Limit Cycles Bifurcating from a 2-Dimensional Isochronous Torus in R ³

In this paper we illustrate the explicit implementation of a method for computing limit cycles which bifurcate from a 2-dimensional isochronous set contained in R3, when we perturb it inside a class of differential systems. This method is based in the averaging theory. As far as we know all applications of this method have been made perturbing noncompact surfaces, as for instance a plane or a cylinder in R3. Here we consider polynomial perturbations of degree d of an isochronous torus. We prove that, up to first order in the perturbation, at most 2(d+1) limit cycles can bifurcate from a such torus and that there exist polynomial perturbations of degree d of the torus such that exactly ν limit cycles bifurcate from such a torus for every ν ∈ {2, 4, . . . , 2(d + 1)}.