Hamilton系统的Lyapunov型不等式、稳定性及特征值问题

项目名称： Hamilton系统的Lyapunov型不等式、稳定性及特征值问题

项目编号： No.11201138

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 张启明

作者单位： 湖南工业大学

项目金额： 23万元

中文摘要： 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。

中文关键词： Hamilton系统；Lyapunov型不等式；稳定性；同宿轨；边值问题

英文摘要： 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

英文关键词： Hamilton system；Lyapunov type inequalities；stability；homoclinic orbit；boundary value problem