哈密顿系统与辛几何中的闭轨道

项目名称： 哈密顿系统与辛几何中的闭轨道

项目编号： No.11271200

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： 张端智

作者单位： 南开大学

项目金额： 60万元

中文摘要： 周期解、闸轨道、同宿轨是Hamilton系统中的重要研究对象，辛道路Maslov型指标理论在Hamilton系统周期解和流形上闭测地线的多重性稳定性研究中发挥了重要作用。在之前的工作中我们建立了闸轨道的Maslov型指标及其迭代理论并在Seifert猜想和与之相关的闸轨道的多重存在性和稳定性的研究中取得较为重要进展。因为Hamilton系统和辛几何之间有着自然的联系，在本项目中我们将致力于Maslov型指标理论的进一步发展以及在上述问题研究中的应用，同时研究辛几何中一些相关的问题。具体如下：1.偶数维欧氏空间中紧凸超曲面上闭特征的多重性和稳定性问题；2.偶数维欧氏空间中可逆紧凸超曲面上闸轨道的多重性和稳定性问题以及与Seifert猜想相关的问题；3.Hamilton系统中的同宿轨问题；4.辛几何中和Weinstein猜想和Arnold猜想相关的问题。我们力争在上述问题研究中取得突破性进展。

中文关键词： 哈密顿系统；闸轨道；闭特征；辛流形；Maslov型指标

英文摘要： Periodic solutions, brake orbits, homoclinic orbits are important research objects in Hamiltonian systems. The Maslov_type index theory of symplectic paths has played great poles in the studies of multilplicities and stabilities of periodic solutions in Hamiltonian systems and closed geodesics on Riemannian manifolds. In our previous works we have established the Maslov_type index and its iteration theory for brake orbits and made important progress in the study of Seifert conjecture and related problems. Since there is natural relation between Hamiltonian systems and symplectic geometry, in this program we will try to further study Maslov index theory and its applications in the study of the above problems, meanwhile we will study problems related to Weinstein conjecture and Arnold conjecture as follows: 1. Multiplicity and stability problems of closed characteristics on compact convex hypersurfaces in even dimensional Euclidian space. 2. Multiplicity and stability problems of brake orbits on revserble compact convex hypersurfaces in even dimensional Euclidian space and problems related to Seifert conjecture. 3. Homoclinic orbits problems in Hamiltonian systems. 4. Problems related to Weinstein conjecture and Arnold conjecture in symplectic geometry.

英文关键词： Hamiltonian systems；brake orbits；closed characteristics；symplectic manifolds；Maslov_type index