Hopf Bifurcation And Rank - Chaotic Attractor For Delay Differential Equations

Retard differential equations refers to the differential equations with time de-lays, which can be used to describe the evolution systems dependent on both the present state and the past state. By considering the influence of the past history, delay differential equations are applied to many fields such as physics, mechanic-s, control theory, biology, medicine and economics. Hopf Bifurcation correspond to the self-excited vibration in dynamical systems. This kind of vibration phe-nomenon can sometimes cause unnecessary loss in our life and productive activity. Hence,the systematic and deep investigation of the bifurcation theory is of great significance in both intense practical and theoretical background. Chaos theory is another important topic in the study of delay differential equations. Recently, some researches find that bifurcating periodic solution of ordinary differential equation adding an external periodic force as an input can result in rank one attractors. Multiscroll chaotic attractor generating theory research and chaotic circuit imple-mentation have become a new direction of chaotic systems research. Due to the bifurcating periodic solutions in some time-delayed systems, the existence of rank one chaotic attractor of delay differential equations has become a very attractive and challenging problem.Based on the instruction of Hopf bifurcation theory for functional differen-tial equations and rank one theory for ordinary differential equations, this paper deals with the issues of Hopf bifurcation and rank one chaotic attractors of delay differential equations, and the major works is stated as follows.In Chapter1, the research background, research developments, main methods and achievements of delay differential equations and Hopf bifurcation are summa-rized. The discovery, research approaches and recent advance of rank one chaotic attractors are introduced.In Chapter2, we briefly introduce local Hopf bifurcation theory and global Hopf bifurcation of delay differential equations, and rank one chaos theory for ordinary differential equations. In Chapter3, we study a model of a hybrid ratio-dependent three species food chain with time delayed intervention in middle-predator. By regarding the time delay as parameter, we obtain the conditions of the local stability of the positive equilibrium and the existence of Hopf bifurcation, and determine the direction of Hopf bifurcation and the stability of periodic solutions bifurcating from positive equilibrium at critical point. Our research shows that the stability of system depends on the time delay. When the time delay passes through some critical values, the positive equilibrium loses (or regains) its stability and Hopf bifurcations occur. In numerical simulation, stable, periodic or chaotic solution are obtained in abundance for the ranges of parameters showing the rich dynamics of system. From our research, we see also that the time delayed intervention could affect the bifurcating and chaotic motions of the system. This suggests that the time delayed intervention can be used to control chaos, or to anticontrol (or generate) chaos.In Chapter4, we proposes and studies an SIRS computer virus propagation model with a saturation incidence rate and a time delay describing the latent and temporary immune period. The coefficients of this model dependent on time delay. We derived an explicit formula for the reproduction number Ro. By linearizing the system at the virus equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the virus equilibrium is investigated and Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. The global stability of the virus-free equilibrium and the virus equilibrium of system (1.3) was studied by comparison arguments and the iteration scheme, respectively. These results may help understand the laws governing the spread of computer virus over a computer network. Finally, numerical simulations are carried out to illustrate the main theoretical results.In Chapter5, we consider a predator-prey system with Michaelis-Menten type functional response and two delays. we focus on the case with two unequal and non-zero delays present in the model, and study the local stability of the equilibria and the existence of Hopf bifurcation. when the delays τ1≠τ2,we obtain explic-it formulas to determine the properties of Hopf bifurcation by using the normal form method and center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcation.In Chapter6, a partial dependent prey-predator model with discrete and distributed delays is studied by using the theory of functional differential equation and Hassard’s method, the conditions on which positive equilibrium exists and Hopf bifurcation occurs are given, finally, numerical simulations are also included.In Chapter7, based on rank one theory for ordinary differential equations and theory of functional differential equations, we firstly try to develop rank one theory for delay differential equations. As we know, no prior work exists on rank one theory for delay differential equations. Secondly, we study the existence of rank one chaotic attractors in time-delayed system. We consider Chen system with time-delay, the conditions under which a supercritical Hopf bifurcation occurs are given by using the normal form method and center manifold theorem. Then we add an external periodic force as an input and observe rank one chaotic attractors. Finally, several numerical simulations supporting the theoretical analysis are also given. Up to now, it is the first time to find rank one chaotic attractors in delay differential equations. Our investigation should be useful to develop rank one theory for delay differential equations.