The Research On The Bifurcation Of A Four-dime Neural Model With Delays

This paper investigate mainly bifurcation problem in a four-dimensional neural network model with delays, they can be divided into two main parts: One is the model no self-connections: Another is the model with self-connections.This paper consists of six chapters.Chapter 1 is introduction, covers the history of the construction of neural network model and some scholars' research.In Chapter 2. there are preliminary knowledge and prepared works. We also give 3 main theorems, which are used in this paper: the distribution theorem of the zeros of the exponential polynomial, global Hopf bifurcation theorem for FDE and Bendixson's criterion for high-dimensional ODE.In Chapter 3. we study the distribution of the zeros of a fourth degree exponential polynomial by the distribution theorem of the zeros of the exponential polynomial.In Chapter 4, we introduce the Hopf bifurcation of the four-dimensional neural network model with delays and no self-connections. We obtain a group of conditions that guarantee the model have the local Hopf bifurcation, and the bifurcation set are drawn in the parameter space. Meanwhile, basing on the normal form theory and the center manifold theorem, we derive the formula for determining the properties of Hopf bifurcation, such as the direction of Hopf bifurcation stability of the bifurcating periodic solutions and so on. Using the global Hopf bifurcation theorem for FDE and Bendixson's criterion for high-dimensional ODE, we obtain the global existence of periodic solutions. Numerical simulations are presented to illustrate the some results.In Chapter 5, we discuss the Pitchfork bifurcation of the four-dimensional neural network model with delays and no self-connections. It is found that thePitchfork bifurcation curve and the stability of the equilibrium.Chapter G research the bifurcation of the four-dimensional neural network model with delays and self-connections. The characteristic equation of the linearized system at the zero solution is investigated by applying the distribution theorem of the zeros of the exponential polynomial, and then get, the stable region of the zero solution and the conditions on the bifurcation. The bifurcation set are provided in the appropriate parameter plane.