Study On Some Chaotic Problems Of Non-autonomous Discrete Dynamical Systems

Chaos,a universal form of motion in nonlinear dynamical systems,is one of the central topics of nonlinear scientific research.A lot of results have been obtained for chaos theory of autonomous discrete dynamical systems now.However,many complex systems occurring in the real world problems such as physical,biological,and economical problems are necessarily described by non-autonomous discrete dynamical systems,since the parameters in these systems are often disturbed by external factors and generally change over time.Thus,many scholars focused on complexity of non-autonomous discrete dynamical systems in the recent years.In this paper,we study several chaotic problems of non-autonomous discrete dynamical systems,including some properties of Lyapunov exponents,especially the relationships of Lyapunov exponents with sensitivity and stability;distributional chaos in non-autonomous discrete dy-namical systems;and some relationships among several chaotic properties of non-autonomous discrete dynamical systems.We first study some properties of Lyapunov exponents for non-autonomous discrete dynamical systems.It is well known that Lyapunov exponents,like topological entropy,can characterize the complexity of dynamical systems in a quantitative perspective.For autonomous discrete dynamical systems,it is proved that systems of positive entropy are chaotic in the sense of Li-Yorke[12].If we use Lyapunov exponents to describe complexity of dynamical sys-tems,does positive Lyapunov exponents at some point imply sensitivity and negative Lyapunov exponents at some point imply stability?It was always taken for granted that the answers are positive.However,Demir et al.proved that this conclusion may not hold for general interval maps by two examples in 2001[20].Hence,it is very interesting to investigate relationships of Lyapunov exponents with sensitivity and stability.In 2010,Kocak and Palmer obtained that this conclusion hold for some differentiable interval maps under certain conditions[40].Motivated by their works,we shall investigate the answers of this question for non-autonomous discrete dynamical systems in this paper.We introduce the concept of Lyapunov exponents and investigate relationship-s of Lyapunov exponents with sensitivity and stability for non-autonomous discrete dynamical systems in this paper.We then study distributional chaos in non-autonomous discrete dynam-ical systems in this paper.Currently there is no unified definition of chaos,and several kinds of common definitions of chaos in discrete dynamical sys-tems are Li-Yorke chaos,Devaney chaos,Wiggins chaos,generic chaos,dense chaos,distributional chaos and so on.There have been some developmen-t in the research of chaos theory for non-autonomous discrete systems now.For example,Tian and Chen extended the concept of Devaney chaos to non-autonomous discrete dynamical systems,and studied some properties of it in 2006[84].In 2009,Shi and Chen generalized related concepts of chaos,such as topological transitivity,sensitivity,chaos in the sense of Li-Yorke,Wiggin-s,and Devaney to general non-autonomous discrete dynamical systems,and established a criterion of Li-Yorke chaos induced by strict coupled-expansion for a certain irreducible transitive matrix[74].However,there are only a few results about distributional chaos for non-autonomous discrete dynamical sys-tems.To the best of our knowledge,there are no criteria of distributional chaos established for non-autonomous discrete dynamical systems in the cur-rent literature.We shall study distributional chaos and several weak versions of distributional chaos in non-autonomous discrete dynamical systems and es-tablish several criteria in this paper.In the field of chaos theory of discrete dynamical systems,the two most common problems studied by scholars are establishing criteria of chaos and investigating some relationships among several chaotic properties.We shall study some relationships among several c:haotic properties of non-autonomous discrete dynamical systems in this paper.We mainly discuss some relationships among weak mixing,topologically weak mixing,generic chaos,dense chaos,sensitivity,and Li-Yorke sensitivity and give some equivalent conditions of sensitivity.The following is the organization of this paper.This paper is divided into five chapters.Chapter 1 is preliminaries.First-ly,we introduce the research objects of this paper;that is,the non-autonomous discrete dynamical systems,and point out the differences with autonomous discrete dynamical systems,and the difficulties in the study.Then we recall some current development in the field of chaos theory of autonomous discrete dynamical systems and non-autonomous discrete dynamical systems,respec-tively.Secondly,we introduce some basic concepts about non-autonomous discrete dynamical systems.Thirdly,we introduce several kinds of relations in non-autonomous discrete dynamical systems,investigate some properties of them,and give some useful lemmas.Finally,we give some concepts and lemmas about density of a sequence and recall some properties of one-sided symbolic cdynamical systems.In Chapter 2,we shall study some properties of Lyapunov exponents,espe-cially some relationships of Lyapunov exponents with sensitivity and stability for non-autonomous discrete dynamical systems.Some new concepts are in-troduced for non-autonomous discrete dynamical systems,including Lyapunov exponents,strong sensitivity at a point and in a set,Lyapunov stability,and exponential asymptotical stability.We prove that the positive Lyapunov ex-ponent at some point implies strong sensitivity for a class of non-autonomous discrete dynamical systems.With a similar method,we show that the uni-formly positive Lyapunov exponents in a totally invariant set imply strong sensitivity in this set under certain conditions.We also prove that the nega-tive Lyapunov exponent at a point implies exponential asymptotical stability for a class of non-autonomous discrete dynamical systems.Finally,we give an example about a non-autonomous logistic system for illustration.In Chapter 3,we shall study distributional chaos in non-autonomous dis-crete dynamical systems.In the case that the metric space is compact,we first prove that a system is Li-Yorke ?-chaotic if and only if it is distributionally?'-chaotic in a sequence;then we give three criteria of distributional ?-chaos,which are induced by topologically weak mixing,asymptotic average shadow-ing property,and some expanding condition,respectively,where ? and ?' are two positive constants.In a general case,we give a criterion of distributional chaos in a sequence induced by a Xiong chaotic set.Since the concept of dis-tributional chaos was proposed,three weak versions of distributional chaos for maps in metric spaces,briefly DC1,DC2,and DC3 have been introduced.We shall consider some properties of several weak versions of distributional chaos in this chapter.We prove that DCl,DC2,and DC21/2 are iterate invariants.We also prove that DC1,DC2,and DC21 are preserved with respect to topological equi-conjugacy.These results generalize some existing results of autonomous discrete dynamical systems,some of which relax the corresponding conditions.In Chapter 4,we shall consider some relationships among several chaot-ic properties of non-autonomous discrete dynamical systems.We first study some relationships among weak mixing,topologically weak mixing,generic chaos,dense chaos,and sensitivity.For example,we prove that weak mixing implies topologically weak mixing for measurable non-autonomous discrete dynamical systems with a fully supported measure,but the converse does not hold;topologically weak mixing implies generic chaos for non-autonomous dis-crete dynamical systems in compact metric spaces,but the converse is false;generic 5-chaos and dense ?-chaos are equivalent in complete metric spaces;dense chaos implies sensitivity;and topologically weak mixing implies sensi-tivity for general non-autonomous discrete dynamical systems.Then we give some equivalent conditions of sensitivity and discuss the relationships between sensitivity and Li-Yorke sensitivity.In Chapter 5,we shall give some conclusions of the thesis and prospects for future works.