Hamilton系统的Lyapunov型不等式、稳定性及特征值问题

Hamilton systems are extensively applied in such fields as Mathematical Science, Life Science and Social Science. Lyapunov-type inequalities are referred to all kinds of inequalities which are improved and generalized from the so-called classical Lyapunov inequality, first derived by Lyapunov, a Russian scientist specializing in mathematics and mechanics.This project will establish some Lyapunov-type inequalities as their explicit representations which are directly used by the potential functions for some Hamilton systems, and try to generalize these inequalities to the higher dimensional cases. It will give the necessary conditions about the existence of nontrivial homoclinic orbits for Hamilton systems, and further give their non-existence conditions of nontrivial homoclinic orbits. Moreover, by taking advantage of the relations between homoclinic orbits and solitary wave solutions, it will find some classes of wave equations which have the time and space structure, but don't have any solitary wave solution. It will give the sufficient conditions of the elliptic stability for the planar linear periodic Hamilton systems, and further find the stability conditions to describe these systems comprehensively. Furthermore, it will establish some stability criterions for several classes of Hamilton systems. Finally, it will discuss the relevant qualities of eigenvalue for linear Hamilton eigenvalue problems. The above research on the nature of the solutions will help us to further study the essential characteristic of Hamilton systems, enrich the relevant theorems of these Hamilton systems and Lyapunov inequalities, and further prompt the development of the qualitative theorems of differential equations.

Hamilton系统广泛应用于数理科学、生命科学及社会科学的各个领域。Lyapunov型不等式是指最先由俄国数学力学家Lyapunov得出的所谓经典Lyapunov不等式经不断改进和推广所得各种形式。本项目拟建立Hamilton系统能直接用位势函数显式表出的Lyapunov型不等式，并推广到高维情形；给出Hamilton系统存在非平凡同宿轨的必要条件，进而给出其非平凡同宿轨的不存在性条件，并利用同宿轨与孤立波的关系，找出不存在孤立波的具有时空结构的波动方程；给出平面线性周期Hamilton系统椭圆型稳定的充分条件，找到全面刻划该系统稳定性的条件，并建立一些Hamilton系统的稳定性准则；探讨线性Hamilton特征值问题特征值的有关性质。以上对解的性态研究将进一步探究Hamilton系统的本质特征，丰富Hamilton系统及Lyapunov不等式的相关理论，并推动微分方程定性理论的发展。

Hamilton系统应用非常广泛，具有较强的物理力学背景。对Hamilton系统的解的性态的研究已成为备受关注的热点问题，取得了很多引人入胜的成果。本项目主要建立了Hamilton系统等微差分系统的Lyapunov型不等式，并得到了解的不共扼性准则；研究了同宿轨的存在性、多重性及不存在性条件；研究了几类生态微分系统反周期解的存在性及稳定性。同时，我们也获得了有关微分方程初边值问题的整体解的存在性，多重性以及平衡解的分支等成果等。所得结果丰富和促进了微分方程定性理论的发展，为生物学、物理学及化学等学科领域出现的相关微分方程模型的研究和处理提供了理论依据。