离散Hamilton系统的复杂轨道问题

The minimax methods in critical point theory has been proved to be an effective tool to deal with complex orbits(e.g. period solutions, homoclinic orbits and heteroclinic orbits) of discrete Hamilton systems with variational structure. However, as an elaborated developed part of critical point theory, Morse theory are rarely used to deal with these problems. To the end, this project is devoted to the following three problems. Firstly, this project shall establish the quantity relation between the minimal period and the Morse index of the critical points to obtain the periodic solutions with minimal period to discrete Hamilton systems. Secondly, the subharmonic solutions of the corresponding systems are shown to be uniformly bounded through Morse index estimates. Then, the existence of a homoclinic orbit is established by taking limits. Finally,we shall study the existence and multiplicity of homoclinic orbits as well as heteroclinic orbits of discrete Hamilton systems by directly using the infinite-dimensional Morse theory.This is completely new.These research will motivate the development of qualitative theory of discrete systems and further develop and perfect the discrete variational theory.

对于具有变分结构的离散Hamilton系统的复杂轨道（如周期解、同宿轨与异宿轨）的研究，临界点理论中的极小极大方法已经成为了一个有效的工具。而运用更精细的临界点理论--Morse理论来处理这类问题的研究成果却极少。为此，本项目将致力于以下三个问题的研究：(1) 建立临界点的最小周期与其Morse指标的数量关系而得到离散Hamilton系统的最小周期解；(2) 通过估计相应系统的次调和解的Morse指标而证明其一致有界性，通过求极限获得同宿轨的存在性。(3)直接利用无穷维Morse理论讨论离散Hamilton系统同宿轨与异宿轨的存在性与多重性，这是一项全新的工作。这些研究将对离散系统定性理论的发展具有重要的促进作用，将进一步发展并完善离散变分理论。

对于具有变分结构的离散Hamilton系统的复杂轨道（如周期解、同宿轨与异宿轨）的研究，临界点理论中的极小极大方法已经成为了一个有效的工具。而运用更精细的临界点理论——Morse理论来处理这类问题的研究成果却极少。此项目首先研究了离散-Laplacian系统的周期解的存在性。通过Clark对偶并计算临界群，我们找到了一个简单易于判断系统非常数周期解的存在性的充分条件。其次，利用Galerkin逼近法和临界点理论中的S1指标理论, 我们将Kaplan-Yorke类型的时滞微分方程中一些著名结果推广到了高阶情形，从而证明了高阶情形下的Kaplan-Yorke猜想也成立。最后，我们建立了一个时滞微分方程模型来研究Wolbachia在蚊群中的传播动力学行为。我们找到了保证种群成功替换的阈值。这个结果对释放策略具有重要的指导作用。