 6995754
zbMATH Keywords: separable Hamiltonian systems
zbMATH Keywords: linear type centers
zbMATH Keywords: phase portraits
zbMATH Keywords: quartic vector field
zbMATH Keywords: stability
MaRDI profile type: Publication
full work available at URL: dcds.2019004
OpenAlex ID: W2898423197
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DOI: 10.3934/DCDS.2019004
description: scientific article; zbMATH DE number 6995754