Global Hopf Bifurcation Analysis Of A Class Of Biochemical Reaction Model With Enzymatic Catalysis

Nonlinear dynamical system for researchers in the field, is one of the most important issues. A lot of dynamical system will involve complex bifurcationing problem. In recent years, due to the continuous development of science and technology, there are many practical problems are emerging. Bifurcationing problem often appear in nonlinear differential systems which with a parameter,so it is very important to study of it, especially for the existence of periodic solution and bifurcation stability. With the development of the power system, many natural basic disciplines such as physics, ecology, social economics, biology and many other disciplines, most of the mathematical model they created, in the course of the study of the model will encounter bifurcation problems, and the Hopf bifurcation is the most common. This paper studies two types of Hopf bifurcation problem.One of them is the existence and the bifurcation direction of the Hopf bifurcation of nonlinear dynamical system of the enzymatic reaction, and discussed the existence of the global Hopf.The other one is a local Hopf bifurcation of continuous innovation diffusion model for delay differential systems.The knowledge used in this article are Hopf bifurcation theory、Hurwitz criterion and the center manifold theory.the text is divided into five parts.The first part is an introduction,describes the significance of biochemistry research background,delay dynamic dystem research,the Bass model research status and problems studied in this paper.The second part is mainly used in this prior knowledge.The third part study of the existence of the Hopf bifurcation of nonlinear dynamical system of the enzymatic reaction, and further study of the direction of Hopf bifurcation and the stability of this system,solve the cycle oscillations relationship in enzymatic reaction which everyone is attention.Studies have shown that when the concentration of enzyme and reactants satisfies a certain relation, system appear periodic oscillation.The fourth part is established model of innovation diffusion with continuous delay in the evaluation stage, study the Hopf bifurcation whose parameter is the actual contact rate the potential adopters who adopt the products,the calculated threshold and bifurcation direction and the stability of the limit cycle. The fifth part is a summary of this article, and pointed out the question which need further research.