Planar Semi-quasi Homogeneous Polynomial differential systems with a given degree

Abstract: This paper study the planar semi-quasi homogeneous polynomial differential systems (short for PSQHPDS), which can be regard as a generalization of semi-homogeneous and of quasi-homogeneous systems. By using the algebraic skills, several important properties of PSQHPDS are derived and are employed to establish an algorithm for obtaining all the explicit expressions of PSQHPDS with a given degree. Afterward, we apply this algorithm to research the center problem of quadratic and cubic PSQHPDS. It is proved that the quadratic one hasn't center, and, that the cubic one has center if and only if it can be written as $\dot{x}=x^2-y^3, \dot{y}=x$ after a linear transformation of coordinate and a rescaling of time.