On the cyclicity of the period annulus of quasi-homogeneous polynomial vector fields

Abstract

In this article, we study the number of limit cycles, bifurcating from the period annulus of any quasi-homogeneous polynomial vector fields with a center, under a one-parameter polynomial perturbation. We first recharacterize quasi-homogeneous polynomial vector fields and its global center, then we establish an upper bound formula for the number of the isolated zeros of the $k$th order Melnikov function in terms of $k$, $max \{s_1,s_2\}$ and the degree $n$ of the perturbation by applying adapted Francoise's algorithm in conjunction with combinatorial techniques, where $(s_1,s_2)$ is the weight exponent of the quasi-homogeneous polynomial vector field. This extends relevant results presented in the literature [JDDE,21(2009)133-152] and [JDE, 276(2021)1-24]. As an application, we completely solve the limit cycle bifurcation problem of a perturbated quasi-homogeneous polynomial vector field.