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, singular loop) under a perturbation. The conditions for a plane polynomial quadratic vector field to have a center are known since the beginning of the last century. In the space of all quadratic systems, the systems with a center form a union of four irreducible affine algebraic sets: • Hamiltonian ( ) • reversible ( ) • generalized Lotka-Volterra ( ) • codimension-four ( ) The subscripts indicate the co-dimension of each algebraic subset. A quadratic center is said to be generic, if it does not belong simultaneously to two of the above algebraic sets. The cyclicity of the open annuli in the generic Lotka-Volterra case ( ) and Hamiltonian ( ) was completely solved by several authors. However, almost nothing is known about the generic riversible case ( ). In this thesis, we study a stratum in the set of all quadratic differential systems , with a center, known as the codimension-four case . It has a center and a node and a rational first integral. In fact we set an upper bound of the number of limit cycles produced from the period annulus around the center. The limit cyclies under small quadratic perturbations in the system are determined by the zeros of the first Poincaré-Pontryagin-Melnikov integral I . We first show that the orbits of the unperturbed system are elliptic curves, and I is a complete elliptic integral. Then using Picard-Fuchs equtions and property of Chebyshev systems, we show that the cyclicity of the period annulus of under quadratic perturbations is greater than or equal to 3 and is less than or equal to 5.