时滞系统普遍存在于各工程领域，具有复杂的动力学行为，其建模分析存在很大的困难。Birkhoff系统是更一般的动力学系统，在复杂系统研究中具有明显的优势。本项目拟研究时滞Birkhoff系统动力学建模及其积分方法，研究将时滞非保守系统和时滞非完整系统表示为时滞Birkhoff系统的方法，从而可运用时滞Birkhoff系统的理论解决一般时滞系统的问题；研究时滞Birkhoff系统的积分方法，包括变换理论，Hamilton-Jacobi方法，Poisson方法和对称性方法，得到方程的解析解或守恒量，使其在应用中提高计算效率，深入了解动力学特性；由于时滞系统的记忆特性和复杂性，基于分数阶模型和时间尺度模型，拓展研究时滞Birkhoff系统动力学建模与积分方法。本项目研究对于深化和完善经典Birkhoff系统动力学理论有重要意义，对科学和工程更具有重要的应用价值，促进分析力学和新兴交叉学科的发展。

Time-dealy system generally exists in engineering. It has complex dynamic behaviors, which is very difficult in modeling and analysis. Birkhoffian system is a more general dynamic system, which has obvious advantages in the study of complex systems. In this project, we intend to study the dynamic modeling of time-delay Birkhoffian system and its integral methods. The representation of time-delay nonconservative system and time-delay nonholonomic system as time-delay Birkhoffian system will be studied, so that the problems of general time-delay systems can be solved by using the theory of time-delay Birkhoffian system. The integral methods of time-delay Birkhoffian system, including transformation theory, Hamilton-Jacobi method, Poisson method and symmetry method, will be studied to obtain the analytical solutions or the conserved quantities of the equation, so as to improve the calculation efficiency in applications and understand the dynamic characteristics of the system deeply. Because of the memory characteristics and complexity of time-delay system, the dynamic modeling of time-delay Birkhoffian system and its integral methods will be further developed based on the fractional order model and the time scale model. The research of this project is of great significance to deepen and perfect the classical theory of dynamics of Birkhoffian system, and has more important application value to science and engineering, as well as promotes the development of analytical mechanics and new interdisciplinary subjects.