Dynamical System Analysis On Two Classes Of Second-Order Quadratic Difference Equation

In this paper, in connection with different situations of parameters of the common second-order difference equation, we can simplify it into two different types of nonlinear mapping systems for research and discussion. The two types of nonlinear mapping system of local dynamics properties and global dynamics are mainly analyzed. First, we should study this article a brief overview of the problems as well as some related research status and background, and explained the relevant concepts related to this article and basic theoretical knowledge in order to study the problem we have a preliminary understanding and awareness. Second, two types of mapping system were analyzed and researched by making use of mainly the theory of critical curves and prefocal curves of focal point.According to a class of second-order difference equations changed into noninvertible iteration map, its dynamic behavior is investigated by using of two-dimensional irreversible continuous iteration mapping dynamic system theory. Firstly, the stability and the local bifurcations of the fixed points are analyzed, and then the global bifurcations of the basin of all bounded attractors for the mapping system are studied by critical set theory of irreversible iteration map.According to the class of second-order difference equations changed into nonlinear mapping system with a vanishing denominator, Firstly, the stability and the local bifurcations of the fixed points are analyzed. Finally, the structures of attractor and topological structures of the basin for the mapping system are studied and explained by singular set theory with a vanishing denominator.Finally, on the basis of different situations of other parameters of the common second-order difference equation, we can simplify into its various other nonlinear mapping system, and on this basis as we pointed out the direction for further discussion and research.