Bifurcation of a class of extended quasi-homogeneous polynomial differential systems

Abstract

In this paper, we study the bifurcation of a class of extended quasi-homogeneous planar polynomial differential systems of degree $3$, prove that this system has no limit cycle in $(a,b,c)\in\mathbb{R}^3$, and provide its global portraits by using quasi-homogeneous blow-up, Poincare-Lyapunov compactification etc.