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.

翻译：实平面微分系统定性理论中的主要开放问题之一是极限环的研究。本文提出了一种算法方法，用于检测在给定多项式微分中心被扰动到一类多项式微分系统时，通过平均方法可从其周期轨道分岔出多少个极限环。我们提出了四种实现平均方法的符号算法。第一种算法基于极坐标变换，可将所考虑的微分系统转化为平均法的规范型。第二种算法用于推导与平均法规范型中未扰动项相关的某些微分系统的解。第三种算法利用部分贝尔多项式，可计算任意阶平均函数的积分公式。最后一种算法基于前述算法，并确定所考虑微分系统平均函数的精确表达式。通过多个算例讨论并评估了算法的实现。实验结果扩展了针对某些微分系统类别的现有相关结论。