Stability And Bifurcation Analysis On Several Classes Of Nonlinear Time-delay Systems

With the development of science and technology, a large number of nonlinear models described by delay differential equations spring up in the engineering, ecology, economy and social science. In this dissertation, by improving the existed models, we propose several nonlinear models with delays, which are three classes of ring neural networks with different coupling forms and one predator-prey system with stage structure. Some important dynamics such as stability of the equilibria, Hopf bifurcations are studied by employing stability theory for functional differential equations, bifurcation theory, center manifold reduction, normal form approach and a comparison theorem in differential equations. This thesis consists of six parts.In Chapter 1, we give an outline of nonlinear science and nonlinear time-delay systems, introduce the background of the concerned models, and present the main research contents.In Chapter 2, we list the basic definitions and important lemmas in this dissertation.In Chapter 3, we discuss a four-neuron ring with two loops. By analyzing the distribu-tion of the roots of the associated characteristic equation, we study the linear stability of the equilibrium and prove the existence of Hopf bifurcation under some conditions. By using the normal form method and the center manifold reduction, we work out an algorithm for determining the property of the bifurcated periodic solutions. Our results complement some earlier ones on the four-neuron ring. Furthermore, the method of verifying the transversality condition can be applied to general characteristic equation with single exponential term.In Chapter 4, the symmetry coupling of two classes of ring neural networks is con-cerned. For the first model, we factorize the associated characteristic equation by using the theory of cyclic matrix. The sufficient conditions of the asymptotical stability of the equi-libria are deduced. We not only present a detailed discussion about the Spatio-temporal Patterns of the bifurcated periodic solutions, but also obtain the criterion about the direction and stability of the bifurcated periodic solutions basing on the normal form approach, which is different from that in Chapter 3. An example illustrates that the coupling manner of this model may give rise to many complex dynamics. For the second model, we emphatically analyze the effect of module coupling on dynamics of the model. We establish the existence of multiple periodic solutions by employing the theory of equivariant Hopf bifurcation. Nu-merical simulation results are given to support the theoretical predictions.In Chapter 5, we propose a predator-prey system with Lotka-Volterra type and stage structure for prey. By using the method of characteristic root analysis, a comparison theo- rem in differential equations and an iteration technique, we discuss the global asymptotical stability of the equilibria when the parameters of the models are in different value ranges. In addition, we analyze the effect of the delays on dynamics of the model and derive the sufficient conditions occurring Hopf bifurcation about the positive equilibrium. Numerical simulations are carried out to illustrate our theoretical results. The main results reveal the evolution of the predator and prey species, and also imply that the rate of transition from immature prey to mature prey is the key factor for the coexistence of the predator and prey species.Last part is the summary of our work and the prospect of our future research.