Bifurcation of Limit Cycles from a Polynomial Non-global Center

Abstract

Consider the planar ordinary differential equation \({\dot x=-yF(x,y), \dot y {=}xF(x,y)}\) , where the set \({\{F(x,y)=0\}}\) consists of k non-zero points. In this paper we perturb this vector field with a general polynomial perturbation of degree n and study how many limit cycles bifurcate from the period annulus of the origin in terms of k and n. One of the key points of our approach is that the Abelian integral that controls the bifurcation can be explicitly obtained as an application of the integral representation formula of harmonic functions through the Poisson kernel.