脉冲微分系统的极小周期与概周期问题

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.

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。

本项目研究脉冲微分系统的动力学性态。借助变分方法和 群指标理论考察了具有混合位势的二阶脉冲哈密顿系统和二阶中立型泛函微分系统周期解的存在性；通过运用Mawhin 和Gaines的延拓定理得到了一阶奇异脉冲微分方程和奇异高阶时滞微分方程周期正解存在的新结果。采用极限思想、Lyusternik-Schnirelmann 范畴论和变分法给出了具时变时滞非线性扰动非牛顿流体方程、奇异非牛顿流体方程和时滞中立型神经网络的行波解和周期波解存在的若干充分条件。利用环绕定理和最小作用原理研究了二阶脉冲哈密顿系统和非常值脉冲强迫摆方程的极小周期问题。通过山路引理和逼近技巧得到了带p-Laplace算子的脉冲哈密顿系统同宿轨存在的条件；借助比较原理和正极限集的不变性，讨论了具临界情形的中立型泛函微分方程系统的渐近性。利用压缩映像和广义Gronwall-Bellman不等式，我们得到了一类广义脉冲时滞中立型神经网络分段伪概周期解的存在性、唯一性和指数稳定性的新结果；我们也讨论了带反馈控制的脉冲Lotka-Volterra模型和造血细胞模型的概周期解的存在性和稳定性。通过使用度理论、上下解方法和单调迭代技巧，我们研究了带p(r)-Laplacian算子的微分方程多点边值问题、非线性分数阶微分方程三点边值问题和无穷区间上分数阶微分方程边值问题解的存在性。我们还利用Dhage多值映射原理和非光滑临界点定理得得到了二阶脉冲中立型泛函微分包含和二阶脉冲半线性微分包含解的存在性结果。