Chaos for Discrete Dynamical System

Since Li and Yorke first gave the definition of chaos by using strict mathematical language in 1975 [1], the research on chaos has greatly influenced modern science, not just natural sciences but also several social sciences, such as economics, sociology, and philosophy. The theory of chaos convinced scientists that a simple definite system can produce complicated features and a complex system instead possibly follows a simple law. However, scientists in different fields, finding different chaotic connotations, gave different definitions of chaos such as Li-Yorke chaos, distributional chaos, and Devaney chaos. In order to establish a satisfactory definitional and terminological framework for complex dynamical systems that are based on strict mathematical definitions, these concepts with less ambiguous are necessary, and their interdependence has to be clarified. There is no doubt that the mathematical definition of Li-Yorke chaos has a large influence than any other one, whereas distributional chaos possesses some statistical connotations besides the uncertainty of long-term behaviors. So, comparing distributional chaos with Li-Yorke chaos is a meaningful and significant problem. In order to reveal the inner relations between Li-Yorke chaos and distributional chaos, the author brought up the definition of distributional chaos in a sequence in [2]. In this paper, we mainly prove the relations between some different chaoses in discrete dynamical systems. The main theorems are stated as follows. Theorem 1. If a dynamical system (X, f) exhibits transitive distributional chaos in a sequence, then,