Chaos Theory Of Discrete Dynamical Systems

Chaos is one of the focuses of nonlinear science, which is a special dynamical behavior of nonlinear dynamical systems. Chaos extensively exists in nature (such as physics, chemistry, biology, geology), engineering technology, social science and many other fields. In general, chaos means a random-like behavior (intrinsic randomness) in deterministic 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 long-term perspective, future behaviors of a chaotic system are unpredictable.Chaos science is a new cross subject, which develops with the rapid development of modern technology, especially based on the occurrence and wide applications of computer technology. Chaos breaks down the boundaries among different fields and is a new science concerning the whole essence of systems. The famous physicist J. Ford claimed that chaos is the third revolution in physics after quantum mechanics and relativity theory in the 20th century.Although chaos has been seriously studied by scientists for near half a century, there is still no widely accepted mathematical definition. The main reason is that chaotic systems are indeed very complicated and there are different understandings from different viewpoints. The usually used definitions in mathematics are the following: chaos in the sense of Li-Yorke [67], Devaney [26], and Wiggins [119] for discrete dynamical systems; and chaos in the sense of Smale's horseshoe [105] for continuous dynamical systems. The usually used criteria of chaos in physics and engineering are boundedness of solutions and the existence of positive Lyapunov exponents or positive topological entropy. Generally speaking, chaos has the following main characteristics: determinacy, boundedness, highly sensitive dependence on initial conditions, long-term unpredictability, positive maximum Lyapunov exponents, infinitely wide frequency power spectrum and ergodicity. etc. [19, 74. 75].From directions of chaos study, it mainly falls into four studying directions: criteria of chaos, control of chaos, anti-control of chaos (or chaotification), synchronization of chaos, where the last three ones belong to a generalized chaos control. The study of deterministic chaos experiences the following three periods: the first period is from order to chaos, when researchers study the conditions, mechanisms and methods that cause chaos; the second period is the order in chaos, when researchers study the universality, statistical characteristics and fractal structures of chaos, etc.; the third period is from chaos to order, when researchers initiatively control chaos to order. In this meaning, we can consider that the study of chaos has gone into a new period. Controlling chaos is the first step of study of chaos turning to its applications. We will eliminate chaos when it is harmful and create chaos when it is useful.In the pursuit of criteria of chaos, many elegant results have been obtained. For example, for continuous interval maps there's a well-known result: "period 3 implies chaos" given by Li and Yorke [67], and further it has been proved that nonpower of 2 period, turbulence, and positive topological entropy all imply chaos in the sense of Li-Yorke [10]. For higher-dimensional maps, there are the famous Marotto theorem [79, 99] and the snap-back repeller theory under weaker conditions [103], and the heteroclinical repellers theory established by Lin and Chen [70]. For general Banach spaces and complete metric spaces, there are the generalized snap-back repeller theory and the coupled-expansion theory established by Shi et al. [98-103], and the heteroclinic cycles connecting repellers theory just established by us [68], etc. For continuous systems, there are the well-known Shilnikov theorem [96, 97] and the Smale's horseshoe theory [105], and the Melniknov method for judging existence of horseshoe (see [73, 82]), etc.For control of chaos, many efficient controlling methods have been found to eliminate chaos when it is harmful, such as the OGY method, the occasional proportion feedback technique (OPF technique), the pulse controlling method, the sliding modular controlling method, the adaptive controlling method, the linear and nonlinear controlling method, the self-feedback controlling method, etc. [19, 24, 36, 124, 125]. When chaos is useful, some researchers also have found many methods to create chaos. For discrete systems, it has been made a great success in anti-control of chaos. For example, Chen and Lai firstly proposed a method of state feedback control for chaotification for any finite-dimensional discrete systems, which can be linear or nonlinear [15-17]. Recently, Shi et al. extended this method to general Banach spaces [101]. For the history and more detailed developments about the anti-control of chaos for discrete systems, we refer to [21, 23]. Anti-control of chaos for continuous systems is more complicated, but it has been made some progress. For example, Yang et al. studied the anti-control problem for continuous systems with limit cycles [121]. Chen et al. studied the anti-control problem for the family of Lorenz systems [24]. Wang et al. showed that it could make a system with minimum phase chaotic by the method of feedback control [117]. Since L. M. Pecora and T. L. Carroll first proposed the concept and method of chaotic synchronization in 1990 [86], it has been made a great progress. It has been found that chaotic synchronization is useful