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.

以下是该研究项目下获得的主要结果的摘要1。在对哈密顿系统的研究中，ITO概括了哈密顿系统完全集成到通用矢量领域的概念。他证明了一个分析矢量场的整合等于存在非谐音和椭圆形的相等点附近的收敛归一化转换。它为庞加雷中心问题提供了答案。2。在沿阵行理论的研究中，莫里塔研究了与二维散射台球问题相关的ZETA功能。他成功地将其meromorthine扩展到了一个半平面，其实际部分大于某些负常数。3。在动态系统分叉理论的研究中，Kokubu研究了Conley索引理论的概括，以呈现为单一扰动的向量场的慢速系统。当缓慢变量具有维度1时，他定义了过渡物质，并获得了提供周期性或杂斜轨道存在的一般方法。4。通过使用奇异哈密顿系统的变分方法，田中证明了（1）散射类型等轨道的存在。（2）-1/γ^2类扰动类别下的周期轨道；（3）对于潜在的有两个单数点的系统的无界和混乱的动作。5。在符号/接触几何形状的研究中，Ono成功地使用整数系数构建了浮子同源性。 Nakai从叶面理论和网络几何学的角度研究了第一阶PDE。特别是，他用有限的类型定义了对PDE的仿射连接，并用它们来研究与解决方案定义的叶面相关的奇异性。