Number and Amplitude of Limit Cycles emerging from {\it Topologically   Equivalent} Perturbed Centers
  We consider three examples of weekly perturbed centers which do not have {\it geometrical equivalence}: a linear center, a degenerate center and a non-hamiltonian center. In each case the number and amplitude of the limit cycles emerging from the period annulus are calculated following the same strategy: we reduce of all of them to locally equivalent perturbed integrable systems of the form: $dH(x,y)+\epsilon(f(x,y)dy-g(x,y)dx)=0$, with $H(x,y)={1/2}(x^2+y^2)$. This reduction allows us to find the Melnikov function, $M(h)=\int_{H=h}fdy-gdx$, associated to each particular problem. We obtain the information on the bifurcation curves of the limit cycles by solving explicitly the equation $M(h)=0$ in each case. 
Jose-Luis Lopez and Ricardo Lopez-Ruiz