Abstract In this paper, we are interested in how the local cyclicity of a family of centers depends on the parameters. This fact was pointed out in [21] , to prove that there exists a family of cubic centers, labeled by C D 31 12 in [25] , with more local cyclicity than expected. In this family, there is a special center such that at least twelve limit cycles of small amplitude bifurcate from the origin when we perturb it in the cubic polynomial general class. The original proof has some crucial missing points in the arguments that we correct here. We take advantage of a better understanding of the bifurcation phenomenon in nongeneric cases to show two new cubic systems exhibiting 11 limit cycles and another exhibiting 12. Finally, using the same techniques, we study the local cyclicity of holomorphic quartic centers, proving that 21 limit cycles of small amplitude bifurcate from the origin, when we perturb in the class of quartic polynomial vector fields.

中文翻译：

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中心族局部循环的下限

摘要 在本文中，我们对中心族的局部循环如何依赖于参数感兴趣。这个事实在[21]中被指出，以证明存在一个立方中心族，在[25]中被标记为CD 31 12，其局部周期性比预期的要多。在这个族中，有一个特殊的中心，当我们在三次多项式一般类中扰动它时，至少有十二个小幅度的极限环从原点分叉。原始证明在我们这里纠正的论点中有一些关键的缺失点。我们利用对非泛型情况下分岔现象的更好理解，展示了两个新的立方系统，展示了 11 个极限环，另一个展示了 12 个。 最后，使用相同的技术，我们研究了全纯四次中心的局部循环，