This thesis deals with the bifurcations of limit cycle by perturbing some Hamiltonian systems . In many applications the number and positions of limit cycles are important to understand the dynamical behavior of the system .Using the idea of Poincar\\'{e} map and associated displacement function built on a proper segment transversal to the period annulus of the unperturbed system , the problem is reduced to the problem of finding maximum numbers of isolated zeros of some special Abelian integral , so-called first order Melnikov function. We present a criterion based on papers by M. Grau, F. Ma?osas and J. Villadelprat that provides an easy sufficient conditionin order for a collection of Abelian integrals to have the Chebyshev property.This condition involves the functions in the integrand of the Abelian integrals and can be checked, in many cases, in a purely algebraic way. U sing this criterion, several known results are obtained in a shorter way and some new results, which could not be tackled by the known standard methods, can also be deduced. In the literature there are many papers dealing with zeros of Abelian integrals . In many cases, it is essential to show that a collection of Abelian integrals has some kind of Chebyshev property. The techniques and arguments to tackle these problems are usually very long and highly non-trivial. For instance, in some papers the authors study the geometrical properties of the so-called centroid curve using the fact that it verifies a Riccati equation (which itself is deduced from a Picard-Fuchs system). In other papers, the authors use complex analysis and algebraic topology (analytic continuation, argument principle, monodromy, Picard-Lefschetz formula, etc.). Certainly, the criterion that we present here cannot be applied to all the situations (since the Abelian integrals need to have a specific structure), and, even in the case where it is possible to apply it, sometimes the sufficient condition that we provide is not verified. However we want to stress that, when it works, it enables us to extremely simplify the solution. To illustrate this fact, we reprove with our criterion the main results of three different papers. We are also convinced that this criterion will be useful to obtain new results on the issue. In the remainder of the thesis, we proved the following new results: The first result: The asymptotic expansion of first-order Melnikov function of a heteroclinic loop connecting a cusp of order two and a hyperbolic saddle for a planar near-Hamiltonian system are given. Next, we considered the limit cycle bifurcations of a hyper-elliptic Lienard system with this kind of heteroclinic loop and we studied the least upper bound of limit cycles bifurcated from the annulus inside the heteroclinic loop and also from the heteroclinic loop and center. We f oun d that at most three limit cycles can be bifurcated from the period annulus, also we gave different distribution of the bifurcated limit cycles. The second result : We studied two justify; LINE-HEIGHT: normal; MARGIN: 0cm 0cm 0pt" And finally we introduced a a specific Lienard system of type (6,8) with cuspidal loop that is surrounded by a homoclinic loop with a nillpotent saddle. Then we studied its bifurcation diagram and proved that the maximum number of limit cycles that can be bifurcated this system is at most equal to nine.