Stability And Bifurcation Of The Two Types Of Nonlinear Discrete Dynamical Systems Analysis

In this thesis, we investigate the stability and bifurcations of a discrete variable-territory predator-prey system, and a discrete predator-prey system of Holling-type. It consists of three chapters.In chapter 1, the development of dynamical system is firstly given. Then we present the basic theory of bifurcation including the easiest condition that bifurcation occurs, the norm and generic form of Flip bifurcation and Neimark-Sacker bifurcation, introduce the methods that can be used to study the bifurcations.In chapter 2,the dynamics of a discrete variable-territory predator-prey system is investigated in the first quadrant. It is shown that the system undergoes flip bifurcation and Neimark-Sacker bifurcation by using center manifold theorem and bifurcation theory. Numerical simulation are presented not only to illustrate our result with analysis, but also to exhibit the complex dynamical behaviors such as period-9, 17, 18, 35 orbits, cascade of period-doubling bifurcation in period-2, 4, 8, quasi-periodic orbits and the chaotic set according to the bifurcation phase, the phrase portraits and the maximum Lyapunov exponent phase of the system.In chapter 3, we discuss the Flip and Neimark-Sacker bifurcation of a discrete predator-prey system of Holling-type, and then study the parameters condition that the Flip and Neimark-Sacker bifurcation occurs. After that discuss the direction and the stability of period-2 orbit of flip bifurcation, the direction and the stability of invariant cycle of Neimark-Sacker bifurcation by using the norm form theory. Lastly we prove the accurate of the analysis according to numeric simulation.