IWCFTA2012 Keynote Speech V - Several notions of chaos for discrete systems

Summary form only given, as follows. In recent more than 30 years, to illustrate the complex degree of a discrete system, some authors have defined chaos from various angles. In order reveal the essences of chaos, it is necessary to clear the relations between district notions of chaos. In this brief survey, we relate to topological chaos, Li-Yorke chaos, Devaney chaos, distributional chaos and (topological) mixing, which are several notions of chaos attracting the attention of many authors. The relations between each two of these notions are given. Indeed, one can see that Li-Yorke chaos is implied by the others, but the inverse does not hold, and topological chaos, Devaney chaos, distributional chaos and mixing are mutually independent.