Criteria And Perturbations Of Chaos In Discrete Dynamical Systems

Chaos is one of central topics of research on nonlinear science and is a general dynamical behavior of nonlinear dynamical systems. Meanwhile, it has a global and essential effect on the development of nonlinear dynamics, and some original research works of nonlinear dynamics connected with chaos. However, before the end of 1950:s and the establishment of chaos theory, the concept of chaos was very ambiguously. Even now, there are different understandings of chaos in different fields. In general, chaos means a random-like behavior (intrinsic randomness) in deterministic systems without 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 of a chaotic system are unpredictable.Research on chaos in dynamical systems has attracted a lot of attention from many scientists and mathematicians. In 1975, Li and Yorke [33] investigated a continuous map on an interval and obtained the well-known result:"period 3 implies chaos". This crite-rion plays an important role in studying problems on chaos of one-dimensional discrete dynamical svstems. They are the first ones who introduced a mathematical definition of chaos. Later, there appeared several different definitions of chaos [6,15,50,61];some are stronger and some are weaker, depending on requirements in studying different problems. In 1978, F. R. Marotto was inspired by the work of Li-Yorke, generalized the Li-Yorke's theorem to n-dimensional spaces, and introduced the concepts of expanding fixed point and snap-back repeller. The Marotto's theorem [46, Theorem 3.1] proved that a snap-back repeller implies chaos in the sense of Li-Yorke. In 1998. Chen et al. [11] found an error in [46]. In 2004, Shi and Chen gave a modified version of Marotto's theorem [67, Theorem 4.5]. In the same year, Shi and Chen captured the essential meanings of the expanding fixed point and snap-back repeller, and generalized these two concepts for the continuously differentiable maps in Rn to maps in general metric spaces [66]. And Shi with her coauthors established several criteria of chaos in discrete dynamical systems on complete metric spaces [66,67,73]. In 2006, Lin and Chen es-tablished some criteria of chaos induced by heteroclinical repellers [41]. In 2008, Li, Shi and Zhang extended the concept of heteroclinical repeller to maps in metric spaces [35]. In order to make the name of the concept more intuitive to reflect the relations of those repellers, they called it a heteroclinic cycle connecting repellers instead of heteroclinical repellers. And they established some criteria of chaos induced by heteroclinic cycles connecting repellers in general Banach spaces and complete metric spaces [34-38].In 1992, Block and Coppel introduced the concept of turbulence in the study of continuous interval maps [10]. It has been proved that if a map f is strictly turbulent, then the map f is semiconjugate to a one-sided symbolic dynamical system in a compact invariant set [10, Chapter?, Proposition 15]. Since the one-sided symbolic dynamical system has a positive topological entropy,f has a positive topological entropy and is chaotic in the sense of both Devaney and Li-Yorke [9,10,32]. In 2004, Yang and Tang extended this one-dimensional result to maps in metric spaces. In 2006, Shi and Chen extended the concept of turbulence to maps in general metric spaces. In order to avoid possible confusion with the term "turbulence" in fluid mechanics, they gave a new name as a coupled-expanding map. Later, Shi with her coauthors established some criteria of chaos induced by coupled-expansion for maps not only in compact subsets of metric spaces but also in bounded and closed sets (not required to be compact) of complete metric spaces [66-68,71-74,92].In applications, a system is usually influenced by various perturbations, so it is important to study the perturbation problem. In [47,48], Marotto studied the pertur-bations of snap-back repellers and showed that if the scalar problem xn+1=f(xn,0) has a snap-back repeller, then the map xn+1= f{xn,?xn-1) has a transversal homoclinic orbit for all|?|<?for some?>0, where f:R2?R is C1. Later, Li and Lyu showed that if a map has a snap-back repeller in Rn, then so is the perturbed map under sufficiently small C1 perturbation, and hence is chaotic in the sense of Li-Yorke [31]. It is noted that the perturbation problems discussed in all the above references were restricted within finite-dimensional maps. In this dissertation, we consider pertur-bation problems of snap-back repellers in general Banach spaces. We shall show that if a map has a regular and nondegenerate snap-back repeller in a Banach space, then the map still has a regular and nondegenerate snap-back repeller under small perturbation, and consequently is chaotic in the sense of both Devaney and Li-Yorke.Structural stability is an important topic in dynamical systems. It means that the topological behaviors of a system is unaffected under sufficiently small perturbations. The idea of structural stability should go back to the work of Poincare on the three-body