Analysis of dynamical systems and related topics in geometry

动力系统分析及几何相关主题

• 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
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440054/
• 关键词: Hamiltonian system; integrable system; variational method; zeta function; Conley index; complex dynamical system; symplectic geometry; 力学系ゼータ関数; コンレイ指数; 接触幾何学

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.

以下是本课题主要研究成果的摘要。在哈密顿系统的研究中，伊藤将哈密顿系统的完全可积性推广到一般向量场的完全可积性。他证明了解析向量场的可积性等价于平衡点附近非共振椭圆型收敛归一化变换的存在性。它给出了庞加莱中心问题的答案。在遍历理论的研究中，Morita研究了与二维散射台球问题相关的zeta函数。他成功地将其亚纯推广到一个实部大于某个负常数的半平面上。在动力系统分岔理论的研究中，Kokubu研究了Conley指标理论在含奇摄动矢量场的慢速系统中的推广。他定义了慢变量为1维时的跃迁矩阵，并给出了证明周期轨道或异斜轨道存在的一般方法。Tanaka利用变分方法证明了奇异哈密顿系统存在如下轨道：(1)散射型；(2) -1/γ^2型摄动下的周期轨道；(3)势有两个奇点的系统的无界混沌运动。在辛/接触几何的研究中，Ono成功地构造了整数系数的Floer同调。Nakai从叶理理论和Web几何的角度研究了一阶偏微分方程。特别地，他定义了有限型偏微分方程的仿射连接，并用它来研究其解所定义的叶理的奇异性。

项目成果

Gedeon, T., Kokubu, H., Mischaikow, K., Oka, H.and Reineck, J.: "Conley index for fast-slow systems I : One-dimensional slow variable"Journal of Dynamics and Differential Equations. 11. 427-470 (1999)

Gedeon, T.、Kokubu, H.、Mischaikow, K.、Oka, H. 和 Reineck, J.：“快慢系统的康利指数 I：一维慢变量”动力学与微分方程杂志。

Kokubu,H.,Mischaihov.K.and Oka,H.: "Directional transition matrix"Banach Center Publ.. 47. 133-144 (1999)

Kokubu,H.、Mischaihov.K. 和 Oka,H.：“方向转换矩阵”Banach Center Publ.. 47. 133-144 (1999)

Ohta, H., Ono, K: "Simple singularities and topology of symplectically filling 4-manifolds"Commentarii Mathematici Helvetici. 74. 575-590 (1999)

Ohta, H.，Ono, K：“辛填充 4 流形的简单奇点和拓扑”Helvetici 数学评论。

H.Ohta and K.Ono: "Simple singularities and topology of symplectically filling 4-manifolds"Commentarii Mathematici Helvetici. 74. 575-590 (1999)

H.Ohta 和 K.Ono：“辛填充 4 流形的简单奇点和拓扑”Commentarii Mathematici Helvetici。

H. Shiga and H. Tanigawa: "Trans. Amer. Math. Soc."Projective structures on Riemann surfaces with discontinuous holonomies.

H. Shiga 和 H. Tanikawa：“Trans. Amer. Math. Soc.”具有不连续完整的黎曼曲面上的射影结构。