脉冲微分系统的极小周期与概周期问题

项目名称： 脉冲微分系统的极小周期与概周期问题

项目编号： No.11471109

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 罗治国

作者单位： 湖南师范大学

项目金额： 62万元

中文摘要： 项目主要研究脉冲微分系统极小周期问题与概周期问题的动力学性态，主要内容有：通过运用直接变分法、Morse指标理论以及极大极小理论来研究Hamilton系统极小周期解的存在性与多重性结果；通过运用临界点理论中的变分法和多个临界点存在定理等非线性工具，研究二阶脉冲微分方程和中立型脉冲泛函微分方程的周期解或二阶脉冲边值问题解的多重性；利用几何方法和非线性方法将一些重要的概周期解性结果推广到脉冲微分系统，探索综合运用Mather的扭转映射与Poincaré-Birkhoff不动点定理研究脉冲微分方程概周期解的存在性，并探讨其稳定性，揭示脉冲扰动的本质特点和产生新的定性行为的脉冲扰动机制。这些都是脉冲微分方程理论中较新颖，具有重要理论意义和明确应用前景的研究课题。这些问题的研究将推动微分方程理论和其他相关学科的发展。

中文关键词： 极小周期；概周期；脉冲微分系统；多重性；动力学性质

英文摘要： This project mainly studies the dynamical peoperty of minimal period problem and almost periodic problem of impulsive differential systems,which is organized as follows by three parts. By using the direct variational method, Morse index theory and minimax theory , we investigate the existence of single and multiple periodic solutions with minimal period for Hamiltonian system. By applying the critical point theory inculding variational methods and three-critical-points theorem, we investigate the multiple periodic solutions of second order impulsive differential equations and impulsive neutral functional differential equations, and the multiple solutions of boundray value problem for second order impulsive differential equations. By using the method of geometry and nonlinear analysis, we generalize some important almost periodicity results to the impulsive differential systems，inculding the discuss of the existence of almost periodic solutions for impulsive differential equations by applying Mather's twist mapping and Poincaré-Birkhoff theorem and its stability, and show the essential role of impulsive perturbation in producing and destroying the existence of periodic solutions and almost periodic solutions.These problems are all new, having very important theoretical meanings and definitely practical prospects in the theory of impulsive differential equations. The studying of these problems will promote the development of the theory of differential equations and other relevant subjects.

英文关键词： Minimal period;Almost period;Impulsive differential system;Multiplicity;Dynamic property