微分方程的分支理论

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 generators, the Picard-Fuchs equations, Petrov argument principle and try to find some new methods to solve this problem..The third problem is on the number of isolated zeros of Abelian integrals of quadratic reversible systems under quadratic perturbations. This is the most difficult part for quadratic systems, and there are many cases to be considered. We will discuss some of the cases where the level curves are not algebraic or the level curves are not of genus 1. We will study this problem by the method in the second problem..The last problem is on the bifurcation of piecewise smooth differential systems. Piecewise smooth dynamical systems have appeared in many disciplines such as in electronics, mechanical systems, control theory, medicine, biology,information and economics. We will investigate the bifurcation phenomenon of this kind of system for the situation where the orbits do not intersect the manifold transversally.The number of limit cycles will be studied by calculating the expression of Poincaré map,which is very complicated..In one word, the problems of our project are important and challenging, and the method of solving these problem will be creative.

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.

本项目研究了任意多项式系统所具有的小环的个数$M(m)$, 得到了$M(6)-M(14)$的下界, 证明了如果 $m\ge 23$, 则$M(m)\ge m^2$, 以及当$m\to \infty$时, $M(m)$的增长速度为$\frac {18}{25}\cdot \frac {1}{2\ln 2 }(m+2)^2\ln (m+2)$; 研究了五次系统的极限环的个数和分布问题, 发现了具有23个极限环的一种新分布;.研究了6次对称超椭圆函数在任意多项式扰动下的周期环域的环性, 以及其中五种类型(双鱼型, 双八字形, 两点异宿, 过幂零鞍点的双同宿, 过双曲鞍点的双同宿)在Lienard扰动下阿贝尔积分零点的个数问题；研究了一类具有过幂零鞍点的双同宿的4次椭圆函数在任意多项式扰动下周期环域的环性; 研究了一类具有过尖点且围绕幂零中心异宿环在Lienard扰动下阿贝尔积分零点的个数问题; 研究了几类分段光滑系统的极限环分支问题, 我们得到了极限环个数的上界和下界, 发现了一些新的分支现象.