Global Phase Portraits of Homogeneous Polynomial Planar Hamiltonian Systems with Finitely Many Isotropic Points

The global phase portrait (GPP) classification of polynomial planar Hamiltonian systems with finitely many isotropic points is a challenging problem. Only homogeneous Hamiltonian systems of degrees up to five have been dealt with in existing literature. In this paper, through a polar coordinate compactification, we prove that the GPP of a homogeneous planar Hamiltonian system is uniquely determined by the phase portrait around its isotropic point, referred to as the local phase portrait (LPP). Thus, the global classification can be reduced to the local classification. Secondly, two distinct approaches, topological index analysis and algebraic factorization, are proposed to establish both the local classification and the global one. And finally, the corresponding physical flows are discussed, and the consistency of results from the two approaches is validated through four examples.