Tian, Yuzhou; Zhao, Yulin Global phase portraits and bifurcation diagrams for Hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the \(y\)-axis. (English) Zbl 1442.37070 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 192, Article ID 111658, 27 p. (2020). In this work, which completes the paper [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.” Reviewer: Cristian Lăzureanu (Timisoara) Cited in 3 Documents MSC: 37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems 70K05 Phase plane analysis, limit cycles for nonlinear problems in mechanics 70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations Keywords:Hamiltonian system; symmetry; quartic polynomial system; global phase portraits Citations:Zbl 1440.34031 PDFBibTeX XMLCite \textit{Y. 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