Phase portraits of linear type centers of polynomial Hamiltonian systems with Hamiltonian function of degree 5 of the form \(H = H_1(x)+H_2(y)\)
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.
scientific article; zbMATH DE number 6995754
scientific article