Table of Contents
Journal of Discrete Mathematics
Volume 2014, Article ID 538423, 5 pages
http://dx.doi.org/10.1155/2014/538423
Research Article

Chaotification for Partial Difference Equations via Controllers

1School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, China
2Department of Mathematics, Shandong University, Jinan, Shandong 250100, China
3Department of Mathematics, Shandong Jianzhu University, Jinan, Shandong 250100, China

Received 1 December 2013; Accepted 22 January 2014; Published 13 March 2014

Academic Editor: Zhan Zhou

Copyright © 2014 Wei Liang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 [68]. 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 [11].

Definition 3 (see [6]). A point is called a fixed point of (1) with (4) if ; that is, it is a fixed point of its induced system (7).

It follows from Definition 3 that is a fixed point of (1) with (4) if and only if it satisfies

Definition 4 (see [6]). Equation (1) with (4) is said to be chaotic in the sense of Li-Yorke (or Devaney) on if its induced system (7) is chaotic in the sense of Li-Yorke (or Devaney) on .

Recently, some chaotification schemes of the discrete system (3) were established in [7]; we list them as follows. For convenience, let be the set of all the maps that are times continuously differentiable in .

Lemma 5 (see [7]). Consider the controlled system in . Assume that (i) is a fixed point of and there exist positive constants and such that , , and for any ;(ii) satisfies the following conditions:(iia) and with ;(iib) is a fixed point of and there exists a point such that ;(iic) is an invertible linear operator for each and there exists a positive constant such that for any , Then, for any constant satisfying where , and for any neighborhood of , there exist a positive integer and a Cantor set such that is topologically conjugate to the symbolic dynamical system , where . Consequently, there exists a compact and perfect invariant set containing a Cantor set such that the controlled system is chaotic on in the sense of both Devaney and Li-Yorke.
A map is said to be an invertible linear map if it is a bounded linear map and bijective and if it has a bounded linear inverse map [6].

Lemma 6 (see [7]). Consider the controlled system where . Assume that (i)assumption (i) in Lemma 5 holds;(ii) satisfies the following conditions: (iia) and with ;(iib) is a fixed point of and there exists a point such that ;(iic) is an invertible linear operator for each and there exists a positive constant such that (12) holds for any . Then, for each constant satisfying all the results in Lemma 5 hold for therein.

3. Chaotification Problems for (1)

Assume that for . Let and be the first-order partial derivatives of for the 1st and the 2nd variables at . Let

Theorem 7. Consider the following system: with (4), where is a map and is a constant. Assume that (i) and ;(ii), and for any , where ;(iii) and there is a point satisfying ;(iv) and . Then, for and for any neighborhood of , there exist a Cantor set and a perfect as well as compact invariant set containing such that system (17) with (4) is chaotic on in the sense of both Li-Yorke and Devaney, where , is given in (16), and .

Proof. Assume that in the proof. System (17) with (4) can be rewritten as where is defined by (8), and
By assumptions (i), (iv), and Definition 3, is a fixed point of (1) with (4), and then , for , and . Further, for any ,where for with . So, for , Therefore,
Now, we prove that satisfies condition (ii) in Lemma 5. By (iii), , where . Furthermore, it follows from condition (ii) that and Obviously, is an invertible map, and it follows from condition (ii) that its inverse is Hence, for any , one can obtain that Therefore, is an invertible linear map. Hence,
In summary, both and meet all the conditions in Lemma 5. So this theorem holds.

Theorem 8. Assume that (1) and ;(2), for , and , for any ;(3) and there is a point satisfying ;(4) and . Then, for and for any neighborhood of , there exist a Cantor set and a perfect and compact invariant set containing such that with (4) being chaotic on in the sense of both Li-Yorke and Devaney, where ; is defined by (16), and .

Proof. The system induced by system (29) is where and are defined by (8) and (20), respectively. Similar to the proof of Theorem 7, it can be proved that and meet all the conditions of Lemma 6. Hence, Theorem 8 holds by Lemma 6.

4. An Example

Consider the controlled system (29) with (4), which is a special case of the discrete heat equation (see in [12]): where denotes the temperature at time and position of the rod. In system (29), By Corollary 5.1 [6], the original system is stable near the origin (see Figure 1(a)). In addition, , , and satisfy all the conditions of Theorem 8 with , , , , , , . Therefore, it follows from Theorem 8 that system (29) with (4) is chaotic in the sense of both Li-Yorke and Devaney for .

fig1
Figure 1: Computer simulation results, where , , and the initial value is , , and . (a) Simple dynamical behaviors for the original system (33); (b) simulation results of the system (29) for , which shows that there is a dense orbit around the origin and then there are complex dynamical behaviors in (29).

We take , for computer simulation. The simulation result is shown in Figure 1(b), which indicates that system (29) with (4) has a dense orbit around the origin and then has very complicated dynamical behaviors near the origin.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the RFDP of Higher Education of China (Grant 20100131110024), the NSFC (Grants 11126120, 11101246), the NNSF of Shandong Province (Grant ZR2011AM002), and the RFDP of Henan Polytechnic University (Grant B2011-032).

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