Applications Of Bifurcation Theory In Delay Systems

This paper considered the applications of the bifurcation theories in delayed dynamical systems. There are a number of retarded differential equations in physics, ecology, epidemiology, social-economy, neural network and many other fields. So that it is important to realize the dynamical behaviors of these model.First, Hale's theory of dividing functional space were discussed and used to Hassard's calculating method of the stability and the bifurcation direction of Hopf bifurcation periodic solution. Since Hale's theory is some difficulty to understand, the work is meaningful to simplify the theory and to use it to Hassard's calculating method.Next, an important delayed neural network introduced by Hopfield was discussed. The conditions of Hopf bifurcation of n neuron were given. For three-unit network system, the general formulas for the stability and the bifurcation direction of periodic solution were calculated and the estimating formula of period for bifurcation periodic solution was given. It is one of the earliest application of Hassard method to three-order delayed differential equation.Then a four-order of delayed immunity model was considered. The conditions of the existence of equilibria and Hopf bifurcation were given, and the bifurcation direction and the stability of the periodic solution were determined. The phenomenon verified on the theory that the virus can persist at a very low level in the body in the face of the host immunity. To our knowledge, up to now, there is no report of using Hassard method to four-order delayed differential equation.At last, a tow-species competitive system with the stage structure and irregular fishery was studied. The life stages of the tow-species are immature and mature, only in the mature, the species have competition. The harvesting effort of the tow mature population is proportioned to the rate of net economic revenue. This is a six-order delayed differential equation. The non-negativity and boundedness of the solutions were studied, the parameter conditions for nine equilibria and their stabilities was studied. Under that...