A limit cycle is an isolated periodic solutions in the set of all periodic solutions of a system of differential equations. Periodic behaviors in the nature often appear as a limit cycle in their corresponding mathematical models and therefore, knowing the number and position of limit cycles are important to understand the dynamical behavior of the system. This thesis deals with bifurcation of limit cycles from planar polynomial near-integrable systems. Here, we use the asymptotic expansions of the first order Melnikov function and Chebyshev criterion for studying bifurcation of limit cycles for small perturbations of some Hamiltonian systems with n-polycycles, n ? 3, in their phase portraits. Moreover, a new family of centers of planar polynomial differential systems of arbitrary even degree is introduced and classified the global phase portraits of the centers of this family having degree 2, 4 and 6 in the Poincaré disc. Also, we have studied existence or non-existence of limit cycles for Higgins-Selkov system and classified the phase portraits of these systems in the Poincaré disc for different values of parameters.
