论文标题
用于研究平面差分系统的定期轨道的算法平均
Algorithmic Averaging for Studying Periodic Orbits of Planar Differential Systems
论文作者
论文摘要
实际平面差异系统定性理论中的主要开放问题之一是极限周期的研究。在本文中，我们提出了一种算法方法，用于检测多少个限制周期可以通过平均方法在一类多项式差异系统中扰动给定的多项式差异中心的周期性轨道分叉。我们提出了四种符号算法来实现平均方法。第一种算法是基于极坐标的变化，该层坐标允许人们将所考虑的差异系统转换为平均正常形式。第二算法用于得出与平均正常的不扰动项相关的某些差分系统的解。第三算法利用了部分钟形多项式，并允许一个人按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，并确定所考虑差异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的算法的实现。实验结果已扩展了某些类别差异系统的现有相关结果。
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.
