由于四元数值泛函微分方程在物理、通信、计算机图形学、神经网络等领域均有重要应用。而目前关于四元数值微分方程的定性研究结果还很少。同时，概周期现象比周期现象更普遍，概周期性是微分方程定性理论研究中的重要对象，几乎自守性是概周期性的推广又对理解概周期性起十分重要的作用；分数阶微分方程非常适于刻画具有记忆和遗传性质的材料和过程，已成为复杂力学与物理过程数学建模的重要工具之一。因此，研究四元数值泛函微分方程、分数阶四元数值微分方程的理论具有重要的理论和应用价值。本项目旨在研究四元数值泛函微分方程的若干定性理论，并主要研究：1、初值问题解的存在性、唯一性及解的稳定性等。2、四元数值泛函微分（差分）方程的周期解、反周期解、概周期、几乎自守解的存在性等。3、 分数阶四元数值微分方程的概周期解、几乎自守解的存在性及稳定性。4、应用本项目新发展起来的理论研究四元数值神经网络模型的各种定性性态。

Quaternion-valued functional differential equations have important applications in the fields of physics, communication, computer graphics, and neural networks and so on. However, there are few qualitative results about quaternion-valued functional differential equations at present. As everyone knows, almost periodic phenomenon is more widespread than the periodic phenomenon, almost periodicity is an important object in the study of the qualitative theory of differential equations, the almost automorphy is a generalization of the almost periodicity and plays a very important role in understanding the almost periodicity; fractional calculus is a generalization of integer order calculus, fractional order differential equations are very suitable for characterization of materials and processes with the properties of memory and hereditary, which have been become one of the important tools for mathematically modelling complex mechanical and physical process. Therefore, it is of great theoretical and applied value to study the theory of quaternion-valued functional differential equations and fractional quaternion-valued differential equations. The purpose of this project is to study some qualitative theories of quaternion-valued functional differential equations, and mainly research: .1. The existence and uniqueness of solutions of initial value problems and the stability of solutions. 2. The existence of quaternion-valued differential (difference) equations of periodic solutions, anti-periodic solutions, almost periodic solutions and almost automorphic solutions etc. 3. The existence and stability of almost periodic solutions and almost automorphic solutions of fractional quaternion-valued differential equations. 4. Applying the new theory of this project to study the qualitative characteristics of quaternion-valued neural network models.