In this thesis, we make a complete study on small perturbations of Hamiltonian vector field with a hyper-elliptic Hamiltonian of degree five, which is a Liénard system of the form , with and olynomials of degree respectively 4 and 3. It is shown that this system can undergo degenerated Hopf bifurcation, which emerges at most three limit cycles in the plane and also this system can undergo Poincaré bifurcation, which emerges at most four limit cycles in the plane for sufficiently small positive and the limit cycles can encompass only an equilibrium inside, i.e. the configuration ( 4 , 0 ) of limit cycles can appear for some values of parameters, where ( 4 , 0 ) stands for four limit cycles surrounding an equilibrium and no limit cycles surrounding two equilibria.