Rank One Strange Attractors In Some Delayed Systems

Delayed differential equations as a kind of special differential equations, the derivative of the unknown function in determining time is determined by the func-tion on previous moments. By considering the influences of the past history, delayed differential equations are applied to many fields such as mechanics, physics, biology, control theory, medicine and economics.As made a break-through in the research of structurally stable systems, the re-search in structurally unstable systems (that is, the bifurcate theory) have attracted more and more attention. Chaos is another important topic in the study of bifurca-tion theory in delayed differential equations. Recently, the researchers discovered that adding an external periodic force in the bifurcating periodic solution of ordinary dif-ferential equations as an input can create rank one strange attractors. Because of the bifurcating periodic solutions in a class of delayed systems, the existence of a strange attractor with SRB measure in delay systems has become a very attractive and chal-lenging topic.Based on Hopf bifurcation theory for delayed differential equations, this paper investigated the existence and decision problem of rank one chaotic theory in delayed differential equations, and applied the rank one chaotic theory to delayed differential equations. The main work is described as follows:In Chapter 1, the research background, research developments, main methods and achievements of delayed differential equations and Hopf bifurcation theory are summarized. The discovery, research approaches and recent advances of rank one chaotic attractors are introduced.In Chapter 2, we briefly introduce SRB measure theory, and the theory of rank one strange attractors in ordinary differential equations with Hopf bifurcations limit cycles under periodical kick is given.In Chapter 3, based on rank one theory for ordinary differential equations and Hopf bifurcation theory for functional differential equations, we try to develop rank one theory for delayed differential equations, rank one chaos existence theorem for delayed systems is given in the chapter.In Chapter 4, using the rank one chaotic theory for delayed differential equations which has been developed, we consider Chua system with time-delay, the conditions under which a supercritical Hopf bifurcation occurs are given. Then we add an peri- odic kick as an input and observe rank one chaotic attractors. Finally, we use Matlab software to programming and get numerical simulation, the numerical simulation is consistent with the theoretical results, the programming we using is a common single footwork (Runge-Kutta method). This will provide a instance for rank one chaotic theory from ordinary differential equations to delayed differential equations.In Chapter 5, we applied the rank one chaotic theory to Lorenz system with time-delay. The Lorenz system is more general since the equilibrium is not at the origin. Then we add an periodic kick as an input and observe rank one chaotic attractors. Fi-nally, we use Matlab software to program and get numerical simulation, the numerical simulation is consistent with the theoretical resultsIn Chapter 6, we used the rank one chaotic theory of delayed differential equa-tions to ecological system, and consider the predator-prey 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 by adding an periodic kick as an input we observe rank one chaotic attractors. Finally, we use Matlab software to program and get numerical simulation, the numerical simulation is consistent with the theoretical results.