Perturbation theory of the quadratic Lotka-Volterra double center

We revisit the bifurcation theory of the Lotka-Volterra quadratic system X-0 : {(x) over dot = -y - x(2) + y(2), (y) over dot = x - 2xy with respect to arbitrary quadratic deformations. The system has a double center, which is moreover isochronous. We show that the deformed system can have at most two limit cycles on the finite plane, with possible distribution (i, j), where i+j <= 2. Our approach is based on the study of pairs of bifurcation functions associated to the centers, expressed in terms of iterated path integrals of length two.