Rank - Chaotic Attractor And Bogdanov - Takens Branch Of Several Delay Differential Equations

Retard differential equations refer the differential equations with time delays, which can be used to describe the evolution system dependent on both the present state and the past state. By fully considering the influence of past history, delay differ-ential equations are applied to many fields, such as physics, mechanics, control theory, biology, medicine, and economics.As made a break through in the research of structurally stable systems, the re-search in structurally unstable systems has attracted more and more attention. Chaos is another important topic in the study of bifurcation theory in delayed differential e-quations. Recently, some researchers find that bifurcation periodic solution of ordinary differential equations adding an external periodic force as an input can result in rank one chaos. Particularly, to study the change of the number of biological populations by observing rank one chaotic attractor in biological systems, is important significance in the actual production, and become a new direction of chaotic system research. The Bogdanov-Takens theory of delayed differential equations is another important top-ic. Investigating Bogdanov-akens bifurcations in delayed systems helps revealing the complicated dynamical behavior, including stable limit cycle、heteroclinic orbits connecting two hyperbolic saddles、double homoclinic loop、subcritical Hopf bifur-cating periodic orbits and coexistence of two limit cycles.Based on the instruction of Hopf bifurcation theory for functional differential e-quations and rank one chaos theory for ordinary differential equations this paper deals with the issue of the rank one chaotic attractors of delayed differential equations. In addition, by using center manifold reduction and normal form theory, dynamical clas-sification near B-T point can be completely figured out in terms of the second and third derivatives of delay term evaluated at the zero equilibrium. The main work is described as follows:1. The research background, research development, main methods and achieve-ments of delay differential equations with Hopf bifurcation are summarized. The discovery, research approaches and recent advance of rank one chaotic attractors are introduced. Using center manifold theory and normal form theory to study Bogdanov-Takens bifurcation are described.2. Based on rank one theory for ordinary differential equations, we try to develop rank one theory for delayed differential equations, rank one chaos existence theorem for delayed system is given in the chapter. The Bogdanov-Takens bifurcation theory of delay differential equations is introduced.3. We use the rank one theory for delayed differential equations to classics eco-logical system, we consider the Lotka-Volterra 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 periodically kick as an input and observe rank one strange attractors. Finally, we get numerical simulation, which is consistent with the theoretical results.4. We applied the rank one theory to predator-prey system with discrete and distributed 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 periodically kick as an input and observe rank one strange attractors. Finally, we get numerical simulation, which is consistent with the theoretical results.5. We study Bogdanov-Takens bifurcation in a harmonic oscillator with negative damping and delayed feedback. Based on center manifold, we obtain the condition and give the proof with appearance of Bogdanov-Takens bifurcation. The parameter-s condition of the corresponding saddle-node bifurcation、a Hopf bifurcation and a homoclinic bifurcation are obtained.