Hamilton系统的概周期解和闸轨道问题研究

Hamilton系统是数理科学、生命科学以及社会科学领域中一类非常重要的系统, 因其各种解的存在性问题研究关系到动力系统的动力学行为和分支理论的推广和发展，所以长期以来备受数学家和物理学家的关注。本项目拟使用临界点理论来研究Hamilton系统的概周期解、最小周期闸轨道和次调和闸轨道问题。具体地，构建合适的Banach空间并进行空间分解，分析Hamilton系统所对应的微分算子的谱和特征空间的性质，发展和应用临界点理论中的Minimax原理、 Morse理论、畴数理论和指标理论等工具来建立一些新的临界点的存在性、唯一性、多重性以及不存在性定理，并利用这些定理来研究Hamilton系统的概周期解、最小周期闸轨道和次调和闸轨道的存在性、唯一性、多重性以及不存在性问题。 . 本项目的研究不仅有助于Hamilton系统理论的发展，而且还将有助于临界点理论的补充和完善，具有较高的学术价值。

Hamiltonian system is a very important system in the fields of mathematical and physical sciences, life sciences and social sciences. Studies on the existence of various solutions for this system are related to the extension and development of dynamical behavior and bifurcation theory for dynamical systems. Hence, it has attracted mathematicians and physicists for a long time. In this project, we will use the critical point theory to study those problems on almost periodic solutions, minimal periodic brake orbits and subharmonic brake orbits of Hamiltonian systems. To be precise, we will construct some suitable Banach spaces and decompose them, analyze those properties of spectrum and eigenspace for differential operators corresponding to Hamiltonian systems, develop and apply those tools in critical point theory such as Minimax principle, Morse theory,Category theory and Index theory to construct some new existence, uniqueness, multiplicity and non-existence theorems on critical points, and then use these theorems to study those problems on existence, uniqueness, multiplicity and non-existence of almost periodic solutions, minimal periodic brake orbits and subharmonic brake orbits for Hamiltonian systems. . The study of this project can contribute to not only the development of the theory on Hamiltonian systems, but also the complement and perfection of critical point theory. It has the higher academic value.