Coupled-expanding Maps And Chaos

In the development of chaos theory in discrete dynamical systems, A. Sharkovskii's amazing discovery [46], as well as T. Li and J. Yorke's famous work that introduced the concept of chaos [28], have activated the rapid advancement of the frontier research. For different purposes of studies, new definitions of chaos were given, specially the Devaney chaos [12] and the Wiggins chaos [60]. The objectives of such studies have been under what conditions a map will imply chaos in the sense of Li-Yorke or Devaney, i.e., what the criteria of chaos are, and how to understand the relationships among different definitions of chaos. For continuous interval maps, i.e., a continuous map whose domain and range are the same real interval, many good results have been obtained: transitivity implies Devaney chaos [58]; positive topological entropy is equivalent to Devaney chaos [27], which also implies Li-Yorke chaos [15]; a map with zero topological entropy can imply Li-Yorke chaos under certain conditions [54], and so on. In general, Devaney chaos is stronger than Li-Yorke chaos. For instance, W. Huang and X. Ye proved that Devaney chaos implies Li-Yorke chaos under certain conditions for a map defined on a compact metric space [22]. F. Blanchard et al. positively answered the question whether a continuous and surjective map with positive entropy, defined on compact metric space, is chaotic in the sense of Li-Yorke [6].In the advancement of dynamical systems, coupled-expanding maps (or horseshoe maps) have been studied by many mathematicians using various methods. M. Misiurewicz showed that for a continuous interval map with positive entropy, there exists a finite forward iteration of the map such that the iterated map is strictly coupled-expanding, and there are infinitely many of such iterations [33]. L. Block, J. Guckenheimer, M. Misiurewicz and L. Young proved that for a generalized A-coupled-expanding continuous interval map, its topological entropy is no less than the logarithmic value of the maximal eigenvalue of the matrix A [8]. L. Block and W. Coppel introduced the concept of turbulent map (called coupled-expanding map lately) in the study of continuous interval maps, and discussed the relationship between turbulent map and period points, topological entropy, symbolic dynamical systems, chaos, and so on [7]. S. Ruette summarized some important works on chaos induced by continuous interval maps, with many discussions about coupled-expanding maps [45]. Y. Shi and G. Chen proved that a strictly coupled-expanding map, with 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 [47]. These important results have greatly extended our understanding of this kind of maps. The concept of coupled-expanding map in a metric space was first introduced by Y. Shi and G. Chen in [49]. Recently, this concept has been further extended to coupled-expanding maps associated with a transitive matrix [50]. In this thesis, we use the term "coupled-expending map" for convenience. It should be remarked that the coupled-expansion theory can be applied to prove chaos induced by snap-back repellers, which was first introduced by F. Marotto [31]. For a generalization of the concept of snap-back repeller in metric spaces and the improvement of Marotto's theorem, the reader is referred to [47, 48, 51, 52].Structural stability is one of major subjects in the theory of dynamical systems. The notion of dynamical systems dates back to H. Poincare's research about the motion of celestial bodies [39, 40]. In 1892, A. Liapunov brought forward his remarkable first and second methods, in which the second method has far-reaching consequences on the research of the stability of ordinary differential equations [29]. Later, G. Birkhoff developed H. Poincare's ideas [5]. A fundamental concept in the study of differentia] dynamical systems is diffeomorphism defined on a smooth manifold. A diffeomorphism is called structurally stable if there exists a C~1-neighborhood of this map such that this diffeomorphism is topologically conjugate to any diffeomorphism in this C~1-neighborhood. This concept was introduced by A. Andronov and L. Pontrjagin in 1937 (they did not use the term "structural stability" in their time) [2]. In the translation of A. Andronov and S. Khaikin's work from Russian to English, S. Lefschetz used this term "structural stability" [1]. A. Andronov and L. Pontrjagin's work [2] was then generalized by M. Peixoto [37, 38]. In the study of higher-dimensional diffeomorphisms, S. Smale brought forward the now-famous Axiom-A systems [53]. Later, J. Palis and S. Smale conjectured that an Axiom-A system together with a strong transversahty condition is equivalent to the structural stability of the underlying system [36]. Based on the works of J. Robbin, C. Ron-binson, R. Marie, S. Hayashi, S. Hu, etc., this conjecture has been positively solved finally [18, 21, 30, 41, 42, 43].It is important to study the perturbation problem or structural stability problem of chaotic systems. In practical applications, a system is usually influenced by many different kinds of perturbations. If a system is not structurally stable, then it might display different dynamical behaviors rather than those of the original system. An interesting question is whether an A-coupled-expanding map is still an A-coupled-expanding map, and still is chaotic, under a small perturbation. There are very few, if any, results on this important problem. In this thesis, we study the perturbation problem of chaotic A-coupled-expanding maps.Nowadays, the rapid development of chaos theory has gained significant impacts on many scientific subjects. Chaos theory has become one vigor branch of dynamical systems [60]. In physics, chaos can be applied to describe complex phenomena such as fluid motions [14], planetary orbits [34], etc. In chemistry, the amounts of chemical compounds were analyzed based on chaos theory in [56]. In engineering, control of chaos [10] and anti-control of chaos [11] constitute two main directions of research in chaos control theory; one-dimensional chaotic maps are used as random number