Stability And Bifurcation Problems On Several Types Of Delayed Dynamical Systems

The study of stability and bifurcation on delayed dynamical systems plays a veryimportant role in the developing of practical background fields. Concretely, the study ofstability on infinite dimensional system, especially the study of global stability is morecomprehensive and deep to show the dynamics of the system. The bifurcation problemis also an important concern, which mainly studies the essential changes of the topologyproperty caused by the changing of parameters at the critical point. To study the delayeddynamical systems with practical background, it needs the necessary tools besides lotsof classical dynamical theories, topology, algebra, nonlinear analysis, etc., also needs thedeep understanding of epidemiology, oncology, clinical medicine, biology and networks.In this thesis, using some theories and methods including Lyapunov stability theoremcombining LaSalle invariance principle, the center manifold theorem, norm form methodand global Hopf bifurcation theorem, we deeply investigate some problems on the localand global stability, local and global Hopf bifurcation of several types of delayed dynam-ical models with strongly practical background. The main work is summarized as thefollowing:(1). The dynamics of an oncolytic virotherapy model with lytic cycle is studied.The viral lytic cycle is considered, which is ignored by most mathematical models foroncolytic virotherapy. Under the diferent ranges of the burst size, using the method ofLyapunov functional combining LaSalle Invariance Principle, we investigate the globalstability of the corresponding equilibrium. We analyze the change on stability and theoccurring of Hopf bifurcation of the positive equilibrium. Moreover, an important clinicimplication is given according to the results.(2). The global dynamics of a cholera model with delay is considered. Under thediferent ranges of the threshold value R0, using the method of Lyapunov functional andapplying the LaSalle Invariance Principle, we establish the global stability for the twoequilibria of the system, the results respectively mean that the cholera dies out and theinfection persists. The maintaining of the global stability as the ODE system indicatesthat the delay according to the infection term does not lead to periodic oscillations, thatis, it is not the decisive factor in causing the seasonal oscillations of cholera. (3). The dynamics of a class of cyclic neural networks with negative feedback anddelay efect is studied. We give a dichotomy of the global dynamics by applying the workby Enciso. Global asymptotic stability of the unique equilibrium is derived by using thismethod and combining the results on the distribution of the roots. The direction of theHopf bifurcation and stability of the bifurcating periodic solutions at the first critical valueare obtained; in this way, we give an answer to the two conjectures proposed by Enciso.Finally, two examples and their numerical simulations support our results.(4). The dynamics of a viral infection model with cell mediated immunity and twointracellular delays is investigated. We prove the dynamics of the system is determined bytwo threshold values. Using the methods of Lyapunov functional and combining LaSalleInvariance Principle，we obtain the global stabilities of the two boundary equilibria. Inthe end, the sufcient conditions of the stability and Hopf bifurcation for the positiveequilibrium are investigated by using the geometric criterion of Beretta and Kuang.