几类Hamilton系统与耗散系统的脉冲分支及相关问题研究

Impulsive hamiltonian systems and dissipative systems are a class of common mathematical models, which described the development process of things with a discontinuity or instant mutation characteristics. It is widely used in various fields of science and technology, such as space technology, mechanics, control system, biological and medical fields. Impulse is a key factor to cause the dynamic properties changing for this kind of differential system. Recent research shows that, with the impulse, a hamiltonian system without periodic solutions and heteroclinic orbit, admits periodic solutions and heteroclinic orbit. To reveal the influence principle of the impulsive dynamic properties for the differential system is a key step for the impulse control. However, how to influence on the impulsive differential system's dynamic properties are still unclear. The project focuses on some impulsive hamiltonian systems and dissipative systems as the research object. We investigate the impulse bifurcation of these hamiltonian systems and dissipative systems by using the variational method, the new fixed point theorems and the discrete system theory, and explore the influence principle for impulse to dynamical properties of the periodic and homoclinic and heteroclinic orbit. This research would provide theoretical support for the design to the Chinese emergency orbit maneuver. With impulse bifurcation parameters, there are no international research report on the equilibrium point and the closed orbit bifurcations of impulsive hamiltonian system and dissipative system.

脉冲Hamilton系统与耗散系统是描述事物状态发展过程具有不连续或瞬间突变特性的常见数学模型，是广泛用于各科技领域中的重要模型，如航天技术、力学、控制系统、生物学以及医学等领域。脉冲是导致这类微分系统动力学性质发生变化的重要因素。研究显示，在脉冲作用下，没有周期解与异宿轨的Hamilton系统出现了周期解与异宿轨。揭示脉冲对微分系统动力学性质的影响规律，是实现脉冲控制的关键。然而至今对于脉冲如何影响微分系统的动力学性质仍不清楚。本项目以几类带脉冲作用的Hamilton系统和耗散系统为研究对象，依托变分法、新型不动点定理以及离散系统理论等工具，研究这些Hamilton系统及耗散系统的脉冲分支，探索脉冲对其周期解及同宿异宿轨等动力学性质的影响规律。本项目的研究可为我国设计空间快速机动轨道提供理论支撑。以脉冲为分支参数研究脉冲Hamilton系统与耗散系统的平衡点与闭轨分支，国内外尚未见报道。

脉冲Hamilton系统与耗散系统是描述事物状态发展过程具有不连续或瞬间突变特性的常见数学模型，是广泛用于各科技领域中的重要模型，如航天技术、力学、控制系统、生物学以及医学等领域。本项目研究了带脉冲作用的Hamilton系统和耗散系统为研究对象，依托变分法、矩阵分析、新型不动点定理以及离散系统理论等工具，研究这些Hamilton系统及耗散系统的脉冲分支，探索脉冲对其周期解及同宿异宿轨等动力学性质的影响规律。作为应用，研究了时滞自组织系统和带指令的自组织系统的集群模式和集群速度，获得了时滞与自组织系统的集群速度的复杂非线性关系，回答了马里兰大学S. Motsch和E. Tadmor 2011年提出的公开问题。