Number and amplitude of limit cycles emerging from topologically equivalent perturbed centers

We consider three examples of weekly perturbed centers which do not have geometrical equivalence: a linear center, a degenerate center and a nonhamiltonian 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)+ (f(x,y) dy−g(x,y) dx)=0, with H(x,y)=(1/2)(x2+y2). This reduction allows us to find the Melnikov function, M(h)=∫H=hf dy−g dx, 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.