In this paper we prove, that under certain hypotheses, the planar differential equation: ˙x = X₁(x, y) + X₂(x, y), ˙y = Y₁(x, y) + Y₂(x, y), where (X_i, Y_i), i = 1, 2, are quasi-homogeneous vector fields, has at most two limit cycles. The main tools used in the proof are the generalized polar coordinates, introduced by Lyapunov to study the stability of degenerate critical points, and the analysis of the derivatives of the Poincar´e return map. Our results generalize those obtained for polynomial systems with homogeneous non-linearities.

In this paper we provide the best lower bounds, that are known up to now, for the Hilbert numbers of polynomial vector fields of degree N , , for small values of N . These limit cycles appear bifurcating from symmetric Darboux reversible centers with very high simultaneous cyclicity. The considered systems have, at least, three centers, one on the reversibility straight line and two symmetric outside it. More concretely, the limit cycles are in a three nests configuration and the total number of limit cycles is at least 2 n + m , for some values of n and m . The new lower bounds are obtained using simultaneous degenerate Hopf bifurcations. In particular, and

Quadratic perturbations of a one-parameter family of quadratic reversible systems with two centers (without other singularities in finite plane) are studied. The exact upper bound of the number of limit cycles, the configurations of limit cycles, and the bifurcation diagrams for different range of the parameter are given.