Research Article | Open Access

Zongcheng Li, "Anticontrol of Chaos for a Class of Delay Difference Equations Based on Heteroclinic Cycles Connecting Repellers", *Abstract and Applied Analysis*, vol. 2014, Article ID 260150, 8 pages, 2014. https://doi.org/10.1155/2014/260150

# Anticontrol of Chaos for a Class of Delay Difference Equations Based on Heteroclinic Cycles Connecting Repellers

**Academic Editor:**Ryan Loxton

#### Abstract

This paper is concerned with anticontrol of chaos for a class of delay difference equations via the feedback control technique. The controlled system is first reformulated into a high-dimensional discrete dynamical system. Then, a chaotification theorem based on the heteroclinic cycles connecting repellers for maps is established. The controlled system is proved to be chaotic in the sense of both Devaney and Li-Yorke. An illustrative example is provided with computer simulations.

#### 1. Introduction

Anticontrol of chaos (or called chaotification) is a process that makes a nonchaotic system chaotic or enhances a chaotic system to produce a stronger or different type of chaos. In recent years, it has been found that chaos can actually be useful under some circumstances, for example, in human brain analysis [1, 2], heartbeat regulation [3, 4], encryption [5], digital communications [6], and so forth. So, sometimes it is useful and even important to make a system chaotic or create new types of chaos. This has attracted increasing interest in research on chaotification of dynamical systems due to the great potential of chaos in many nontraditional applications.

In the pursuit of chaotifying discrete dynamical systems, a simple yet mathematically rigorous chaotification method was first developed by Chen and Lai [7–9] from a feedback control approach. After that, many chaotification schemes appeared for discrete dynamical systems based on the feedback control approach. The reader is referred to Chen and Shi [10] and Wang and Chen [11] for a survey of chaotification of discrete dynamical systems, as well as some references cited therein.

It is well known that the time delay appears in many realistic systems with feedback in science and engineering. Meanwhile, it has been shown that introducing delays to an undelayed system can be beneficial, especially for chaotic systems. This is the delayed feedback control method, which is widely used in chaos control. For continuous-time control systems, we refer to [12] and the references therein. In [12], the authors developed a unified computational approach for solving optimal state-delay control problems and proved that the approach was very effective for parameter identification and delayed feedback control. For discrete-time control systems, we refer to [13] and the references therein. In [13], the authors obtained the necessary and sufficient conditions for stabilizability of discrete-time systems via delayed feedback control.

To the best of our knowledge, there are few results on chaotification of delay difference equations. Motivated by the delayed feedback control method, we studied the chaotification problem for a class of delay difference equations with at least two fixed points. Since the sawtooth function and the sine function have some favourable properties, some of which are similar, they are often used as controllers; see [10, 11, 14–16] and so forth. Particularly, we succeeded in using the sine function as a controller to chaotify linear delay difference equations in [16]. This motivates us to use the sine function as the controller and employ a feedback control approach to study the chaotification problem for a class of delay difference equations. It is proved that the controlled system is chaotic in the sense of both Devaney and Li-Yorke, by applying the result of heteroclinic cycles connecting repellers; see [17] for the result and some references therein.

The rest of the paper is organized as follows. In Section 2, the chaotification problem under investigation is described, and some concepts, lemmas, and reformulation of the controlled system are introduced. In Section 3, the chaotification problem is studied and a chaotification scheme is established. An example is provided to illustrate the theoretical result with computer simulations in Section 4. Finally, Section 5 concludes the paper.

#### 2. Preliminaries

In this section, we describe the chaotification problem, give a reformulation of the delay difference equation, and introduce some fundamental concepts and lemmas, which will be used in the next section.

##### 2.1. Description of Chaotification Problem

In this paper, we consider chaotification of the following delay difference equation: where is a fixed integer and is a map. Equation (1) can be viewed as a discrete analogue of many one-dimensional delay differential equations by using the forward Euler scheme, such as the well-known Mackey-Glass equation where , is the delay, and is a one-dimensional nonlinear function. Equation (2) is a prototype for a retarded functional differential equation which has many applications in sciences. Special cases of (2) or its discretization have been studied by many authors; for instance, see [18–22] and the references therein.

From the above discussion, we see that the delay difference equation (1) is indeed very general. There exist many papers which are concerned with the existence of chaotic behavior for special forms of functions . For example, see [21, 22] and references therein. However, (1) cannot be chaotic for a more general class of functions . The object of this paper is to design a simple control input sequence such that the output of the controlled system exhibits chaos in the sense of both Devaney and Li-Yorke for a more general class of functions . The controller to be designed in this paper is in the form of where and are two undetermined positive parameters.

