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.

中文翻译：

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用于研究平面微分系统周期轨道的算法平均

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。在本文中，我们提出了一种算法方法，用于通过平均方法检测给定多项式微分中心在一类多项式微分系统中受到扰动时从其周期轨道可以分叉出多少个极限环。我们提出了四种符号算法来实现平均方法。第一种算法基于极坐标的变化，允许将所考虑的微分系统转换为平均的正常形式。第二种算法用于推导出与平均法线的未扰动项相关联的某些微分系统的解。第三种算法利用部分贝尔多项式并允许以任何顺序计算平均函数的积分公式。最后一个算法基于上述算法，并确定所考虑的微分系统的平均函数的精确表达式。使用几个示例讨论和评估了我们算法的实现。实验结果扩展了某些类别的微分系统的现有相关结果。