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

TL;DR

This paper introduces an algorithm to classify planar semi-quasi homogeneous polynomial differential systems of a given degree and applies it to analyze the center problem for quadratic and cubic cases.

Contribution

It develops a new algebraic method to explicitly characterize all PSQHPDS of a specified degree and investigates their center properties.

Findings

Quadratic PSQHPDS do not have centers.

Cubic PSQHPDS have a center if and only if they are equivalent to a specific form.

The paper provides a systematic approach for classifying PSQHPDS based on 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.