Analysis Of Bifurcation And Chaos In A Class Of 2-D Quadratic Maps

Many systems in the real world can be represented by nonlinear difference equations, which can be changed into nonlinear map systems. In biology, natural populations, whose generations are nonoverlapping, can be represented by two dimensional(2-D) quadratic map systems. In economics, a Cournot duopoly game with adaptive expectations can be described by a 2-D quadratic map. In addition, 2-D quadratic maps are often used as Poincaré maps of 3-D ODEs to reduce the study of continuous time systems. Thus, both from the point of view of fundamental studies and of applications, it should be interesting to analyze the 2-D quadratic maps.In Chap 1, we describe the present situation of 2-D quadratic maps; the research content, research method and the organization of this paper are briefly given so that we have a first impression on this paper.In Chap 2, we determine the existence and local stability of the nonnegative fixed points; the necessary and sufficient conditions for local bifurcations such as fold, Flip, and Neimark-Sacker are determined by Jury condition and local bifurcation theorem.In Chap 3 and Chap 4, we depict the stability of positive fixed point at Flip and Neimark-Sacker bifurcations with center manifold and normal forms respectively, and get the critical stability condition for those bifurcations.In Chap 5, numerical simulations are displayed to verify our results above.In Chap 6, we prove that the system is really chaotic with appropriate parameters after a series of Flip bifurcations by finding the topological horseshoe in a chaotic attractor of the system with the topological horseshoe theory and HsTool.