We apply the averaging method in a class of planar systems given by a linear center perturbed by a sum of continuous homogeneous vector fields, to study lower bounds for their number of limit cycles. Our results can be applied to models where the smoothness is lost on the set $\Sigma =\{xy=0\}$. They also motivate to consider a variant of Hilbert 16th problem, where the goal is to bound the number of limit cycles in terms of the number of monomials of a family of polynomial vector fields, instead of doing this in terms of their degrees.

中文翻译：

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一些平滑和非平滑平面系统系列的极限环

我们将平均方法应用于由线性中心给定的一类平面系统，该线性中心被连续的均匀矢量场之和所扰动，以研究其极限环数的下界。我们的结果可以应用于在光滑度损失的模型上$\Sigma =\{Xÿ=0\}$。他们还激发考虑希尔伯特第16个问题的一种变体，该问题的目的是根据多项式矢量场族的单项式的数目来限制极限环的数目，而不是根据其次数来进行限制。