Research On Complexity Of The Dynamical System

Dynamical system is a developmental system according to time, it studies the basic motion laws of things, the core of the subject is to study asymptotic properties and topological structures of the orbits of points. Study on chaos is not only one of the most active branch for modern dynamical system but also one of the main subjects for nonlinear scientific field. For the present, study on chaos has been considered the best tool for researching complex problems in some fields of society, it is applied widely in physical, chemical, biology, medicine, economics and engineering etc, and has become a hot spot for study.Chaos is that similar random behavior is also appear (intrinsic randomness) in the deterministic nonlinear system even where any additional random factor is absent, it seems a kind of random motion.The best feature of chaotic system is that the evolution of the system is sensitive to initial conditions. Therefore the future state of the system is unpredictable for long-term observations. In 1975, Li and Yorke published their famous paper " Period three implies chaos " on American Mathematical Monthly which first gave the definition of chaos with strict mathematical language and acutely demonstrated the evolution from order to chaos. From then on, scholars from various fields gave different definitions of chaos according to their own understandings such as Devaney chaos, distributional chaos and distributional chaos in a sequence ect. So discussing the relations among different descriptions of chaos is the most principal task of the study on chaos theory. In the late 30 years, many scholars have discussed the relations among chaos, between chaos and the other dynamic properties such as positive topological entropy and topological mixing. A series of fruition is achieved.Impact of chaos upon modern science is not confined to natural science, it is alsouseful in studying some problems arising from economics, social sciences and otherfields. Therefore making in-depth and careful studies on chaos is of far-reaching signif-icance. In recent years, research has shown that the theory of the dynamical systemhas a major development tendency, which is mutual permeation and blending betweentopologically dynamical system and ergodic theory, and the research of chaotic phe-nomena in close combination with topological entropy. In this way, we can get to knowthe mathematical and physical essence of chaos better, which is the purpose of thispaper.In this paper, we make a deep study on the relation between distributional chaosand mixing, the conditions on the existence of the distributively chaotic set, the rela-tion between distributional chaos in a sequence and weak mixing, the relation betweendistributional chaos in a sequence and Devaney chaos and the conditions on the exis-tence of the set which is distributively chaotic in a sequence etc, these are a focus thesedays. The results obtained in this paper are significant theoretically to reveal innermotion law of chaos and clarify the essence of chaotic motion. A detailed introductionis as follows:In chapter 1, we introduce some preliminary knowledge in topologically dynamicalsystem and symbolic space, which will be used in this paper.In chapter 2, we study the relation between mixing and distributional chaos andprove that there exists a shift on the one-sided symbolic space (Σ,ρ) which is mixingbut not distributively chaotic. This demonstrates that a map which is mixing or weakmixing does not mean be distributively chaotic; For compact system (X, f ), if f hasregular shift invariant set, f is distributively chaotic on the set whose every point isrecurrent but not weakly almost periodic. For interval continuous self-map f , if it'stopological entropy is positive, it is also distributively chaotic on the set whose everypoint is recurrent but not weakly almost periodic; We introduce the definition of the coupled map lattices and give the condition on the existence of distributively chaoticset for coupled map lattices; We also introduce the definition of the delayed shift whichis a special shift and discuss it's topological properties and distributional chaos.In chapter 3, We discuss the relations between weak mixing, mixing and distri-butional chaos in a sequence, prove that mixing or weak mixing implies distributionalchaos in a sequence on the locally compact metric space containing at least two points,moreover, they also imply Li-Yorke chaos; For the complete metric space (X, d) with-out isolated points, if f : X→X is transitive and has a periodic point of period p,we prove that f is distributively chaotic in a sequence, moreover, Devaney chaos isstronger than distributional chaos in a sequence; For generally compact system, wegive the conditions on the existence of the distributively chaotic set in a sequence inwhich every point is weakly almost periodic but not almost periodic.In chapter 4, we construct a kind of interval self-map of finite type such that theHausdorf dimension for its non-wandering set is arbitarily given decimal s, and provethat the map restricted in the non-wandering set has some complex dynamic behaviourssuch as positive entropy, distributional chaos, Devaney chaos, Li-Yorke sensitivity andmixing etc; We study the uniformly convergent sequence of continuous functions andgive the conditions that the uniform limit function of function sequence is weak mixing,Li-Yorke sensitive and distributively chaotic in a sequence.Combining topological and analytic method with statistical thought, this paperprobes into all the questions discussed in theoretical aspect. The results obtained inthis paper provide the important theoretical basis for the application of chaos in variousfields.