Phase portraits of linear type centers of polynomial Hamiltonian systems with Hamiltonian function of degree 5 of the form \\begin{document}$ H = H_1(x)+H_2(y)$\\end{document}

Abstract

We study the phase portraits on the Poincare disc for all the linear type centers of polynomial Hamiltonian systems of degree \\begin{document} $5$ \\end{document} with Hamiltonian function \\begin{document} $H(x,y) = H_1(x)+H_2(y)$ \\end{document} , where \\begin{document} $H_1(x) = \\frac{1}{2} x^2+\\frac{a_3}{3}x^3+ \\frac{a_4}{4}x^4+ \\frac{a_5}{5}x^5$ \\end{document} and \\begin{document} $H_2(y) = \\frac{1}{2} y^2+ \\frac{b_3}{3}y^3+ \\frac{b_4}{4}y^4+ \\frac{b_5}{5}y^5$ \\end{document} as function of the six real parameters \\begin{document} $a_3, a_4, a_5, b_3, b_4$ \\end{document} and \\begin{document} $b_5$ \\end{document} with \\begin{document} $a_5 b_5≠ 0$ \\end{document} . We characterize the type and multiplicity of the roots of the polynomials \\begin{document} $\\hat{p}(y) = 1+b_3y + b_4 y^2+b_5y^3$ \\end{document} and \\begin{document} $\\hat{q}(x) = 1+a_3x+a_4x^2+a_5x^3$ \\end{document} and we prove that the finite equilibria are saddles, centers, cusps or the union of two hyperbolic sectors. For the infinite equilibria we found that there only exist two nodes on the Poincare disc with opposite stability. We also characterize the separatrices of the equilibria and analyze the possible connections between them. As a complement we use the energy level to complete the global phase portrait.