Weakly Almost Periodicity And Distributional Chaos In A Sequence

The first strict mathematical definition of chaos is Li-Yorke chaos, which has bigger influence than any others. It has sensitive dependence on initial condition, but it does not describe measure and stability.The definition of distributional chaos which first occurred in 1994, later than Li-Yorke chaos, has statistical laws besides sensitive dependence on initial condition. So distributional chaos has more actual significance. In order to reveal the inner relations between Li-Yorke chaos and distributional chaos, professor Lidong Wang brought up the definition of distributional chaos in a sequence. Also, the condition of distributional chaos in a sequence is much weaker than distributional chaos. So we can study chaotic properties on more kinds of compact system in the sense of distributional chaos in a sequence.In the view of ergodic theory, absolute zero measure set can be neglected. So Zhou introduced the definition of the center of measure. And the close of weakly almost periodic points is the center of measure. Also, he pointed out that almost all primary dynamical properties are focused on center of measure.Consequently, it is meaningful to discuss the problems of chaos in the set of weakly almost periodic points. Although nearly all the chaotic definitions are indeterminant in long-term actions, chaotic phenomenon of them are not equal to each other. Different chaotic definitions have different sense in actual analysis.The main purposes of this paper are as following:1 Construct a distributively chaotic set in a sequence on the symbolic dynamical system, such that every point of the distributively chaotic set is weakly almost periodic point. Then extend the result to compact system by means of topological semi-conjugacy and almost shift invariant set.2 Discuss the chaotic properities of a kink of single-species model, then obtain the model is distributively chaotic, Martelli-chaotic, Devaney chaotic and Block-Coppel chaotic, thereby providing theoretical guidance for application of the model in the field of ecology.