The Two Classes Of Nonlinear Discrete Dynamical Systems, Bifurcation Analysis

In this thesis, we investigate the stability and bifurcation of a discrete delay Mackey-Glass system, and a class of two dimensional discrete dynamical systems. It consists of three chapters.In Chapter 1, the development of dynamical system are given.In Chapter 2, a discrete delay Mackey-Glass system obtained by Trapezoidal method is investigated. Firstly, the stability of the model is studied. It is found that there exist Neimark-Sacker bifurcations when the delay passes a sequence of critical values. Then the explicit algorithm for determining the direction of the Neimark-Sacker bifurcation and the stability of the closed invariant curve bifurcating from the positive equilibrium are derived by using the center manifold theorem and the normal form method. Finally, computer simulations are performed to illustrate the analytical results found.In Chapter 3, the local stability of positive equilibrium of a class of two dimensional discrete dynamical systems is considered. The direction and stability of Neimark-Sacker bifurcation of the system is also discussed of by using the norm form theory. Finally, computer simulations are provided to illustrate to the analytical results founded.