| Dynamical systems are an important part of nonlinear science and con-cerned with limit behaviors of the nature phenomena with evolution of time. Due to the foundation and development established by Poincare, Lyapunov. Birkhoff et al., they have become one of the most significant branches of the modern mathematics.In the study of dynamical systems, their entropy is not only an important description of their complexity but also a most essential non-negative invari-ant value to classify dynamical systems at the moment. Thus the theory of entropy is of a significant position in the domain of dynamical systems and becomes a hot subject studied by many scholars. The concept of entropy is first introduced by Clausius who is a famous physicist in the mid-19century. He used this term to describe the "Second Law of Thermodynamics". In1948, based on the work of Clausius and Maxwell, the "Father of Thermo-dynamics" Shannon extended the concept of entropy in thermodynamics to information theory, applied it as a basic value to describe the size of uncertain-ty and gave a formula to compute the information entropy. In1958, in order to solve some classic problems in ergodic theory, Kolmogorov drew the same idea of Shannon and extended the concept of entropy to ergodie theory. In the1960s, in order to study the topological dynamical systems, Adler, Konhelm and McAndrew referred to the definition of measure entropy and introduced the concept of topological entropy for continuous maps defined in compact metric spaces in terms of open covers; In1971, Bowen gave; another definition of topological entropy for uniformly continuous maps defined in metric spaces in terms of (n, ε)-separated sets and (n, ε)-spanning sets. He proved that these two definitions are equivalent for compact systems. This work gave a clear and intuitive understanding for that the topological entropy reveals. Afterwards, in1996, Kolyada and Snoha introduced the concept of topological entropy for time-varying (non-autonomous) dynamical systems in [20]. They also used open covers,(n,ε)-separated sets and (n, ε)-spanning sets to give two different definitions, and proved that these two definitions are equivalent for compact systems.So far it has been found that the topological entropy is a unique non-negative invariant value of topological conjugacy, and every compact system has an identified topological entropy. It provides a numerical measure for the complexity of movement caused by the continuous map’s action on the basic space. Computation and estimation of the topological entropy of a compact system have become very important subjects. Many good results have been obtained about the estimations of topological entropy of compact systems. For instance, in1969, Goodwyn proved that the topological entropy of a continu-ous map defined in a compact metric space is no less than its Borel invariant measure entropy [33]. In1970, Bowen got a series of good results about the estimations of topological entropy. He proved that the topological entropy of a continuous map defined in a compact metric space equals the topological en-tropy of the map restricted to its non-wandering set, gave an estimation of the topological entropy of a homeomorphism with expanding properties in distance defined in a compact metric space, and obtained an estimation of the topolog-ical entropy of a homeomorphism defined in a smooth compact manifold [32]. In the same year, Ito obtained an estimation of upper bound of the topological entropy of a homeomorphism defined in an n-dimensional compact Riemanni-an manifold [34]. In1971, Bowen gave an estimation of upper bound of the topological entropy of a diffeomorphism defined in an n-dimensional compact Riemannian manifold and then generalized Ito’s result [40]. In the same year, Goodman obtained the variational principle which bridges the measure entropy and topological entropy [38]. Later, Dinaburg obtained the variational princi ple for a homeomorphism defined in a finite-dimensional space [42]. In1975, Manning obtained an estimation of lower bound of the topological entropy of a continuous map defined in a compact and connected smooth manifold [41]. Misiurewicz and Przytycki gave a relationship between the topological entropy and the degree of a smooth map [43]. For continuous maps defined on compact intervals, many good results have also been obtained. For example, Block, Guckenheimer, Misiurewicz and Young proved that for an A-coupled-expanding continuous interval map, its topological entropy is no less than the logarithmic value of the maximal eigenvalue of the matrix A [1].In1975, Li and Yorke investigated continuous interval maps and obtained the well-known result:period3implies chaos [12], and then the concept of chaos was first introduced. From then on, a lot of attention from many scien-tists and mathematicians have been got to the the study on chaos in dynamical systems. Chaos is one of the most significant contents of nonlinear science, an inherent property of nonlinear dynamical systems, and a common phenomenon of non-linear systems. In general, chaos means a random-like behavior of deter-ministic systems without adding any stochastic factors. The most important characteristic of a chaotic system is that its evolution has highly sensitive dependence on initial conditions. So, from a long-term perspective, future behaviors are unpredictable. Afterwards, for different purposes of studies, sev-eral new different definitions of chaos were given, specially such as the Devaney chaos [14] and the Wiggins chaos [4,35]. People started to study under what conditions a map will imply chaos in the sense of Li-Yorke or Devaney, i.e. establish criteria of chaos.For autonomous discrete dynamical systems, many good results about chaos have been obtained. Such as, for continuous interval maps, i.e., con-tinuous maps whose domains and ranges are the same real intervals, positive topological entropy implies Li-Yorke chaos [35]; transitivity implies Devaney chaos [36]; a map with zero topological entropy can imply Li-Yorke chaos under certain conditions [37], and so forth. Recently, Blanchard et al. proved that a continuous and surjective map with positive entropy, defined on a compact metric space, is chaotic in the sense of Li-Yorke [21]. It has been found that coupled-expanding maps (or horseshoe maps) are one of the most powerful tool to determine whether a dynamical system is chaotic or not. In1992. Block and Coppel introduced the concept of turbulence in the study of continuous interval maps, and discussed the relationships between turbulent maps and topological entropy, symbolic dynamical systems, period points, chaos and so on. They proved that if a map f is strictly turbulent, then the map in a compact invariant set is semi-conjugate to a one-sided symbolic dynamical system. Thus, the