generators [57], which can be applied to spread-spectrum communications [25] and cryptosystems for image encoding [4] and data encryption [23]. Hence, it is practically very useful to find some methods for constructing simple chaotic maps. Because of the simple expressions of polynomial maps, they are very convenient to use in engineering and technological applications. Notably, D. Dubois has constructed a class of one-dimensional chaotic polynomials of arbitrary degrees [13].This thesis deals with three basic problems: the first is under what conditions a strictly A-coupled-expanding map is chaotic, where A is an irreducible transitive matrix with one row-sum larger than or equal to 2; the second is the perturbation problem of A-coupled-expanding maps, and the third is the construction of chaotic polynomial maps. Corresponding to these problems, we divide the thesis into three chapters.In Chapter 1, we discuss strictly A-coupled-expanding maps in complete metric spaces, or on compact subsets of metric spaces, where A is an irreducible transitive matrix with the one row-sum larger than or equal to 2. A useful result in symbolic dynamical systems is obtained via some detailed analysis on matrix A in Section 2. Recently, symbolic dynamical systems have been applied to establishing several criteria of chaos for strictly coupled-expanding maps, by Y. Shi, G. Chen, P. Yu, etc. For instance, it has been proved that (1) a strictly coupled-expanding map satisfying certain conditions is chaotic in the sense of both Li-Yorke and Wiggins [51, Theorem 3.1]; further, if the coupled-expanding map is expanded in distance on one subset, then it is chaotic in the sense of both Li-Yorke and Devaney [51, Theorem 3.2]; (2) if a strictly A-coupled-expanding map, defined on compact subsets of a metric space, is expanded in distance on each subset, then it is chaotic in the sense of both Li-Yorke and Devaney [50, Theorem 5.2]; (3) a strictly A-coupled-expanding map on closed and bounded subsets of a metric space, satisfying certain conditions, is chaotic in the sense of both Li-Yorke and Devaney [50, Theorem 5.6]. In Section 3 of Chapter 1, we investigate strictly coupled-expanding maps in detail, based on the previous works, and generalize some existing results while weaken the conditions of some theorems; more precisely: (1) Theorems 1.3.1-1.3.2 extend the results of Theorems 3.1 and 3.2 in [51] to general A-coupled-expanding maps (see Remarks 1.3.1 and 1.3.3); (2) Theorem 1.3.2 weakens the requirement that the map is expanded in distance on each subset in [50, Theorem 5.2] (see Remark 1.3.3); (3) Theorem 1.3.3 generalizes the results of Theorem 5.6 in [50], which is about an A-coupled-expanding map for a special kind of transitive matrices, to an A-coupled-expanding map associated with a general transitive matrix (see Remark 1.3.4). L. Block and W. Coppel showed that if a continuous interval map is strictly coupled-expanding, defined on two nondegenerate compact intervals, then there exists an uncountable invariant subset on which the map is topologically semi-conjugate to the shift on two symbols [7, Chapter II, Proposition 15]. Further, S. Ruette proved that this kind of maps is mixing, sensitive on initial conditions, and has dense periodic points [45, Proposition 6.1.3]. We study strictly A-coupled-expanding maps defined on nondegenerate compact intervals, in Section 4, and obtain Propositions 1.4.1 and 1.4.2, which generalize Block and Copple's and Ruette's results, respectively (see Remarks 1.4.1 and 1.4.3). The result of Theorem 1.4.2 extends X. Yang and Y. Tang's result in [61, Theorem 1] to a strictly A-coupled-expanding map defined on compact subsets of a metric space (see Remark 1.4.4). In Section 5, an example defined on the plane is given with computer simulation to illustrate the results obtained in Section 3.In Chapter 2, we study C~r-perturbation problems of strictly A-coupled-expanding maps in Euclidean spaces, where r = 0 or 1 and A is an irreducible transitive matrix with one row-sum larger than or equal to 2. In Section 3, we show that a special class of A-coupled-expanding maps remains chaotic in the sense of Li-Yorke under a small C~1-perturbation. We prove that two kinds of strictly A-coupled-expanding maps defined on locally convex subsets of a Euclidean space remains chaotic in the sense of both Li-Yorke and Devaney under a small C~1-perturbation. Further, we show that two types of strictly A-coupled-expanding maps are structurally stable. In Section 3, we show by applying the Brouwer degree theory that a special kind of A-coupled-expanding maps remains chaotic in the sense of Li-Yorke under a small C~0-perturbation. In Section 5 we discuss an example to illustrate the results obtained in Section 4.In Chapter 3, we apply the coupled-expansion theory to construct chaotic polynomial maps, and verify that the constructed maps are chaotic in the sense of both Li-Yorke and Devaney. In Section 3, for an arbitrarily given nonzero real polynomial, we apply the coupled-expansion theory to construct chaotic polynomial maps by simply multiplying the given polynomial with a suitably designed quadratic polynomial, and then prove that the constructed polynomial is chaotic in the sense of both Li-Yorke and Devaney. With the aid from the coupled-expansion theory, we also verify that a polynomial with at least two different non-negative zeros or nonpositive zeros can be chaotic in the sense of both Li-Yorke and Devaney by multiplying a suitable real constant. Under some weaker conditions, Theorem 3.3.1 obtains part of the results of Theorem 3.1 in [62] (see Remark 3.3.1) and Theorem 3.3.2 generalizes Theorem 4.1 in [62] (see Remark 3.3.3). In Section 4, by applying the interpolation theory to constructing polynomials to approximate a class of chaotic maps, we prove that if the approximation error is small enough then the constructed polynomial maps are chaotic in the sense of both Li-Yorke and Devaney. One illustrative example is provided in Section 5 with computer simulations, which show that the constructed polynomial maps have quite complex dynamical behaviors.