The Research On Local Bifurcation Of Some Classes Of Discrete Systems

The concept of dynamical systems, which describes how one sys-tem with time evolution and change of a kind of rule, originates from the qualitative theory of ordinary differential equations. The theory mainly studies dynamical behaviors of the determined time-evolution systems. Nonlinear dynamics has been widely applied to many fields such as physics, chemistry, biomathematics, medicine and economy since the 1970s. In dynamics, there are two kinds of mathematical models:one is the continue-time models described by differential equations or dynamical systems, and the other is the discrete-time models described by difference equations, discrete dynamical sys-tems or iterative maps. With the development and application of computer, the research on discrete system attracts more and more people's" attention and plays a very important role in view of practical significance and application value. The main problems for discrete dynamical systems involves its stability, branch and chaos etc.In this paper, some classes of discrete dynamical systems are mainly considered and their dynamical behaviors, especially the problem of local bifurcation, are investigated and discussed. By using Matlab and Maple softwares, Numerical and symbolic calculations are carried out. The correctness of theoretical analysis results has been confirmed by numerical simulations.Chapter 1 introduces the background and phylogeny of the dis-crete dynamical systems, contains the basic concepts and some the-oretical knowledge.Chapter 2 considers one-dimensional discrete time-delay IWC system. The stability of the fixed points of the system is investigated by analyzing the distribution of the roots of characteristic equation according to the corresponding linear system. Branch phenomena are studied by applying the normal form theory and the center man-ifold theorem. Chapter 3 considers one-dimensional discrete time-delay Logis-tic system, which is obtained by Euler method. N-S bifurcation is investigated by using the delay as a bifurcation parameter.Chapter 4 considers two-dimensional discrete time-delay Holling-Tanner predator-prey system, which is obtained by Euler method. The stability and bifurcation phenomena of the discrete system are investigated.Chapter 5 considers the Hopf bifurcation problem of one-dimensional system with discrete time-delay.