Chaos Theory Of Discrete Dynamical Systems And Its Applications

Dynamical systems that study the limit behaviors of the nature phenomena with evolution of time are an important parts of nonlinear science. Due to the foundation and development done by Poincare, Lyapunov and Birkhoff, they have become one of the important branches of the modern mathematics. Chaos is a universal dynamical behavior of nonlinear dynamical systems and one of the central topics of research on nonlinear science. Meanwhile, it has a global and essential effect on the development of nonlinear dynamics. But, what is chaos? Even now, there is not a unified defi-nition of chaos because the scholars from different fields have different understanding about chaos. In general, chaos means a random-like behavior (intrinsic randomness) in deterministic systems without any stochastic factors.The mathematical definitions of chaos commonly used are chaos in the sense of Li-Yorke [54], Devaney [25], and Wiggins [107] for discrete dynamical systems; and chaos in the sense of Smale's horseshoe for continuous dynamical systems [96]. Sensitivity is a key ingredient of chaos. Therefore, the research on sensitivity has drawn lots of scholars' attention [1,8,35,38,40,52,53,109]. In1992, Banks ct al. proved that the two former conditions (transitivity and dense periodic points set) of Devaney chaos imply the last condition sensitivity [8]. In1993, Glasner and Weiss got a stronger result:any transitive and non-minimal dynamical system that almost periodic points are dense in the phase space is sensitive [35]. In2002, Abraham et al. studied the sensitivity from the viewpoint of ergodicity [1]. They proved that if a measure-preserving map T with full measure support on a metric probability space is either strong-mixing or topologically mixing, weak-mixing and satisfies other conditions, then it is sensitive. In2010, Li and Shi introduced a definition of topologically strongly ergodic and proved that if a measure-preserving map T with full measure support on a metric probability space is topologically strongly crgodic, then T is sensitive [52]. The result of Li and Shi weakened Abraham's conditions. Other results about sensitivity can be seen in [40,109,113].The "largeness" of the time set where sensitivity happens can be thought of as a measure of how sensitive the system is. On account of this reason, in2007, Moothathu proposed three stronger forms of sensitivity:syndetic sensitivity, cofinite sensitivity, and multi-sensitivity [69]. They obtained some sufficient conditions of stronger sensitivity and a counter example which is not syndetically sensitive. Recently, Li and Shi proposed another stronger form of sensitivity—ergodic sensitivity and gave some sufficient conditions of four stronger sensitivity for measure-preserving map and semi-flow on a probability space [53].To determine whether a system is chaotic or has which kinds of chaos is a difficult topic in the research on chaos. Now, there arc plentiful results for autonomous dis-crete dynamical systems. In1975, Li and Yorke investigated continuous interval maps and obtained the well-known result:"period3implies chaos"[54]. In this paper, a descriptive definition of chaos was firstly introduced by them. In1978, Marotto was in-spired by Li and Yorke's work, and generalized the Li-Yorkc's theorem to n-dimensional spaces, and introduced the concepts of expanding fixed point and snap-back repeller, and proved that a snap-back repeller implies chaos in the sense of Li-Yorkc [65, Theo-rem3.1]. In2004, 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 finite-dimensional spaces to general metric spaces [84]. From then on, Shi with her coauthors established several criteria of chaos for autonomous discrete dynamical systems on complete metric spaces [84,85,91]. Other results about criteria of chaos can be seen in [55-59,61].The theory of coupled-expansion is another vigorous criterion of chaos for discrete dynamical systems. In1992, Block and Coppel introduced the concept of turbulence in the study of continuous interval maps [11]. 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. Thus,f is chaotic in the sense of Devaney and Li-Yorke [10,11]. In2006, Shi and Chen generalized 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 [87]. From then on, Shi with her coauthors established some criteria of chaos induced by coupled-expansion [89-91,123].Theory and practice always coexist and supplement each other. Chaos theory comes from practice, and should go back to practice. Research on chaos control has attracted a lot of attention. Chaos control includes two opposite directions:control of chaos and anti-control of chaos (or called chaotification). The process of making a chaotic system nonchaotic or stable is called control of chaos. Chaotification or anti-control of chaos is a process that makes a nonchaotic system chaotic, or enhances a chaotic system to present a stronger or different type of chaos. During the last decade, the research on chaos control of autonomous discrete systems has been developed rapidly on account of the harmness of chaos. We refer to [18,45,49] for more discussions about it. However, it has been found that chaos is very useful in some circumstances, such as in human's brain, robot manufacture, heartbeat