Research On Chaotic Property And Complexity Of Dynamical Systems

Chaos behavior can be described as sensitive dependence on initial conditions and long-term unpredictability. In recent years, the research on chaos theory has become one of the main subjects of nonlinear science. Many scholars have done a lot of meaningful work in the study of chaos, which has prompted chaos theory to be widely used in many fields, such as biology, information science, economics and so on.In the development of chaos theory, scholars have put forward many different kinds of definition of chaos, relevant transitive properties (transitive, weak mixing and mixing). 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 model in many other areas such as Economical system and Biological system also show the chaos characteristics. For example, there are many specific models in biological system and economical system, whose research was based on chaos theory.In the investigation of topological dynamical system, we can generate corresponding dynamical system according to different ways, such as iterative system, inverse limit sys-tem, hyperspace system (set valued map dynamical system), probability measure system and g-fuzzification system. Thus many theories of chaos can be extended to these dynam-ical systems, which provide a better way to study the complexity of the above systems.The investigation on the chaotic property of the original system and the corresponding generation system and the relationship between the different generation systems is al-ways an important direction in the research of topological dynamical system. At the same time, some scholars studied dynamical system in the view of family. They extended some concepts of chaos in general dynamical system to family, which makes a wider range for the chaos theory.This thesis aims at investigating the chaotic properties and complexity. The main content can be divided into three parts. Firstly, we investigate the chaotic properties of different systems. Secondly, we investigate the complexity of several different systems. Thirdly, as an application of the chaos theory, we investigate the chaotic property of Laffer curve. Numerical simulations are used to illustrate the dynamical properties of the model. Specifically speaking,In chapter 1, we mainly introduces the research background of this thesis and the development status at home and abroad and the main work of this thesis.In chapter 2, some basic definitions and result on dynamical system, different kind-s of concepts of chaos and their relationships, and several definitions of chaos on the Furstenberg family are presented in this chapter.In chapter 3, we investigate the chaotic property of several systems. Divided into the following three parts:1. In the compact metric space, we prove that the transitive system with a repelling periodic point is chaotic in the sense of Kato. So, this system, connecting with the existing conclusions, is strong Li-Yorke chaos, strong distributional chaos in a sequence, Rulle-Takens chaos and Martelli chaos. We also prove that weak mixing system is strong Kato chaos. Undoubtedly, weak mixing system is the above mentioned chaos. This notes that weak mixing property is stronger than a repelling periodic point+transitivity.2. On the definition of chaos in the metric space, a new chaos in discrete spatiotem-poral systems is given and one sufficient condition for this system to be distributionally chaotic occuring on measure center is derived.3. The chaotic property of Furstenberg family is studied in this section, we prove that the (g1,g2)-chaoticity and sensitivity of two uniformly topological equiconjugate time-varying discrete systems are equivalent. Moreover, some examples are given to show that Li-Yorke chaos, distributional chaos, and distributional chaos in a sequence are not preserved under topological conjugation.In chapter 4, we investigate the complexities of several systems. Divided into the following three parts:1. An interval map is called finite typal, if the restriction of the map to non-wandering set is topologically conjugate with a subshift of finite type. It is proved that the corre-sponding set valued maps induced by them are topologically conjugate. Moreover, We prove that the set valued map which induced by interval map restricting on non-wandering set is topological double-ergodic. Thus, a conclusion can be obtained immediately that the set valued map induced by sub-shift of finite type is topological double-ergodic.2. We research the chain properties and Li-Yorke sensitivity of Zadeh's extension on the space of upper semi-continuous fuzzy sets and prove that a dynamical system has chain recurrence (resp., chain mixing property, shadowing,h-shadowing) if and only if the Zadeh's extension has chain recurrence (resp., chain mixing property, finite shadowing, h-shadowing) and obtain that if the Zadeh's extension is Li-Yorke sensitive, then the dynamical system is spatiotemporally chaotic.3. We prove that an iterated function system is chain transitive (resp., chain mixing, transitive) if and only if the step skew product corresponding to the iterated function sys-tem is chain transitive (resp., chain mixing, transitive). As an application, it is obtained that an iterated function system with the (asymptotic) average shadowing property is chain mixing, improving the main result [1, Theorem 2.1].In chapter 5, as an application of the chaos theory, we investigate the chaotic prop-erty of Laffer curve on the interval. The range of Laffer curve being topological chaos (distributional chaos, w-chaos, Martelli chaos, Devaney chaos) is given by calculation. Numerical simulations are used to illustrate the dynamical properties of the model.