Classification of Global Phase Portrait of Planar Quintic Quasi-Homogeneous Coprime Polynomial Systems

This paper is devoted to the complete classification of global phase portraits (short for GPP) of quasi-homogeneous but non-homogeneous coprime planar quintic polynomial differential systems (short for QCQS). We firstly study the canonical forms of QCQS. It is shown that these canonical forms can have 0, 1, 2, 4 parameters. Then, we investigate the global topological structures of all canonical forms, by using the quasi-homogeneous blow-up technique for the finite singularities and the Poincaré–Lyapunov compactification for the infinite singularities. We finally perform a topological classification for the set of GPP.