Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems

Abstract

This paper is devoted to the complete classification of global phase portraits of quasi-homogeneous but non-homogeneous coprime planar polynomial differential systems of degree $$4$$ . To prove our result, we firstly study the canonical forms for these systems. Then, the global topological structures of the systems having canonical forms are studied by using the quasi-homogeneous blow-up technique for the finite singularities and the Poincare-Lyapunov compactification for the infinite singularities. Finally we perform a topological classification for the set of global phase portraits.