微分方程的分支理论

项目名称： 微分方程的分支理论

项目编号： No.11271046

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 赵丽琴

作者单位： 北京师范大学

项目金额： 60万元

中文摘要： 本项目研究Hilbert第16问题、弱化Hilbert第16问题和分支理论中的热点、难点问题，这些问题的研究对Hilbert第16问题的解决具有推动作用. 研究难度大,具有挑战性,是国际前沿的研究课题. 具体有以下四方面的内容:(1) 证明6次多项式系统至少具有36个极限环,并给出极限环的分布;(2) 给出一些超椭圆Hamilton函数所对应的向量场的周期环域的环性;(3)研究二次系统中的可逆系统，给出其中1-2种不具有代数曲线或者具有代数曲线但亏格不为1时的阿贝尔积分零点的个数;(4)研究分段光滑系统的切分支现象. 我们将运用全局分支理论, 通过对适当的未扰动系统进行多次扰动、探索有效的变换、研究阿贝尔积分生成元的Chebyshev性质和Picard-Fuchs 方程、应用复域中的Petrov 幅角原理、计算Poincaré映射的表达式等方法, 在应用并发展现有理论的基础上解决这些问题.

中文关键词： 希尔伯特第 16 问题；弱化希尔伯特第 16 问题；周期环域的环性；极限环；

英文摘要： This project intends to solving some interesting but challenging problems on the famous Hilbert's 16th problem, the weak Hilbert's 16th problem and on the bifurcation theory. These problems are crucial and the study on them is very important and useful for the development of the Hilbert's 16th problem and bifurcation theory. There are 4 problems to be investigated in our project. The first one is about the lower bound of number of limit cycles of planar sixth-degree polynomial systems. Up to now, it is proved that there is at least 35 limit cycles for this kind of polynomial systems, and we infer from our research of quintic polynomial systems that the estimation is not good. We will prove that there are at least 36 limit cycles for sixth-degree polynomial systems, and obtain the various configurations of these limit cycles.This will be realized by choosing suitable unperturbed and perturbed systems and by multi-perturbation methods. The second problem is on the cyclicity of period annuli of some hyper-elliptic Hamiltonian systems. It is well known that the difficulty increases sharply for this problem if the degree of the polynomials of x is bigger than 4 because in general there are at least 4 generators in the expressions of Abelian integral. We will investigate the Chebyshev properties of the ge

英文关键词： Hilbert's 16th problem；weakend Hilbert's 16th problem；cyclicity of period annulus；limit cycle；