研究极限环的一种新方法
A new approach for the study of limit cycles
摘要：
证明了任意平面多项式系统的星形极限环既可以看作是一个新的相关平面非自治多项式系统在给定区间上的解，也可以看作是一个三维多项式系统的异宿解.我们用几个例子来说明这些观点。在我们的方法中的一个关键思想是分解的周期解作为两个合适的函数的总和。作为第一个应用，我们使用这些新的方法来证明所有的星形可逆极限环是代数的。作为第二个应用程序，我们引入了一个函数，其零点控制的周期轨道，持续作为极限环，当我们扰动一个星形可逆中心。据我们所知，这是第一次全面解决这个问题。在某种程度上，这个函数起着类似于阿贝尔积分的作用，用于研究哈密顿系统的扰动。
英文摘要：
We prove that star-like limit cycles of any planar polynomial system can also be seen either as solutions defined on a given interval of a new associated planar non-autonomous polynomial system or as heteroclinic solutions of a 3-dimensional polynomial system. We illustrate these points of view with several examples. One of the key ideas in our approach is to decompose the periodic solutions as the sum of two suitable functions. As a first application we use these new approaches to prove that all star-like reversible limit cycles are algebraic. As a second application we introduce a function whose zeroes control the periodic orbits that persist as limit cycles when we perturb a star-like reversible center. As far as we know this is the first time that this question is solved in full generality. Somehow, this function plays a similar role that an Abelian integral for studying perturbations of Hamiltonian systems.
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