Study On Branches Of Several Kinds Of Discrete Dynamical Systems

The bifurcation phenomenon is one of the main forms of complex dynamics in nonlinear dynamical systems, which can be probably found in continuous systems and discrete systems. The discrete systems play an important role in the study of bifurcation problem.This thesis is mainly concerned with the bifurcation problems in several classes of discrete systems, consisting of five chapters, and its main contents are as follows.Chapter 1 introduces the background, significance of the research and the major work.Chapter 2 recalls the basic concepts in discrete dynamical system, the local bifurcation theory, the center manifold theory and Lyapunov exponents.Chapter 3 studies a discrete predator-prey model with a non-monotonic functional response, originally presented in Hu, Teng and Zhang[43]. First, by citing several examples to illustrate the limitations and errors of the local stability of the fixed points E3 and E4 obtained in this article, we formulate an easily verified and complete discrimination criterion for the local stability of the two fixed points. Here, we present a very useful lemma, which is a corrected version of a known result, and a key tool in studying the local stability and bifurcation of a fixed point in a given system. We then study the stability and bifurcation for the fixed point E1 of this system, which has not been considered in any known literature. Unlike known results that present a large number of mathematical formulae that are not easily verified, we formulate easily verified sufficient conditions for flip bifurcation and fold bifurcation, which are explicitly expressed by the coefficients of the system. The center manifold theory and Project Method are the main tools in the analysis of bifurcations. The theoretical results obtained are further illustrated by numerical simulations.In chapter 4, a semi-discrete model is derived for a nonlinear simple population model, and its stability and bifurcation are investigated by invoking a key lemma we present. Our results display that a Neimark-Sacker bifurcation occurs in the positive fixed point of this system under certain parametric conditions. By using the Center Manifold Theorem and bifurcation theory, the stability of invariant closed orbits bifurcated is also obtained. The numerical simulation results not only show the correctness of our theoretical analysis, but also exhibit new and interesting dynamics of this system, which do not exist in its corresponding continuous version.In chapter 5, it is first investigated for a conjecture of trichotomy of period two for a third order rational difference equation, and then the bifurcation of this equation is further considered. The results obtained partially verify a conjecture in a known literature.