高次多项式平面微分系统单值奇点的极限环分支

Bifurcation of limit cycles is a main topic in the qualitative theory and bifurcation theory of planar differential systems. It also play an important role in the study of Hilbert' 16th problem. Due to the fact that the monodromic singularities is very sensitive to the perturbation, the bifurcation of limit cycles of them has attract many interest. However, the traditional research method can only deal with the systems with simple form,thus until now most of the literature are restricted to the systems of low degree, and there is very few result on the systems of high degree. Therefore, the related theory and application problems are need to be solved urgently. In this project we study the limit cycles bifurcating from monodromic singularity of planar polynomial systems of high degree. We will firstly search the conditions which ensure that these systems possess high-order weak focus or high-codimension center for several classes of polynomial systems of high degree. Secondly, by taking advantage of the low order terms with respect to the expansion of Lyapunov quantities, the complexity which arise from many parameters is reduced and thus we can investigate the number of limit cycles producing from the weak focus and the center. For the holomorphic systems, the perturbation technology of the minimal basis of Bautin ideal and the parallelization as well as the idea behind it, which is developed by us recently, will be employed in our study. . The investigation of the above problem will reveal some new geometric properties of the monodromic singularity of polynomial systems of high degree. This investigation will help us to obtain the new lower bounds for the open problem "how many limit cycles can bifurcate from an elementary monodromic singularity of polynomial systems of degree n?".

极限环分支是平面微分系统定性和分支理论的核心内容，对探索Hilbert第16问题具有十分重要的意义。由于单值奇点对摄动非常敏感，其极限环分支问题备受关注。但传统的研究方法只能处理形式简单的系统，乃至迄今绝大多数工作仅限于低次多项式系统，而对高次系统的探讨则少之又少，因此相关的理论与应用问题亟待解决。本项目将研究高次平面系统的几类单值奇点的极限环分支。首先寻求这些系统具有高阶细焦点或高余维中心的条件，再借助Lyapunov量关于摄动参数展开式的低次项来讨论从焦点和中心产生的极限环个数，以此降低多参数导致的复杂性。对其中的Holomorphic系统，拟有机结合Bautin理想的最小基摄动法与我们近期提出的并行算法及其蕴含的思想来展开研究。. 本项目将揭示平面高次系统单值奇点的一些新的几何性质，希望为公开问题“平面n次多项式系统的初等单值奇点在n次摄动下最多产生几个极限环”找到新下界。

本项目研究平面高次多项式微分系统单值奇点的极限环分支问题，其中单值奇点是指中心或焦点。主要研究成果包括：平面半拟齐次多项式系统的规范型（这是研究极限环分支的重要基础）、平面多项式系统高阶细焦点阶数的算法、平面高次多项式系统高余维中心的极限环分支。除此之外，我们还利用阿贝尔方程来研究平面多项式系统的极限环个数、利用庞加莱紧致化爆破法等手段研究平面多项式系统的全局动力学结构。. 对上述问题的研究，有助于我们理解平面高次多项式系统的动力学性质，丰富了平面微分系统定性和分支理论，有一定的学术意义和应用价值。