  One of the main open problems in the qualitative theory of real planar
differential systems is the study of limit cycles. In this article, we present
an algorithmic approach for detecting how many limit cycles can bifurcate from
the periodic orbits of a given polynomial differential center when it is
perturbed inside a class of polynomial differential systems via the averaging
method. We propose four symbolic algorithms to implement the averaging method.
The first algorithm is based on the change of polar coordinates that allows one
to transform a considered differential system to the normal form of averaging.
The second algorithm is used to derive the solutions of certain differential
systems associated to the unperturbed term of the normal of averaging. The
third algorithm exploits the partial Bell polynomials and allows one to compute
the integral formula of the averaged functions at any order. The last algorithm
is based on the aforementioned algorithms and determines the exact expressions
of the averaged functions for the considered differential systems. The
implementation of our algorithms is discussed and evaluated using several
examples. The experimental results have extended the existing relevant results
for certain classes of differential systems.
