Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900 , together with the other 22 problems . The original problem was posed as the problem of the topology of algebraic curves and surfaces . Actually the problem consists of two similar problems in different branches of mathematics: 1 - An investigation of the relative positions of the branches of real algebraic curves of degree n ( and similarly for algebraic surfaces ) 2 - The determination of the upper bound for the number of limit cycles in polynomial vector fields of degree n and an investigation of their relative positions. \\indent Usually , the maximum of the number of limit cycles is denoted by H(n) , and is called the Hilbert number . Recall that a limit cycle is an isolated closed orbit . It is the ?- (forward) or ?- (backward) limit set of nearby orbits . In many application the number and positions of limit cycles are important to understand the dynamical behavior of the system . This problem is still open even for the case n=2 . Limit cycle behavior is observed in many physical and biological systems . As usual , we use the notion of the cyclicity for the total number of limit cycles which can emerge from a configuration of trajectories (center , period annulus , a singular loop) under a perturbation. There are many problems in mechanics , electrical engineering and the theory of automatic control which are described by non-smooth systems . More precisely , we suppose that the unperturbed system dx = H_y , dy = ?H_x. Has a family of periodic orbits L_h around the origin . If h ?1 , L_h approaches the origin which is an elementary center of parabolic-focus type . And if h ?0, L_h ?L_0, where L_0 is a compound homoclinic loop with a saddle S_1(1 , 0). \\indent In this thesis , we study limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with the center at the origin and a homoclinic loop around the origin . By using the first Melnikov function of piecewise near-Hamiltonian systems , we give lower bounds of the maximal number of limit cycles in Hopf and homoclinic bifurcations , and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in ? . In the case when the degree of perturbing terms is low , we obtain a precise result on the number of zeros of the first Melnikov function.