For convenience, define throughout the rest of the paper. Let and denote the first-order partial derivatives of with respect to the first and the second variables at the point , respectively. In the following, by and denote the open and closed balls of radius centered at .

##### 2.2. Reformulation

Here, we reformulate (1) and (3) into two special high-dimensional discrete dynamical systems. The following transform method is used by many researchers; for example, see [16, 21] and some references therein.

By setting equation (1) and the controlled system (3) with controller (4) can be written as the following -dimensional discrete systems on : respectively, where , and the maps are given by

The map is called the map induced by . System (7) is called the system induced by (3) in the Euclidean space . It is evident that a solution of (3) with an initial condition corresponds to a solution of system (7) with an initial condition . We say that the solution of system (7) is induced by the solution of (3). Therefore, we can investigate the dynamical behavior of system (3) by investigating that of its induced system (7) in . There is the same conclusion between systems (6) and (1). The idea in the above definitions is motivated by [21], where the authors say that the induced system and the original system are equivalent.

##### 2.3. Some Basic Concepts and Lemmas

Since Li and Yorke [23] first introduced a precise mathematical definition of chaos, there have been several different definitions of chaos, some stronger and some weaker, depending on the requirements in different problems; see [24–27] and so forth. For convenience, we list two definitions of chaos in the sense of Li-Yorke and Devaney, which are used in this paper.

*Definition 1. *Let be a metric space, a map, and a set of with at least two distinct points. Then is called a scrambled set of if for any two distinct points , (i);(ii).The map is said to be chaotic in the sense of Li-Yorke if there exists an uncountable scrambled set of .

*Remark 2. *There are three conditions in the original characterization of chaos in Li-Yorke’s theorem [23]. Besides conditions (i) and (ii), the third one is that, for all and for all periodic points of ,
But conditions (i) and (ii) together imply that the scrambled set contains at most one point that does not satisfy the above condition. Hence, the third condition is not essential and can be removed.

*Definition 3 (see [24]). *Let be a metric space. A map is said to be chaotic on in the sense of Devaney if (i)the set of the periodic points of is dense in ;(ii) is topologically transitive in ;(iii) has sensitive dependence on initial conditions in .

*Remark 4. *By the result of Banks et al. [28], conditions (i) and (ii) together imply condition (iii) if is continuous in . Consequently, condition (iii) is redundant in the above definition if is continuous in . It has been proved in [29] that, under some conditions, chaos in the sense of Devaney is stronger than that in the sense of Li-Yorke.

For convenience, some definitions of relevant concepts given in [30] are listed below.

*Definition 5 (see [30, Definitions 2.1 and 2.4]). *Let be a metric space and a map. A point is called an expanding fixed point (or a repeller) of in for some constant , if and there exists a constant such that
The constant is called an expanding coefficient of in . Furthermore, is called a regular expanding fixed point of in if is an interior point of . Otherwise, is called a singular expanding fixed point of in .

Now, we introduce some relative concepts for system (3), which are motivated by [15, Definitions 5.1 and 5.2]. There are identical concepts for system (1).

*Definition 6. *Consider the following.(i)A point is called an -periodic point of system (3) if is an -periodic point of its induced system (7); that is, , , and . In the special case of , is called a fixed point or a steady state of system (3).(ii)The concepts of density of periodic points, topological transitivity, sensitive dependence on initial conditions, and the invariant set for system (3) are defined similarly to those for its induced system (7) in .(iii)System (3) is said to be chaotic in the sense of Devaney (or Li-Yorke) on if its induced system (7) is chaotic in the sense of Devaney (or Li-Yorke) on .

The following two lemmas will be used in the next section.

Lemma 7. *Assume that the map in (1) is continuously differentiable in a neighborhood of with and satisfies
**
then the fixed point of system (6) is a regular expanding fixed point in some norm in .*

*Proof. *It follows from that is a fixed point of system (6). Since is continuously differentiable in a neighborhood of , is continuously differentiable in some neighborhood of . The Jacobian matrix of map in system (6) at is
and its characteristic equation is
From , we can show that all the eigenvalues of have absolute values larger than 1. Otherwise, suppose that there exists an eigenvalue of with ; then we get the following inequality:
which is a contradiction. Then it follows from [31, Theorem 4.3] that there exist a positive constant and a norm in such that is continuously differentiable in and is an expanding fixed point of in in the norm ; that is,
where is an expanding coefficient of in . Further, it follows from [30, Lemma 2.2] that is a regular expanding fixed point of system (6) in the norm in . This completes the proof.

