Analysis of dynamical systems and related topics in geometry

动力系统分析及几何相关主题

基本信息

批准号：
11440054
负责人：
ITO Hidekazu
金额：
$ 3.9万
依托单位：
Tokyo Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B).
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

The following is the abstract for the main results obtained under this research project.1. In the research of Hamiltonian systems, Ito generalized the notion of complete integrability for Hamiltonian systems to that for general vector fields. He proved that the integrability of an analytic vector field is equivalent to the existence of a convergent normalizing transformation near an equlibrium point that are non-resonant and elliptic. It gives an answer to the Poincare center problem.2. In the research of ergodic theory, Morita studied the zeta function associated with two dimensional scattering billiards problem. He succeeded in extending it meromorphically to a half plane with its real part greater than some negative constant.3. In the research of bifurcation theory of dynamical systems, Kokubu studied the generalization of Conley index theory to slow-fast systems which are singularly perturbed vector fields. He defined transition matrices when the slow variables are of dimension one, and obtained a general method for proving the existence of periodic or heteroclinic orbits.4. By using variational method for singular Hamiltonian systems, Tanaka proved the existence of orbits such as (1) scattering type ; (2) periodic orbits under the class of perturbation of type -1/γ^2 ; (3) unbounded and chaotic motions for systems whose potential have two singular points.5. In the research of symplectic/contact geometry, Ono succeeded in constructing the Floer homology with integer coefficients. Nakai studied 1st order PDE's from the viewpoint of foliation theory and Web geometry. In particular, he defined affine connections for those PDE's with finite type, and used them to study the singularities associated with the foliation defined by their solutions.

本课题取得的主要成果如下： 1．在哈密顿系统的研究中，伊藤将哈密顿系统的完全可积性概念推广到一般向量场的可积性。场等价于平衡点附近存在一个非共振且椭圆的收敛归一化变换。 2．在遍历理论中，森田研究了与二维散射台球问题相关的zeta函数，成功地将其亚纯扩展到实部大于某个负常数的半平面。 3．将康利指数理论推广到奇扰动向量场的慢-快系统，他定义了慢变量为一维时的转移矩阵，并获得了证明存在性的通用方法。 4.利用奇异哈密顿系统的变分法，田中证明了诸如(1)散射型轨道的存在性；(2)-1/γ^2型扰动类下的周期轨道； 3．势具有两个奇异点的系统的无界混沌运动。5．在辛/接触几何的研究中，小野成功地构造了整数系数的弗洛尔同调。 Nakai 从叶状结构理论和网络几何的角度研究了一阶偏微分方程，特别是，他为那些有限类型的偏微分方程定义了仿射连接，并用它们来研究与由它们的解定义的叶状结构相关的奇点。