不连续跳跃和时滞现象广泛存在于许多动态系统中，脉冲泛函微分方程和更一般的测度泛函微分方程是描述此类现象的数学模型，它们属于无穷维不连续混合动力系统，其动力学研究比一般泛函微分方程更加困难，特别在揭示不连续跳跃和时滞作用于系统动态行为规律方面，仍需要一些新的理论和方法。.本项目以脉冲泛函微分方程和测度泛函微分方程为研究对象。首先，研究两类方程的适定性和渐近性，包括解的存在唯一性、延拓性和连续依赖性，以及稳定性、吸引集(盆)、不变集和吸引子等性质；同时，建立两类方程二分性相关理论，给出指数二分性与三分性、非一致二分性等概念及等价特征，进而分析非线性方程的稳定性、周期性和拟周期性、不变流形等动力行为；最后，应用所获结果解决一些不连续时滞生物系统的动力学分析及其控制问题。项目研究有助于理解不连续跳跃和时滞对系统行为的影响，为不连续动力系统研究提供一些新的理论和方法。

Impulsive functional differential equations (FIDEs) and measure functional differential equations (FMDEs) are two classes of mathematical models to describe some common phenomena such as discontinuous jumps and time delays in many dynamical systems. The study of these differential equations is more complex and difficult than one of general functional differential equations since they belong to infinite dimension discontinuous dynamical systems. It need some new theories and methods to show how jumps and delays affect dynamical behaviors for solutions of the systems. .This project concerns with impulsive functional differential equations and measure functional differential equations. We first investigate the well-posedness and asymptotic behaviors involving the existence and uniqueness, prolongation, continuous dependence, stability, attracting sets, attracting basin, invariant sets and attractors of the FIDEs and FMDEs. Then, we shall establish the related theory of dichotomies of the FIDEs and FMDEs. The concept on exponential dichotomies, exponential trichotomies and nonuniform exponential dichotomy is given, and some equivalent characters of the dichotomies are provided. On this basis, we will analyze some dynamical behaviors such as the stability, periodicity and almost periodicity,invariant manifolds. Lastly, the obtained results will be applied to the analysis of discontinuous delayed biomathematical models and its control. The proposal work should be helpful to understand effects of jumps and delays to some dynamical properties of hybrid systems. This will develop some new theories and methods in the field of discontinuous dynamical systems.