Distributional Chaos Theory And Its Application

The research on chaos theory has become one of the main subjects of nonlinear science.In the development of chaos theory,Many scholars have put forward many different kinds of definition of chaos,relevant transitive properties(including transitive,weak mixing and mixing property),and sensitivity,shadowing property and so on.In order to have a deeper research on chaos theory,it is very important to figure out the relationships among different kinds of chaos,the relationships between relevant transitive properties and different definitions of chaos and conditions of occurrence of chaos.These problems have always been a hot issue in the study of chaos theory.Meanwhile,the models in many other areas such as such as biology,information science,economics also show the chaos characteristics.In this paper,we emphasize some chaotic properties of dynamical systems,particularly transitivity,almost specification property and asymptotic average shadowing property.As an application of the distributional chaos theory,we investigate the chaotic property of Laffer curve.Numerical simulations are used to illustrate the dynamical properties of the model,and obtain the following results:1.We construct a mixing subshift by symbolic dynamic system,with a strong distributionally scrambled set,that is,it is invariant,extremal and transitive.It shows that how"strong"distributional chaos may be,even in a "simple" system.2.The relationships among asymptotic average shadowing property,almost specification property and distributional chaos are studied.Firstly,it is proved that a nontrivial system with AASP(asymptotic average shadowing property)displays uniformly distributional chaos if it has two almost periodic points.Secondly,there exists an uniformly uncountable distributionally scrambled set composed by such points that the orbit closure of every point contains the measure center.Finally,as a corollary,the similar results hold for the system with almost specification property.The above results further the conclusions on the relationship between almost specification property and distributional chaos,which has been proved by predecessors.3.As an application of the distributional chaos theory,we investigate the chaotic property of Laffer curve on the interval.Firstly,the range of Laffer curve being topological chaos(distributional chaos,?-chaos,Martelli chaos,Devaney chaos)is given by calculation.Moreover,numerical simulations are used to illustrate the dynamical properties of the model.