Global phase portraits and bifurcation diagrams for Hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the \(y\)-axis
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.''
scientific article; zbMATH DE number 7187773
scientific article