用于研究平面微分系统周期轨道的算法平均法
Algorithmic Averaging for Studying Periodic Orbits of Planar Differential Systems
AI总结：
针对实平面微分系统极限环研究的开放问题，提出四种实现平均法的符号算法，用于计算多项式微分中心受扰后的周期轨道分岔极限环数，拓展了相关研究结果。
摘要：
实平面微分系统定性理论中的核心开放问题之一是对极限环的研究。本文提出一种算法化方法，用于通过平均法分析给定多项式微分中心在一类多项式微分系统内受到扰动时，可从其周期轨道分岔出的极限环数量。我们提出四种符号算法以实现平均法：第一种算法基于极坐标变换，可将所研究的微分系统转化为平均法标准型；第二种算法用于推导与平均法标准型的未扰项相关的若干微分系统的解；第三种算法利用部分Bell多项式，能够计算任意阶平均函数的积分公式；最后一种算法基于上述算法，确定所研究微分系统的平均函数的精确表达式。我们讨论了算法的实现，并通过多个算例进行了评估。实验结果拓展了现有针对若干类微分系统的相关研究结果。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
