Stability And Bifurcation Analysis On Some Kinds Of Delayed Continuous Dynamic Systems

In the study of delayed continuous dynamic systems, stability, the existence of peri-odic solutions and bifurcations are very significant questions. Among them, the stabilityindicates the balance of structure; periodic solutions re?ects periodic motion law of na-ture; and a bifurcation is a change of the topological type of the systems as its parameterspass through a critical value. The investigation of the above needs to use the theory of dy-namical systems, functional, algebra, topology and graph theory and other related knowl-edge. Therefore, the study has a strong practical background and it is of great theoreticalsignificance.In this thesis, we investigate some kind of retarded and neutral differential equations.Using the theory and methods including LaSalle invariance principle, topological degreetheory, the center manifold theorem, norm form method and global bifurcation theorem,we study the local and global stability, the existence of periodic solutions, fixed pointbifurcation and global Hopf bifurcation for these systems. Details are as follows:When studying global stability and the existence of periodic solutions, the mainmethods adopted in this paper are: combining Lyapunov method with the results in graphtheory, we prove the global stability of the positive equilibrium for a system of popu-lation with stage-structure and multiple group; constructing a Lyapunov functional andthen using LaSalle invariance principle as well as the embedding idea of asymptoticalautonomous semi?ow, we give the sufficient conditions which ensure the zero solution ofthe hematopoietic stem cell dynamics with multiple delays is globally asymptotically sta-ble; constructing a Lyapunov functional and making use of Barbala¨t lemma, we presentsufficient conditions ensuring the global asymptotic stability of zero solution for a kind ofscalar neutral differential equation. On the existence of periodic solution, we mainly usea combination of coincidence degree theory with Hopf bifurcation analysis.In the process of analyzing the bifurcation, the first step is to study the characteristicequation, which is often a transcendental equation for retarded differential equation, of thelinearization of original system at the steady state. Under different types of systems dis-cussed, we employ the Routh-Hurwitz criterion as well as the results provided by Hayes,Ruan and Wei, Beretta and Kuang to analyze the distribution of the roots for the corre- sponding characteristic equations. Then, the stability of the equilibria and the occurrenceof Hopf bifurcation and pitchfork bifurcation are obtained. Secondly, the properties of thedifferent bifurcation, involving the direction of Hopf bifurcation, the stability of bifurcat-ing periodic solution and the stability of the equilibria, are computed using the methodfor normal form calculation given by Hassard et al, Faria and Magilhaes, Wang and Wei.In particular, using the global Hopf bifurcation theorem due to Wu and the Bendixson cri-terion on higher dimensional ordinary differential equations given by Li and Muldowney,we prove the global existence of bifurcating periodic solutions for a delayed model ofasset prices and the hematopoietic stem cell dynamics with multiple delays. The resultsreveal that the connected component of the Hopf branch through the isolated center is un-bounded in the former and is bounded which connects two different centers in the latter.Also obtained is the global continuation of local Hopf branch for a kind of scalar neutraldifferential equation by the global Hopf bifurcation theorem established by Krawcewiczet al, showing that the connected components through the isolated centers are unbounded.