We give a new approach of investigation and approximation of solutions of fractional differential systems (FDS) subjected to periodic boundary conditions. According to the main idea of the numerical–analytic technique, we construct a sequence of functions that it proved to be convergent. It is shown that the limit function of the constructed sequence satisfies a modified FDS and periodic conditions. It is a solution of the given periodic BVP, if the corresponding determined equation has a root. An example of fractional Duffing equation is also presented to illustrate the theory.