Chaotification for Partial Difference Equations via Controllers

Chaotification problems of partial difference equations are studied. Two chaotification schemes are established by utilizing the snap-back repeller theory of general discrete dynamical systems, and all the systems are proved to be chaotic in the sense of both Li-Yorke and Devaney. An example is provided to illustrate the theoretical results with computer simulations. 1. Introduction Consider the following first-order partial difference equation: where is time step, is the lattice point with , and is a map. Equation (1) is a discretization of the partial differential equation where is time variable, is spatial variable, and is a map. Equation (1) often appears in imaging and spatial dynamical systems and so forth [1, 2]. Chen and Liu studied the chaos for (1) in by constructing spatial periodic orbits in 2003 [3]. Chen et al. [4] reformulated (1) to a discrete system: Applying this approach, the second author of the present paper gave several criteria of chaos for (1) [5]. She with her coauthors established some chaotification schemes for (1) and proved all the systems are chaotic [6, 7]. Recently, Li studied the chaotification for delay difference equations [8]. However, only a few papers study the chaotification problems of (1) except for [6–8]. In this paper, the chaotification of (1) is studied. This paper is organized as follows. First, (1) is reformulated to a discrete system, and several concepts and lemmas are listed. Then, we give two chaotification schemes for (1) via controllers and prove that all the systems are chaotic in the sense of both Li-Yorke and Devaney. Finally, we give one example with computer simulation result to verify the theoretical predictions. 2. Preliminaries Consider the following boundary condition for (1): where is a map. For the initial condition where satisfies (4), (1) has a unique solution , and it can be easily proved by iterations. Let then (1) with (4) can be rewritten in the following form: where is said to be the induced map by and , and (7) is called the induced system by (1) with (4). Definition 1 (see [9]). Let be a metric space and let be a map. A subset of is called a scrambled set of if for any two different points , The map is said to be chaotic in the sense of Li-Yorke if there exists an uncountable scrambled set of . Definition 2 (see [10]). A map is said to be chaotic on in the sense of Devaney if (i) is topologically transitive in ;(ii)the periodic points of in are dense in ;(iii) has sensitive dependence on initial conditions in . Chaos of Devaney is stronger than that of Li-Yorke in some conditions