regulation, digital communications, cryptography, fluid mechanics, liquid mixing, etc [15,26,31,82,97]. Therefore, many chaotification schemes have been proposed for autonomous discrete systems. In1996, a mathematically rigorous chaotification method was first developed by Chen and Lai via a feedback control approach [16,17,19]. They showed that the controlled system via modoperation is chaotic in the sense of Devaney when the original system is linear, and is chaotic in a sense of Wiggins when the original system is nonlinear. In [102,103,116,126], Wang et al. applied the Marroto theorem to prove that the controlled systems arc chaotic in the sense of Li-Yorke. Recently, Shi, Chen and Yu extended those methods to infinite-dimensional discrete dynamical systems and presented several chaotification schemes for discrete dynamical systems in Banach spaces [20,86,92]. They proved that the controlled systems arc chaotic in the sense of both Devaney and Li-Yorke.Theory and methods of chaos have been widely used in the fields of vibration, automatic control, system engineering, etc. At the same time, they have a profound influence on some social sciences. In1838, Cournot first established the fundamental theory of oligopoly for a general strategic game. Later, there appeared many other im-proved versions of Cournot model [105]. Recently, some complex dynamics of bounded rationality duopoly games had been widely studied in [2-4,32,75,114,117]. In1991, Puu first found a variety of complex dynamics arising in the Cournot duopoly case in-cluding attractors with fractal dimension [75]. In2003, Yasscn ct al. studied a Cournot duopoly game in which players use different production methods and choose their quan-tities with delayed bounded rationality. In2003and2010, in [3] and [117], the authors studied the Cournot duopoly model which contained two heterogenous players, one was boundedly rational and the other was naive. They computed numerically the largest Lyapunov exponent and the fractal dimension of the chaotic attractor. In2004, Yi et al. studied the evolutionary game of Cournot model with different behavior rules and concluded that the behavior rule of short-sighted eyes couldn't be washed out by the Nash behavior rule [114].Ergodic theory came from analysis of statistical properties for Hamiltonian dy-namical systems. Its aim is to study the long-term average behaviors of systems. It has important and profound applications in many branches of mathematics, such as mathematical physics, probability, number theory, and so on [12,13,14,33,101]. Topological dynamical systems under a great deal of groups have invariant measures corresponding to Borcl a?algebra. Hence, ergodic theory has become a basic method in the study of topological dynamical systems [113]. Ergodic theorem is a major topic of ergodic theory. The first important result of ergodic theory is the pointwise ergodic theorem that was established by Birkhoff in1931. The next important one is the mean ergodic theorem that was established by Von Neumann in1932. Following the original works of Birkhoff and Von Neumann, many mathematicians proposed generalizations and improvements of their results [29,64,70,74,77,101]. In1987, Elton investigated the ergodic theorems for Iterated Function Systems with finite contracted maps on compact metric spaces. In2001, Ma and Zhou improved the Elton theorem and obtained some new ergodic theorems [64]. In2005, O Hyong_chol et al. studied the ergodic theorems for Iterated Function Systems with infinite contracted maps on compact metric spaces [70]. Time-varying (non-autonomous) systems exist in many practical problems. A lot of problems on economy and biology can be described by periodic discrete system [6,23,24,28,41,81]. But for the convenience of study and the restriction of development of science itself, time-varying systems are often simplified as autonomous discrete systems. Since a time-varying discrete system is generated by iteration of a family of maps, its dynamical behaviors arc much more complicated than those of an autonomous discrete system. It is more difficult to study time-varying systems than autonomous systems. There have been a few results about chaos in time-varying discrete system [6,30,47,88,94,95,99,115,119]. In1996, Kolyada and Snoha investigated the entropy of time-varying discrete system [47]. In2003, Frankc and Selgradc discussed the attractor problem for periodic discrete dynamical systems that arc the special cases of time-varying discrete systems [30]. In2006, AlSharawi et al. generalized the Sharkovskii theorem to periodic discrete dynamical systems [6]. In recent years, there arc some progress of chaos criteria for time-varying discrete dynamical systems. In2006, Tian and Chen investigated the dynamical behaviors of a sequence of maps both in the iterative and successive way in a same metric space (X, d)[99]. They gcnerilized the Dcvaney 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 and established some criteria of chaos in the strong sense of Li-Yorke for finite-dimensional linear time-varying discrete systems and a, criterion of chaos in the strong sense of Li-Yorkc for general time-varying discrete systems [88]. Recently, Shi introduced a new type of subsystem of time-varying system and investigated close relationships