Chaotic Behavior Of Interval Mapping Dynamic System

In the late 20th century, there is a great development in nonlinear science, while the chaos of the research develop greatly. With the development of the understanding of chaotic human in natural science, especially the application of chaos, how to define the chaotic connotation and denotation, thus scientific definition chaos and how underground judge was the research exploration of the chaos topic. In 1975, Tien-Yien Li and James A. Yorke gave the chaos of the earliest definition, namely famous Li - Yorke theorem. Then, Devaney summary chaotic behaviors of the dynamic system for topological transitivity, periodic points dense and sensitive dependence on initial data. Chaos of the symbolic dynamical system has been Proved .However there is other way to measure the chaotic nature of the dynamics, they are the Liapunov exponents, topology entropy, differ- rent homoclinic point, and period is not a power of 2. In 2000, Gong Chen who is a American Texas A&M university professor, characterized the chaotic behavior by means of the growth rate of the total variation of iterates on the interval. Researching the relationship between an inte- rval map iterative variation growth trend and sensitive dependence on initial data, we get some good results, but some conclusion is not mature yet. In this paper, first we introduce chaos of the classic definition and criterion. Second, chaotic behaviors of the discrete dynamic system are described concretely. Last, discussing interval map of the chaotic behavior and iteration variation relationship.