Stability and Bifurcation Analysis on Delay Differential Equations

Abstract

Most recent studies on delay differential equations are mainly focused on local stability analysis, stability switches, and local existence of Hopf bifurcations with only one delay included, while the global existence of Hopf bifurcation and stability analysis with two delays are hardly discussed. In this thesis, we numerically explore global behaviors of Hopf branches arising from where the characteristic roots crossing the imaginary axis, and we reveal that there seems to be a strong and simple underlying rule, which is been partly studied by Li and Shu in their recent paper [Li & Shu 2010a]. In addition, stability analysis on delay differential equations with two discrete delays will also be studied. We extend the work by [Guet al.2005], and establish a similar theory on a more general type of models. Finally, we provide some preliminary results for the two-delay models with parameters depending on one delay.