Stability and bifurcations analysis in delay differential equations with applications

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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.