Hopf Bifurcation Of Delayed Predation Systems And Synchronization Of A Chaotic System

In recent years, population models have received widespread attention. In these models,predator-prey model is the basis, which has been studied by mathematicians and ecologists.Then, introducing the time delays into predator-prey model becomes a new research topic,which accords better with the real life. On the other hand, The application of chaoticsynchronization in secure communications makes it a great prospect. In this paper, wemainly talk about the Hopf bifurcation of predator-prey systems with delay, and chaossynchronization of four-dimensional chaotic system. In chapter 1, we give the significance ofinvestigating this thesis, development of this filed and the main work in this thesis.In chapter 2, we investigate the Hopf bifurcation of a delayed predator-prey system withthe mutual slaughter term. Basing on the Lotka-Volterra predator-prey system, we mainlytalks about the mutual slaughter of predator, and the stability of positive equilibrium andthe existence of Hopf bifurcation. Besides, by using the normal form theory and centermanifold argument, we have derived the explicit formulae of the direction of bifurcation.In chapter 3, the Hopf bifurcation of a predator-prey system with time delays and stagestructure for prey is investigated. We investigate the stability of positive equilibrium andthe existence of Hopf bifurcation, and obtain the conditions for Hopf bifurcation.In chapter 4, using the method of active control and adaptive control, we study the chaossynchronization problems of a four-dimensional chaotic system. Based on the Lyapunov sta-bility theory, we get the su?cient condition, under which the two identical systems can reachsynchronization.