between dynamical behaviors of time-varying discrete dynamical system and its subsystems [94]. In [95], Shi with 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 gave several sufficient conditions of non-chaos for periodic discrete systems.As far as we know, although not much and profound research has been done for time-varying discrete systems, there are still a large number of problems of time-varying discrete systems waiting for people to study, which including sensitivity, chaotification, crgidic theory, and so on. In this dissertation, we study the problems on sensitivity for time-varying discrete dynamical systems and its special case—periodic discrete systems, and discuss quite systematically the ergodic theorems for periodic discrete systems. Moreover, we consider the chaotification of time-varying discrete systems in finite-dimensional and Banach spaces and establish several chaotification schemes via feedback control approach by employing the theory of coupled-expansion and snap-back repeller. Thus far, the complex dynamical behaviors of the Cournot models were ob-tained by computer simulations without strictly theoretical proofs. In this dissertation, we establish a new Cournot duopoly model and prove that this model is chaotic in the both sense of Devaney and Li-Yorke by employing the snap-back rcpcllcr theory.In this dissertation, we mainly investigate three chaos problems:sensitivity, chaoti-fication, and applications of the chaos theory in game theory. This dissertation consists of four chapters. Their main contents are briefly introduced as follows.In Chapter1, we summarize the development of the theory of chaos and some of its applications. Some preliminaries, including several definitions of chaos that are often used in mathematics and some other basic concepts in discrete dynamical systems arc given. Some concepts and properties of symbolic dynamical system and concepts of measure spaces and measure-preserving maps arc recalled.In Chapter2, we investigate the sensitivity for time-varying discrete system and its special case—periodic discrete system, and study the ergodic theorems for periodic discrete system. First of all, we study the sensitivity for time-varying system and give two sufficient conditions of sensitivity for general time-varying dynamical systems. At the same time, four stronger forms of sensitivity are introduced for time-varying sys-tems, including syndetic sensitivity, cofinite sensitivity, multi-sensitivity, and ergodic sensitivity. Two sufficient conditions of cofinite sensitivity for general time-varying dy-namical systems arc presented. Then we discuss the sensitivity for periodic discrete system and give a sufficient conditions of sensitivity for measure-preserving periodic discrete systems is given. It is proved that if a measure-preserving periodic discrete system with full measure support is topologically strongly ergodic, then it is sensitive. Some relationships between four stronger forms of sensitivity of periodic discrete sys- tern and its induced autonomous system arc studied. It is proved that if the induced autonomous system has one of the four stronger forms of sensitivity, then so does the periodic system. On the contrary, if a periodic system is uniformly continuous and has one of the four stronger forms of sensitivity, then so does its induced system. Finally, we generalize the mean ergodic theorem in Hilbert space and ergodic theorems for au-tonomous dynamical system to periodic discrete system, including the von Neumann mean ergodic theorem and the Birkhoff pointwise ergodic theorem.In Chapter3, we consider chaotification of time-varying discrete dynamical sys-tems. We mainly consider chaotification of time-varying discrete dynamical systems in finite-dimensional spaces, general and special Banach spaces. Several chaotifica-tion schemes via feedback control techniques arc established, where the controllers are time-invariant for convenient operation in practice, including general controllers and sawtooth function. The range of controlled parameters is given for each chaotification scheme. The general controllers have strictly A-coupled-expanding property or snap-back rcpcllers. It is only required that the original time-varying systems satisfy the uniform Lipschitz condition in some cases. The controlled systems are proved to be chaotic in the strong sense of Li-Yorke by employing the chaos criteria theorem that was established by Shi and Chen [88] for a strictly coupled-expanding time-varying discrete system. At the end, an example with computer simulations is provided for illustration.In Chapter4, we study an application of chaos theory in game theory—Cournot model with the behavior rule of short-sighted eyes. At first, we establish a new nonlinear Cournot model with the behavior rule of short-sighted eyes. Then we analyse the local stability of the equilibria of the Cournot model. It is proved that this model exhibits some chaotic behaviors in a certain range of the output adjusted coefficients, and the system is chaotic in the sense of both Devancy and Li-Yorke by employing the snap-back repeller theory. At last, we give some computer simulations. Especially, it has three different chaotic strange at tractors when the output adjusted coefficients have different values.