求非线性微分系统极限环的数学方法--再论平均法
An Algorithmic Approach to Limit Cycles of Nonlinear Differential Systems: the Averaging Method Revisited
摘要：
本文介绍了一种用平均法分析非线性连续微分系统中心极限环分支的算法。我们开发了三种算法来实现平均方法。第一个算法允许转换所考虑的微分系统的正常形式的平均。在这里，我们将平均的范式的未扰动项限制为零。第二种算法用于推导任意阶平均函数的计算公式。第三个算法是基于前两个算法，确定所考虑的微分系统的平均函数的精确表达式。该方法在Maple中实现，并通过几个例子证明了其有效性。此外，我们报告了一些不正确的结果，在已发表的论文的平均方法。
英文摘要：
This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method. We develop three algorithms to implement the averaging method. The first algorithm allows to transform the considered differential systems to the normal formal of averaging. Here, we restricted the unperturbed term of the normal form of averaging to be identically zero. The second algorithm is used to derive the computational formulae of the averaged functions at any order. The third algorithm is based on the first two algorithms that determines the exact expressions of the averaged functions for the considered differential systems. The proposed approach is implemented in Maple and its effectiveness is shown by several examples. Moreover, we report some incorrect results in published papers on the averaging method.
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