Stability and bifurcations analysis in delay differential equations with applications

时滞动力系统是一类具有重要理论意义和应用价值的系统。在数学上，这些系统由延迟微分方程（DDE）表示。动态微分方程被广泛应用于生物科学、物理学、工程学、经济学等领域的众多现象的建模。稳定性和分岔是动力系统中最重要的性质，因为单个可预测过程只有在相应的自然意义下是稳定或准稳定的才能在物理上实现。中心流形理论和规范形方法在非线性动力系统的研究中具有重要意义，并在有限维常微分方程中得到了广泛的应用。相对于常微分方程稳定性、分支和NF计算的大量研究成果，微分方程的研究成果较少。为了预测和控制真实的世界模型的长期行为，时滞和系统参数如何影响系统的稳定性，以及当系统的稳定性随着时滞和参数的变化而被破坏时，会发生什么样的分岔，这些问题对于从事动力系统研究的数学家来说是一个具有挑战性的问题。因此，有必要扩展理论和方法来深入和广泛地研究动态微分方程，同时需要考虑计算方法的效率。这个建议是关于发展有效的计算方法，减少中心流形和计算的NF的DDE，然后分析的稳定性和分叉的参数和延迟的变化，虽然其他方面也将涉及。预计这项研究将增加我们对系统与时滞合作时动力学行为的理解。本研究所提出的方法将对非线性动力学领域产生很大的潜在影响，不仅为大类动态微分方程的理论发展奠定了基础，而且为解决真实的复杂动力系统提供了实用的工具。

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.