Chaotification For First-Order Partial Difference Equations

Chaos control is one of the important subjects in the research of chaos, which has been developed so far mainly in two different directions: control of chaos and anti-control of chaos. A process of making a chaotic system nonchaotic or stable is called control of chaos. Over the last decade, chaotic motions are regarded as harmful, so the traditional control engineering design always tries to stabilize a chaotic system, therefore, research on control of chaos has been rapidly developed ((?)f. [1～4]). Anti-control of chaos (or called Chaotification) is a process of making an originally non-chaotic dynamical system chaotic, or enhancing a chaotic system to present stronger or different type of chaos. Recently, it has been found that chaos can actually be very useful under some circumstances, such as in encryption [5], digital communications [6]. human brain analysis [7], and heartbeat regulation [8].In the pursuit of chaotifying discrete dynamical systems, a mathematically rigorous chaotification method from a feedback control approach was first developed by Chen and Lai [9～11]. They showed that the Lyapounov exponents of a controlled system are positive [9], and the controlled system via the mod-operation is chaotic in the sense of Devaney when the original system is linear, otherwise, it is chaotic in a weaker sense of Wiggins [12]. Later, Wang and Chen [13, 14] further showed that the Chen-Lai algorithm for chaotification leads to chaos in the sense of Li-Yorke by employing the Marotto theorem [15]. Recently, Shi, Yu, and Chen [16] established some chaotification schemes of discrete dynamical systems in Banach spaces, which can be either finite-dimensional or infinite-dimensional. They proved that the controlled systems are chaotic in the sense of both Devaney and Li-Yorke by applying the coupled-expansion theory and the snap-back repeller theory [17, 18]. In addition, they extended the Chen-Lai and Wang-Chen algorithms via feedback control with mod-operation and sawtooth functions in finite-dimensional real spaces [11, 14] to the Banach space l~∞, and showed that the controlled systems in both R~k (k <∞) and l~∞ are chaotic in the sense of Devaney, Li-Yorke, and Wiggins. Shi and Chen proposed several some chaotification schemes for finite-dimensional systems via piece-continuous controllers [19]. The reader is referred to [20] for a survey of chaotification of discrete dynamic systems and some references cited therein.As we all know, continuous systems can be described by differential equations, while sample systems (such as national income, product's output etc.) can only be described by difference equations. On the other hand, an accurate solution of a general nonlinear differential equation is difficulty solved. So we often compute its approximate solution by applying discrete method. Partial difference equations often appear in engineering applications, particularly in the fields of digital filter, imaging, and spatial dynamical systems (cf. [21, 22] and some references cited therein). Stability of solutions of partial difference equations has been investigated extensively [23, Chapter 6]. Chen and Liu [24] initiated the study of chaos in the sense of Li-Yorke for first-order partial difference equation in R~3 by constructing spatial periodic orbits of specified period. A significant work on first-order partial difference equation is that Chen, Tian, and Shi [25] reformulated the first-order partial difference equation into the following discrete system:So some results and methods for the above system can be applied to first-order partial difference equations. By applying this approach, Shi [26] established some criteria of chaos for first-order partial difference equations. Recently. Shi. Yu. and Chen first established a chaotification scheme of first-order partial difference equations with the sawtooth function, and showed that the controlled system is chaotic in the sense of both Devaney and Li-Yorke [16] by using the above reformulation method. To the best of our knowledge, there are few results about chaotification of partial difference equations expect for [16]. In the present paper, we further study the chaotification problem of first-order partial difference equations.This paper are mainly divided into two parts. One is chaotification schemes of discrete dynamical systems, and the other is chaotification problems for a class of first-order partial difference equations. The later part includes three parts: chaotification schemes in the case that the system size is infinite; chaotification schemes in the case that the system size is finite and the solutions satisfy the periodic boundary condition; chaotification schemes in the case that the system size is finite and the solutions satisfy nonperiodic boundary conditions. Since chaotification schemes in the former two parts and the methods used are similar, and the later two parts are different, we put the former two parts together in Chapter 2, the third part in Chapter 3.In Chapter 1. we discuss chaotification problems of discrete dynamical systems in some special Banach spaces. Section 2 introduces some notations, basic concepts and lemmas. In Section 3. we establish two chaotification schemes of discrete dynamical systems in either finite-dimensional or infinite-dimensional Banach spaces with general controllers by applying the snap-back repeller theory, where the assumptions are weaker than those in Theorems 3.1 and 3.2 in [16], respectively. In Section 4 and 5, we establish three chaotification schemes of discrete dynamical systems in finite-dimensional Banach spaces with the mod-operation and sawtooth functions by applying the coupled-expanding theory, respectively.In Chapter 2, we study chaotification problems of first-order partial difference equations with the periodic boundary condition. We establish nine chaotification schemes for first-order partial difference equations. In some schemes, the map corresponding to the original system is only required to be continuous in a closed neighborhood and satisfy the Lipschitz condition. At the end of this chapter, one example is presented to demonstrate that the controlled system, which is generated by a chaotification scheme established in the paper, has complicated dynamical be- haviors with computer simulations.In Chapter 3, we study chaotification problems of first-order partial difference equations with nonperiodic boundary conditions in the case that the system size is finite. Some restrictions are imposed on the boundary conditions in order to make the controlled systems chaotic. Here we establish five chaotification schemes.