Center-focus determination and limit cycles bifurcation for $p:q$ homogeneous weight singular point

TL;DR

This paper investigates the center-focus problem and limit cycle bifurcations in $p:q$ homogeneous weight singular points of polynomial differential systems, providing methods for computing focal values and demonstrating multiple limit cycles near the origin.

Contribution

It introduces a new approach for analyzing $p:q$ homogeneous weight singular points, including a method for computing focal values and studying bifurcations.

Findings

Existence of three or five limit cycles near the origin for the $2:3$ case.

Method for calculating focal values in non-homogeneous polynomial systems.

Analysis of center-focus determination for weighted singular points.

Abstract

The quasi-homogeneous (and in general non-homogeneous) polynomial differential systems have been studied from many different points of view. In this paper, Center-focus determination and limit cycles bifurcation for $p:q$ homogeneous weight singular point are investigated. Some prosperities of Successive function and focus values are discussed, furthermore, the method of computing focal values is given. As an example, center-focus determination and limit cycle bifurcation for $2:3$ homogeneous weight singular point are studied, three or five limit cycles in the neighborhood of origin can be obtained by different perturbations.