Stability and bifurcations analysis in delay differential equations with applications

具有时滞的动力系统具有很大的理论意义，并且在其应用方面形成了一个重要的类别。在数学上，这些系统由延迟微分方程 (DDE) 表示。 DDE 广泛用于生物科学、物理学、工程、经济学等领域的多种现象的建模。稳定性和分岔是动力系统中最重要的属性，因为单个可预测过程只有在相应的自然意义上稳定或准稳定时才能在物理上实现。中心流形理论和范式（NF）方法在非线性动力系统的研究中具有重要意义，并已广泛应用于有限维常微分方程（ODE）。与大量关于 ODE 稳定性、分岔和 NF 计算研究的文献相比，DDE 方面的研究成果却很少。为了预测和控制现实世界模型的长期行为，对于研究动力系统的数学家来说，具有挑战性的问题包括时滞和系统参数如何影响系统的稳定性，以及当时滞和参数的变化破坏稳定性时会出现什么样的分岔。因此，有必要扩展理论和方法来深入、广泛地研究DDE，并且需要考虑计算方法的效率。该提案涉及开发有效的计算方法来减少中心流形并计算 DDE 的 NF，然后分析参数和延迟变化时的稳定性和分岔，尽管也涉及其他方面。预计这项研究将增加我们对系统与时间延迟合作的动态行为的理解。本研究中开发的方法将对非线性动力学界产生非常高的潜在影响，这不仅会加强大类 DDE 的理论发展基础，而且还会为求解实际复杂动力系统提供实用工具。

The dynamical systems with time delay are of great theoretical interest and form an important class with regards to their applications. Mathematically, these systems are represented by delay differential equations (DDEs). DDEs are used extensively in the modeling of a multitude phenomena in biological sciences, physics, engineering, economics, etc. Stability and bifurcation are the most significant properties in dynamical system since an individual predictable process can be physically realized only if it is stable or quasistable in the corresponding natural sense. Center manifold theory and normal form (NF) method are of fundamental importance in the study of nonlinear dynamical systems, and have been applied in finite-dimensional ordinary differential equations (ODEs) broadly. Compared with the great amount of the publications in the study of stability, bifurcation and NF computation for ODEs, there are only few results in that of DDEs. In order to predict and control the long term behavior of real world models, to the mathematicians working in dynamical system, the challenging problems include how time delay and the system parameters affect the stability of the system, and what kind of bifurcations will occur when the stability is destroyed with the variation of delay and parameters. Therefore, it is necessary to extend the theory and methodology to study DDEs deeply and widely, as well as a need to consider the efficiency of the computational methods. This proposal is concerned with developing efficient computational methods for reducing the center manifold and computing the NF of DDEs, then analyzing the stability and bifurcations as the parameters and delays vary, although other aspects will also be involved. It is anticipated that this proposed research will increase our understanding of the dynamical behavior if a system cooperates with time delays. The methodologies developed in this proposed research will have very high potential impact on the nonlinear dynamics community, which will not only strengthen the foundation for theoretical development in a large class of DDEs, but also provide a practical tool for solving real complex dynamical systems.