Since the result in the following lemma is related to the one-sided symbolic dynamical system , we briefly recall some results of it for convenience. Let with the distance where and . Then is a complete metric space and a Cantor set. The shift map defined by is continuous. The dynamical system defined by is called a one-sided symbolic dynamical system. It has plentiful dynamical behaviors; we refer to [24, 26] for details. Particularly, it is chaotic in the sense of both Devaney and Li-Yorke and has a positive topological entropy.

Lemma 8 (see [17, Theorem 4.3 and Corollary 4.2]). *Let a map have (≥2) different fixed points . Assume that *(i)*for each , , is an expanding fixed point of in some norm ;*(ii)* has a -heteroclinic cycle connecting fixed points and is continuously differentiable in some neighborhood of each point on the cycle satisfying .**Then for each , , and for each neighborhood of , there exist a positive integer and a Cantor set such that is topologically conjugate to the one-sided symbolic dynamical system . Consequently, there exists a compact and perfect invariant set , containing the Cantor set , such that is chaotic on in the sense of Devaney as well as in the sense of Li-Yorke.*

*Remark 9. *Under the conditions in Lemma 8, there exists a positive integer , such that has a regular and nondegenerate -heteroclinic cycle connecting repellers in the Euclidean norm . Therefore, Lemma 8 can be briefly stated as the following: “a regular and nondegenerate heteroclinic cycle connecting repellers in implies chaos in the sense of both Devaney and Li-Yorke." We refer to [17] for details.

#### 3. A Chaotification Scheme

In this section, a chaotification scheme for the controlled system (3) with controller (4) is established for the case that the original system (1) has at least two fixed points. Here, we only need that the map of the original system is continuously differentiable in a region containing two fixed points. In the case that the fixed points are more than two, if two of them satisfy conditions in the following theorem, then we can choose the two fixed points to establish a chaotification scheme by using Lemma 8 for . If none of the two fixed points is the origin , then we can choose a transformation of coordinates such that one fixed point becomes the origin in a new coordinate system. Therefore, without loss of generality, we only discuss the case that the original system (1) has two fixed points and in .

Theorem 10. *Consider the controlled system (3) with controller (4). Assume that *(i)* is continuously differentiable in for some with , which implies that there exist positive constants and such that for any *(ii)*there exists a point with such that .**Then there exist two positive constants and satisfying
**
where is some positive integer, such that, for any and , the controlled system (3) with controller (4) is chaotic in the sense of both Devaney and Li-Yorke.*

*Proof. *In order to prove that system (3) with controller (4) is chaotic in the sense of both Devaney and Li-Yorke, we only need to prove that its reformulated system (7) is chaotic in the sense of both Devaney and Li-Yorke. Lemma 8 is used to prove this theorem. Thus it suffices to show that all the assumptions in Lemma 8 hold for .

For convenience, let and throughout the proof, where is an undetermined integer.

By assumption (i), we find that is continuously differentiable in and is continuously differentiable in . It is clear that and . It can also easily be proved that and are two fixed points of . From the last two relations of (18), it follows that
Similarly, it can also be shown that
Then, it follows from Lemma 7 that and are two regular expanding fixed points of system (7). That is, there exist two positive constants , and two norms , in such that and are two regular expanding fixed points of in in norm and in in norm , respectively. For convenience, we can choose and to be very small such that .

Next, we need to show that has a 2-heteroclinic cycle connecting fixed points and . There exist small intervals containing 0 and containing such that and . Consider the following equation:
Obviously, is continuously differentiable in . It is easy to see from (18) that
which implies that there exists a point with such that by the continuity of . Consider the following two equations:
With a similar method, we can also show that there exist two points with and with such that and . Similarly, the following equation
also has a solution with such that . We can choose a large positive integer such that for any the points , are in and , are in .

Take . It is clear that and . Set for . We can easily show that for , and

Take . It is also clear that and . Set for . It can also easily be shown that for , and
Therefore, has a 2-heteroclinic cycle connecting repellers and .

Finally, we will show that
We use the method of proof by contradiction to prove it. For simplicity, we only prove that condition (28) holds for . Suppose that . A direct calculation shows that, for any ,
Then it follows from (29) that
Hence, . On the other hand, it follows from that . Then, we get the following contradiction:
Therefore, . Similarly, we can prove that condition (28) holds for .

