Simultaneous bifurcation of limit cycles from two nests of periodic orbits

Let be an holomorphic differential equation having a center at p, and consider the following perturbation . We give an integral expression, similar to an Abelian integral, whose zeroes control the limit cycles that bifurcate from the periodic orbits of the period annulus of p. This expression is given in terms of the linearizing map of at p. The result is applied to control the simultaneous bifurcation of limit cycles from the two period annuli of , after a polynomial perturbation.