problem in the celestial mechanics [56]. When Lefschetz translated Andronov and Khaikin's monograph [3] into English, he introduced the term'structural stability', and then it is used as a mathematical definition. Later, the structural stability has been extensively studied, and many elegant results have been obtained [58,61,93]. In [93], Zhang, Shi, and Chen studied the structural stability of a strictly A-coupled-expanding map in Rn. We shall study the structural stability of maps with snap-back repellers in Banach spaces in this thesis. This result is deeper than the persistence of snap-back repellers under small perturbations.In the real world, many mathematical models are described by time-varying (i.e. nonautonomous) discrete systems. But they were often simplified as autonomous dis-crete systems for convenience. Since a time-varying discrete system is generated by a family of maps, its dynamical behaviors are much more complicated than those of an autonomous discrete system. For example, a finite-dimensional linear autonomous discrete system can not be chaotic, but a finite-dimensional linear time-varying discrete system may be chaotic in the sense of Li-Yorke [70, Examples 2.1 and 2.2]. Hence, it is more difficult to study chaos in time-varying discrete systems than in autonomous dis-crete systems. Thus, there have been a few results about chaos in time-varying discrete system [70,79]. In 2009, Shi and Chen generalized the concept of chaos in autonomous discrete systems to time-varying discrete systems and established some criteria of chaos in the sense of Li-Yorke for finite-dimensional linear time-varying discrete systems and a criterion of chaos in the strong sense of Li-Yorke for general time-varying discrete systems [70]. As mentioned above, many time-varying discrete systems are simplified as au-tonomous discrete systems for simplicity. In fact, they can be regarded as small time-varying perturbations of some autonomous discrete systems. In addition, a time-varying discrete system can also be perturbed. In this dissertation, we will study small per-turbations of some chaotic time-varying discrete systems in Banach spaces, which are strictly A-coupled-expanding for a certain transition matrix A. Applying this result, we will further study small time-varying perturbations of some chaotic autonomous discrete systems with regular and nondegenerate snap-back repellers.In this dissertation, we mainly study two chaos problems:criteria of chaos and perturbations of chaotic discrete dynamical systems. This dissertation consists of four chapters. Their main contents are briefly introduced as follows.In Chapter 1, we summarize the development of chaos theory and some of its applications and give some preliminaries, including several definitions of chaos that are often used in mathematics and some other concepts in discrete dynamical systems. And we briefly recall some concepts and properties of symbolic dynamical system.In Chapter 2, we study small perturbations and structural stability of chaotic discrete systems in Banach spaces induced by snap-back repellers. First, if a map has a regular and nondegenerate snap-back repeller, then it still has a regular and nondegenerate snap-back repeller under a sufficiently small perturbation. Consequently, the perturbed system is still chaotic in the sense of both Devaney and Li-Yorke as the original one. We further discuss structural stability of strictly A-coupled-expanding maps in Banach spaces. Applying this result, we show that a map with a regular and nondegenerate snap-back repeller in a Banach space is C1 structurally stable on its chaotic invariant set.In Chapter 3, we consider time-varying discrete dynamical systems in Banach spaces. A criterion of chaos induced by coupled-expansion for time-varying systems is first established and then persistence of coupled-expansion are considered for time-varying systems under small time-varying perturbations, under which the perturbed system is shown chaotic in the strong sense of Li-Yorke. By applying this result, a map with a regular and nondegenerate snap-back repeller is shown to be still chaotic in the strong sense of Li-Yorke under small time-varying perturbations. Finally, we provided an example with computer simulations for illustration.In Chapter 4, we study chaos in periodic discrete systems.we frist consider re-lationships between chaotic dynamic behaviors of a periodic discrete system and its induced autonomous discrete system; then establish a criterion of chaos induced by coupled-expansion for time-varying systems; and next study persistence of coupled-expansion for periodic discrete systems under small periodic perturbations, under which the perturbed system is shown chaotic in the sense of Devaney as well as in the strong sense of Li-Yorke. By applying this result, a map with a regular and nondegenerate snap-back repeller is shown to be still chaotic in the sense of Devaney as well as in the strong sense of Li-Yorke under small periodic perturbations. Finally, we provide two examples with computer simulations for illustration.