Global phase portraits and bifurcation diagrams for Hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the \(y\)-axis
type: Item
instance of: scholarly article
title: Global phase portraits and bifurcation diagrams for Hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the \(y\)-axis
zbMATH Open document ID: 1442.37070
author: Yuzhou Tian
author: Yulin Zhao
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2020-04-07T00:00:00Z
review text: In this work, which completes the paper [\textit{J. Llibre} et al., Discrete Contin. Dyn. Syst., Ser. B 23, No. 2, 887--912 (2018; Zbl 1440.34031)], global phase portraits and bifurcation diagrams for the systems \[\begin{aligned} \begin{cases}\dot{x}=y-ax^4-3bx^2y^2-5cy^4 \\ \dot{y}=x+2bxy^3+4ax^3y,\end{cases}\\ \begin{cases}\dot{x}=y-ax^4-3bx^2y^2-5cy^4 \\ \dot{y}=2bxy^3+4ax^3y,\end{cases}\\ \begin{cases}\dot{x}=-ax^4-3bx^2y^2-5cy^4 \\ \dot{y}=x+2bxy^3+4ax^3y,\end{cases} \end{aligned}\] are presented. In Section 2, some preliminary results, such as Poincaré compactification, types of singular points, topological indices and Neumann's theorem, are recalled. In the others sections, the proofs of the results are given. ``The main ideas of the proofs are to characterize the local phase portraits at the finite and infinite singular points, and to determine their separatrix configurations.''
reviewed by: Cristian Lăzureanu
Mathematics Subject Classification ID: 37J20
Mathematics Subject Classification ID: 70K05
Mathematics Subject Classification ID: 70H33
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C07
zbMATH DE Number: 7187773
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: symmetry
zbMATH Keywords: quartic polynomial system
zbMATH Keywords: global phase portraits
MaRDI profile type: Publication
full work available at URL: j.na.2019.111658
OpenAlex ID: W2981530608
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DOI: 10.1016/J.NA.2019.111658
description: scientific article; zbMATH DE number 7187773