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

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4$$\end{document}. 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 Poincaré-Lyapunov compactification for the infinite singularities. Finally we perform a topological classification for the set of global phase portraits.