The Stability And Hopf Bifurcation Analysis For Two Classes Of Delay Differential Equations

As delay differential equations can explain some problems arising from the ecol-ogy, economics and social science very well, many investigator have been paid their attention to the nonlinear models described by delay differential equations in recent years. In this thesis, we propose two nonlinear models with delays including a delayed economic model and a delayed single species model on the basis of the existed models. By employing stability theory for functional differential equations, bifurcation theory, center manifold and normal form theory, some important dynamics of the two models such as stability of the equilibrium, Hopf bifurcations are detailed studied. This thesis is divided into four parts.In chapter one, we briefly introduce nonlinear time-delay system at first, then we give an outline of the background of the two models and the main research contents.In chapter two, some notations, definitions and lemmas are listed.In chapter three, we investigate a single species model with delay in a polluted environment. We first discuss the positively invariant solution and the existence of Hopf bifurcation. By the analysis the distribution of roots of the characteristic equation to the linearized system, some sufficient conditions to ensure that the asymptotically stability and the existence of Hopf bifurcation near the equilibrium are derived. We also obtain the explicit formulae for determining the stability and the direction of the Hopf bifurcation period solutions by using the normal form theory and center manifold theory. Finally, numerical simulations show the correctness of our conclusion.In chapter four, we generalize a delayed economic model. By taken the delays as parameters,we investigate the stability of equilibrium. The existence of Hopf bifur-cation is also discussed. We provide the explicit formulae for determining the stability and the direction of the Hopf bifurcation period solutions. Numerical simulations are consistent with theoretical analysis.