Symmetric Phase Portraits of Homogeneous Polynomial Hamiltonian Systems of Degree 1, 2, 3, 4, and 5 with Finitely Many Equilibria

Abstract

Roughly speaking, the Poincaré disc D2 is the closed disc centered at the origin of the coordinates of R2, where the whole of R2 is identified with the interior of D2 and the circle of the boundary of D2 is identified with the infinity of R2, because in the plane R2, we can go to infinity in as many directions as points have the circle. The phase portraits of the quadratic Hamiltonian systems in the Poincaré disc were classified in 1994. Since then, no new interesting classes of Hamiltonian systems have been classified on the Poincaré disc. In this paper, we determine the phase portraits in the Poincaré disc of five classes of homogeneous Hamiltonian polynomial differential systems of degrees 1, 2, 3, 4, and 5 with finitely many equilibria. Moreover, all these phase portraits are symmetric with respect to the origin of coordinates. We showed that these polynomial differential systems exhibit precisely 2, 2, 3, 3, and 4 topologically distinct phase portraits in the Poincaré disc. Of course, the new results are for the homogeneous Hamiltonian polynomial differential systems of degrees 3, 4, and 5. The tools used here for obtaining these phase portraits also work for obtaining any phase portrait of a homogeneous Hamiltonian polynomial differential system of arbitrary degree.