under some circumstances. For example, it has been applied to secret communications [36]. Many methods of making different systems synchronized has been found too, such as the driving-response method proposed by Pecora and Carroll, the method of coupled synchronization, the method of pulse synchronization, the method of linear and nonlinear feedback synchronization, the disturbing method of continuous variables, the method of self-adaptive control, the method of noise inductive synchronization, etc. [24, 36, 124, 125].Based on a great deal of research works in recent years, there are more and more closed relationships between chaos and engineering technologies, such as biological engineering, mechanical engineering, electronic engineering, chemical engineering, informational engineering, and computer engineering [19, 20, 22]. The other potential applications of chaos include the following: analysis of human brain's neural motions and heart pulse, image data encryption, secret communications, dynamic analysis and safeguard of electric power and network, eliminating noise in engineering and civil electric instruments, design of electric shaking generators, dynamic analysis and safeguard of engines in spaceflight and aviation, liquid mixing, store and high speed retrieval of information, decision-making and forecasting, identification in systems and models, diagnose of failure in machine vibration, treatment of computer figures, medical science and biology, treatment of signals and communications, controlling systems and optimization, and more and more fields and directions.With the development of natural science and borderline science such as physics, control theory, biology, medical science and economics, many mathematical models described by difference equations are proposed. For example, the well-known population model-logistic map is a difference equation, in which there may exhibit chaotic phenomena. Even some problems described by continuous dynamical systems can also be discretized. For example, we can use the method of Poincare map or numerical computation schemes for differential equations to obtain some difference equations. The qualitative properties of these difference equations can provide a lot of useful information for analyzing the properties of the original differential equations. So, there are important significances for study on chaos of difference equations both in theory and applications.In this dissertation, we mainly study two chaos problems on criteria of chaos and anti-control of chaos in discrete dynamical systems. It consists of four chapters and the main contents are as follows:In Chapter 1, we summarize the development of chaos, and give some preliminaries, including several definitions of chaos that often used in mathematics and some other concepts in discrete dynamical systems. We also recall some existing criteria of chaos in discrete dynamical systems.In Chapter 2, we first give a concept of heteroclinic cycle connecting fixed points in general metric spaces, and give its two classifications: regular and singular; nondegenerate and degenerate. Next, we give the concept of heteroclinic cycle connecting repellers, which can be viewed as a generalization of the concept of heteroclinical repellers proposed by Lin and Chen [70] in Euclidean spaces. Several criteria of chaos are established for maps in complete metric spaces and metric spaces with the compactness in the sense that each bounded and closed set of the space is compact, respectively, by employing the coupled-expansion theory proposed by Shi and Chen et al. [98-103]. The maps in these criteria are proved to be chaotic in the sense of both Devaney and Li-Yorke, which are stronger than the result obtained by Lin and Chen [70]. Especially, we get a criterion of chaos in the sense of Li-Yorke for heteroclinic cycles connecting repellers and saddles in metric spaces with the compactness in the sense that each bounded and closed set of the space is compact. Finally, employing these criteria of chaos obtained in the above to general Banach spaces and Euclidean spaces, we obtain some criteria of chaos, where the maps are also chaotic in the sense of both Devaney and Li-Yorke. In addition, we provide three examples with computer simulations for illustration.In Chapter 3, we mainly study stability of fixed points and criteria of chaos in time-delay difference equations. We first transform these time-delay difference equations into higher-dimensional ordinary difference systems, and then introduce some basic concepts. Next, we study stability of fixed points and establish a criterion of chaos, where the equation is proved to be chaotic in the sense of both Devaney and Li-Yorke. Finally, we discuss stability of fixed points and criteria of chaos for two special classes of time-delay difference equations. In addition, we provide two examples of time-delay difference equations and obtain their bifurcation diagrams.In Chapter 4, we mainly study anti-control of chaos for discrete dynamical systems. Using the criteria of chaos established in Chapter 2 and the method of state feedback control, we study chaotification problems of discrete systems in Banach spaces with at least two fixed points. We also use the snap-back repeller theory and the method of state feedback control to study chaotification problems of linear time-delay difference equations. We provide two examples with computer simulations for illsutration.