Therefore, all the assumptions in Lemma 8 are satisfied for . It follows from Lemma 8 that, for any and , there exists regular and nondegenerate 2-heteroclinic cycle repellers and . Consequently, system (7) and thus system (3) with controller (4) are chaotic in the sense of both Devaney and Li-Yorke. This completes the proof.

*Remark 11. *From the proof of Theorem 10, we see that there exists some positive integer such that, for any , , system (7) is chaotic in the sense of both Devaney and Li-Yorke. However, it is very difficult to determine the particular integer since the expanding area of a fixed point is not easy to obtain, and there are few methods to determine the concrete expanding area of a fixed point in the existing literatures. In practical problems, we can take the positive integer large enough such that controller (4) can be used to chaotify system (1).

*Remark 12. *There are many delay discrete dynamical systems which have more than two fixed points. As all or some of the fixed points satisfy assumptions in Theorem 10, which are not very strict conditions, we can choose two of them and use the chaotification scheme of this paper to chaotify these systems. Since the result of Theorem 10 follows from the result of Lemma 8, there will be many chaotic invariant sets when using the chaotification scheme to chaotify these systems. Therefore, the chaotic behaviors induced by a heteroclinic cycle connecting repellers seemed to be more complex than those induced by a single snap-back repeller. The difference between them will be our future research.

#### 4. An Example

In the last section, we present an example of chaotification for the delay difference equation (1) with computer simulations. The map in (1) is taken as follows:

It is obvious that is continuously differentiable on and satisfies condition (18); that is, for any , where and in condition (18). It is also clear that and there exists a point such that . Therefore, all the assumptions in Theorem 10 are satisfied. It follows from Theorem 10 that there exist two positive constants where is some positive integer, such that, for any and , the controlled system (3) with controller (4) is chaotic in the sense of both Devaney and Li-Yorke.

In fact, it is obvious that the solutions of the uncontrolled system (6) are bounded in for . There are three fixed points for the uncontrolled system (6); that is, , , . One can easily verify that is an unstable fixed point and that and are two stable fixed points. When we take an initial condition , the solution of the uncontrolled system (6) should tend to the stable fixed point when tends to infinity. This conforms to Figure 1, where the curve tends from to the stable fixed point and is taken from 0 to 20000 for simulation. There is a similar conclusion for the case . For simplicity, we omit it.

In order to help better visualize the theoretical result of Theorem 10, we take , and for computer simulations. Both of them satisfy the above conditions. Consequently, the controlled system (7) should be chaotic in the sense of both Devaney and Li-Yorke. It is obvious that the solutions of controlled system (7) are bounded in and for and , respectively. The simulated results in Figures 2 and 4 show that the controlled system (7) indeed has complex behaviors, where is taken from 0 to 20000.

In summary, the simulated results show that the uncontrolled system (6), that is, system (1), has simple dynamical behaviors, and the controlled system (7), that is, system (3) with controller (4), has complex dynamical behaviors; see Figures 1, 2, 3, and 4. The graphs presented indeed display an expected well-behaved structure in one moment, and after the chaotification treatment the referred system presents some behavior that resembles chaos. Nothing is less than expected.

#### 5. Conclusion

In this paper, we consider anticontrol of chaos for a class of delay difference equations via the feedback control technique. Based on the result that a regular and nondegenerate heteroclinic cycle connecting repellers for maps implies chaos, we establish a chaotification theorem. The controlled system is proved to be chaotic in the sense of both Devaney and Li-Yorke. It is noted that there are many delay discrete dynamical systems which have more than two fixed points. As all or some of the fixed points satisfy assumptions in Theorem 10, which are not very strict conditions, we can choose two of them and use the chaotification scheme established in this paper to chaotify these systems. Numerical simulations confirm the theoretical analysis. However, the map of the original system needs to satisfy some conditions in Theorem 10. Therefore, it is very interesting to explore the chaotification scheme for more general maps, which will be our further research.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant nos. 11101246 and 11101247). The author would like to thank the editor and the anonymous referees for their valuable comments and suggestions, which have led to an improvement of this paper.

