Dynamic Properties Of Coupled Time-delay Differential System

The research focus on discuss two kinds of coupled differential systems with delay, based on the theories of delay differential equation and the theories of Hopf bifurcation. Through discussing the distribution of characteristic equations roots, the stability of the delay differential equations (DDE) and the condition of Hopf bifurcation are obtained, and then analyze the dynamic properties of the whole system. This paper has three chapters. In the first chapter, we briefly introduce the research significance, research achievements, research method and the article structure. In the second chapter, the stability problem of mutually delay-coupled semiconductor lasers system is investigated. By analyzing the associated characteristic equation, linear stability is investigated and the conditions of Hopf bifurcation are demonstrated. The phenomenon such as stability switch is found. The existence of multiple periodic solutions and bifurcation form is also discussed by using the Z2-equivariant bifurcation theory. Numerical simulations are presented to illustrate the results in this part. In the third chapter, a global coupled phase oscillator system with delay is considered. The existence of periodic solutions has been discussed.