This thesis deals with the bifurcations of limit cycle by perturbing some Hamiltonian systems. Using the idea of Poincare map built on a proper section transversal to the period annulus of the unperturbed system, the problem is reduced to the problem of finding least upper bound for the number of isolated zeros of a so-called first order Melnikov function or Abelian integral. We use standard methods to consider Hopf bifurcation, Poincar e bifurcation and bifurcation of limit cycle form graphics. These methods are: the method based on Picard-Fuchs equations, the method based on Chebeychev criterion and the method of studying the asymptotic expansion of Abelian integrals.
