The 16th Hilbert problem restricted to circular algebraic limit cycles
The 16th Hilbert problem restricted to circular algebraic limit cycles
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DOI：
10.1016/j.jde.2015.12.019
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发表时间：
2016-04
影响因子：
2.4
通讯作者：
J. Llibre;R. Ramírez;V. Ramírez;N. Sadovskaia
中科院分区：
文献类型：
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作者：
J. Llibre;R. Ramírez;V. Ramírez;N. Sadovskaia
我们证明了以下两个结果。首先，每个具有S个不变圈的S次平面多项式向量场是达布可积的，没有极限环。其次，一个S次的平面多项式向量场至多存在S− 1个不变圈，这些不变圈是代数极限环。特别地，我们解决了第16个希尔伯特问题限制在由圆给出的代数极限环，因为一个S次的平面多项式向量场最多有S− 1个由圆给出的代数极限环，并且达到了这个数目。
We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S− 1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S− 1 algebraic limit cycles given by circles, and this number is reached.