map is chaotic in the sense of Devaney and Li-Yorke [1,21]. In2004, Shi and Chen proved that a strictly coupled-expanding map which satisfies an expanding condition in distance on bounded closed subsets in a complete metric space or compact subsets in a metric space is topologically conjugate to a one-sided symbolic dynamical system [45]. In2006, they gen-eralized the concept of turbulence to maps in general metric spaces [8]. In order to avoid possible confusion with the term turbulence in fluid mechanics, they introduced a new name as a coupled-expanding map. From then on, Shi et al. established several criteria of chaos induced by coupled-expansion. In2006, Shi and Yu established some new criteria of chaos induced by coupled-expansion on bounded closed subsets in a complete metric space [7]. In2009, this concept has been further extended to coupled-expanding maps associat-ed with a transitive matrix by Shi with her coauthors, and they also proved that a strictly coupled-expanding map with an expanding condition in distance on compact subsets in a metric space is chaotic in the sense of Li-Yorke and Devaney, and a strictly coupled-expanding map defined on a complete metric space is chaotic in the sense of Li-Yorke and Devaney under certain conditions [6]. In2010, Zhang and Shi established some. new criteria of chaos induced by A-coupled-expansion on bounded closed subsets in a complete metric space and proved that the map is chaotic in the sense of Li-Yorke and Devaney and chaotic in the sense of Li-Yorke and Wiggins under certain conditions, respec-tively [9]. In2011, Zhang, Shi and Chen studied the C1-perturbation problem of strictly A-coupled-expanding maps, and showed that under certain condi-tions strictly A-coupled-expanding maps are chaotic in the sense of Li-Yorke or Devaney under small C1-perturbations, and further proved that strictly A-coupled-expanding maps are C1structurally stable in their chaotic invariant sets under certain stronger conditions [19].Time-varying discrete dynamical systems appear in many practical prob-lems. Many problems on biology and economy can be described by periodic discrete systems [50-53]. But due to the restriction of development of science itself and also for the convenience of study, time-varying systems are often sim-plified as autonomous discrete systems. Since a time-varying discrete system is generated by iteration of a family of maps in an order, its dynamical behaviors are much more complicated than those of an autonomous discrete system. So, it is more difficult to study time-varying systems than autonomous systems. There have been only a few results about chaos in time-varying discrete system [5,10,15,17,20,46,47,48,49]. In2006, Tian and Chen investigated the dynamical behaviors of a sequence of maps both in the iterative and successive way in the same metric space (X, d)[49]. They generalized the Devaney chaos of autonomous discrete systems to time-varying discrete systems. In2009, Shi and Chen generalized the concepts of chaos in autonomous discrete systems to time-varying discrete systems, established some criteria of chaos in the strong sense of Li-Yorke for finite-dimensional linear time-varying discrete systems and gave a criterion of chaos in the strong sense of Li-Yorke for general time-varying discrete systems [5]. Recently, Shi introduced the concept of induced systems of a time-varying system and investigated close relationships between dynamical behaviors of the original system and its induced systems [10]. In [48], Shi and her coauthors further deeply studied chaos in periodic discrete systems. They obtained several criteria of chaos both in the sense of Devaney and in the strong sense of Li-Yorke and also gave several sufficient conditions of non-chaos for periodic discrete systems.In this dissertation, we mainly investigate the following two problems: one is estimations of upper and lower bounds of topological entropy, the other is under what conditions a time-varying discrete system which is.A-coupled-expanding will imply chaos in the strong sense of Li-Yorke. This dissertation consists of three chapters. Their main contents are briefly introduced as fol-lows.In Chapter1, we summarize the development of the theory of chaos. Some preliminaries, including several basic concepts in autonomous and non-autonomous discrete dynamical systems, are given. Concepts of topological entropy and some fundamental theories of symbolic dynamical systems and coupled-expansion are recalled. In Chapter2, we investigate estimations of upper and lower bounds of topological entropy for time-varying discrete systems. Firstly, we discuss an estimation of lower bound of topological entropy for A-coupled-expanding sys-tems on disjoint, nonempty and closed subsets in compact metric spaces, and prove that its topological entropy is no less than the logarithmic value of the spectral radius of the matrix A. This result extends the result of Block with his coauthors for continuous interval maps to time-varying discrete systems defined on compact metric spaces [1]. Secondly, we study an estimation of upper bound of topological entropy for a time-varying discrete system defined on compact subsets in finite-dimensional Euclidean space, and show that if the sequence of maps satisfy the uniformly Lipschitz condition, then its topological entropy is no more than the product of the logarithmic value of the Lipschitz constant and the dimension of the Euclidean space. This result includes the result of Denker with his coauthors for interval maps [23].In Chapter3, we mainly investigate under what conditions a time-varying discrete system which is A-coupled-expanding will imply chaos in the strong-sense of Li-Yorke, where A is an irreducible transitive matrix with the one row-sum (column-sum) larger than or equal to2. This chapter is divided in-to five sections. Based on some existing works, in Section3, we establish a new criterion of chaos in the strong sense of Li-Yorke induced by strictly A-eoupled-expansion for time-varying systems defined in complete metric spaces (see Theorem3.3.1), which extends some results for autonomous discrete sys-tems in [9] to time-varying discrete systems and weakens some conditions of criteria of chaos given in [5] and [17]. Applying Theorem3.3.1, we establish another criterion of chaos in the strong sense of Li-Yorke induced by coupled-expansion associated with a special irreducible transitive matrix. In Section4, we establish three new criteria of chaos in the strong sense of Li-Yorke induced by strictly coupled-expansion for time-varying systems defined on compact subsets in metric spaces. One of them extends some results for autonomous discrete systems in [6] to time-varying systems. In Section5, we give an ex-ample about a time-varying logistic system for illustration. |
PDF Full Text Request