Local bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations

We study the analytic system of differential equations in the plane which can be written, in a suitable coordinates system, as(x˙,y˙)T=∑i=0∞Fq-p+2is,where p,q∈N,p⩽q,s=(n+1)p-q>0,n∈N and Fi=(Pi,Qi)T are quasi-homogeneous vector fields of type t=(p,q) and degree i, with Fq-p=(y,0)T and Qq-p+2s(1,0)<0. The origin of this system is a nilpotent and monodromic isolated singular point. We show the Taylor expansion of the return map near the origin for this system, which allow us to generate small amplitude limit cycles bifurcating from the critical point. Also, as an application of the theoretical procedure, we characterize the centers and we generate limit cycles of small amplitude from the origin of several families. Finally, we give a new family integrable analytically which includes the centers of the systems studied.