Abstract

Consider the planar ordinary differential equation $x\prime =-yF(x,y),\; y\prime =xF(x,y),$ 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.$$

This is a joint work with A. Gasull and R. Prohens.