#### References

- W. J. Freeman, “Chaos in the brain: possible roles in biological intelligence,”
*International Journal of Intelligent Systems*, vol. 10, pp. 71–88, 1995. View at: Google Scholar - S. J. Schiff, K. Jerger, D. H. Duong, T. Chang, M. L. Spano, and W. L. Ditto, “Controlling chaos in the brain,”
*Nature*, vol. 370, pp. 615–620, 1994. View at: Google Scholar - M. E. Brandt and G. R. Chen, “Bifurcation control of two nonlinear of models of cardiac activity,”
*IEEE Transactions on Circuits and Systems I*, vol. 44, pp. 1031–1034, 1997. View at: Google Scholar - W. L. Ditto, M. L. Spano, J. Nelf et al., “Control of human atrial fibrillation,”
*International Journal of Bifurcation and Chaos*, vol. 10, pp. 593–602, 2000. View at: Google Scholar - G. Jakimoski and L. G. Kocarev, “Chaos and cryptography: block encryption ciphers based on chaotic maps,”
*IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications*, vol. 48, no. 2, pp. 163–169, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. G. Kocarev, M. Maggio, M. Ogorzalek, L. Pecora, and K. Yao, “Special issue on applications of chaos in modern communication systems,”
*IEEE Transactions on Circuits and Systems I*, vol. 48, pp. 1385–1527, 2001. View at: Google Scholar - G. R. Chen and D. J. Lai, “Feedback control of Lyapunov exponents for discrete-time dynamical systems,”
*International Journal of Intelligent Systems*, vol. 6, pp. 1341–1349, 1996. View at: Google Scholar - G. R. Chen and D. J. Lai, “Anticontrol of chaos via feedback,” in
*Proceedings of the IEEE Conference on Decision and Control*, pp. 367–372, San Diego, Calif, USA, 1997. View at: Google Scholar - G. R. Chen and D. J. Lai, “Feedback anticontrol of discrete chaos,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 8, no. 7, pp. 1585–1590, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. R. Chen and Y. M. Shi, “Introduction to anti-control of discrete chaos: theory and applications,”
*Philosophical Transactions of the Royal Society of London A. Mathematical, Physical and Engineering Sciences*, vol. 364, no. 1846, pp. 2433–2447, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X. F. Wang and G. R. Chen, “Chaotification via arbitrarily small feedback controls: theory, method, and applications,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 10, no. 3, pp. 549–570, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Q. Chai, R. Loxton, K. L. Teo, and C. Yang, “A class of optimal state-delay control problems,”
*Nonlinear Analysis. Real World Applications*, vol. 14, no. 3, pp. 1536–1550, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. D. Zhu and Y. P. Tian, “Necessary and sufficient conditions for stabilizability of discretetime systems via delayed feedback control,”
*Physics Letters A*, vol. 343, pp. 95–107, 2005. View at: Google Scholar - X. F. Wang and G. R. Chen, “Chaotifying a stable map via smooth-amplitude high-frequency feedback control,”
*International Journal of Circuit Theory and Applications*, vol. 28, pp. 305–312, 2000. View at: Google Scholar - Y. M. Shi, P. Yu, and G. R. Chen, “Chaotification of discrete dynamical systems in Banach spaces,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 16, no. 9, pp. 2615–2636, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. C. Li, “Chaotification for linear delay difference equations,”
*Advances in Difference Equations*, vol. 2013, article 59, 2013. View at: Google Scholar - Z. C. Li, Y. M. Shi, and W. Liang, “Discrete chaos induced by heteroclinic cycles connecting repellers in Banach spaces,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 72, no. 2, pp. 757–770, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Giannakopoulos and A. Zapp, “Local and global Hopf bifurcation in a scalar delay differential equation,”
*Journal of Mathematical Analysis and Applications*, vol. 237, no. 2, pp. 425–450, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. H. Jiang, Y. Shen, J. G. Jian, and X. X. Liao, “Stability, bifurcation and a new chaos in the logistic differential equation with delay,”
*Physics Letters A*, vol. 350, no. 3-4, pp. 221–227, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. M. He, X. Lai, and A. Y. Hou, “Stability and Neimark-Sacker bifurcation of numerical discretization of delay differential equations,”
*Chaos, Solitons and Fractals*, vol. 41, no. 4, pp. 2010–2017, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Huang and X. Zou, “Co-existence of chaos and stable periodic orbits in a simple discrete neural network,”
*Journal of Nonlinear Science*, vol. 15, no. 5, pp. 291–303, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Peng, J. C. Yu, and X. J. Wang, “Complex dynamics in simple delayed two-parameterized models,”
*Nonlinear Analysis. Real World Applications*, vol. 13, no. 6, pp. 2530–2539, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. Y. Li and J. A. Yorke, “Period three implies chaos,”
*The American Mathematical Monthly*, vol. 82, no. 10, pp. 985–992, 1975. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. L. Devaney,
*An Introduction to Chaotic Dynamical Systems*, Addison-Wesley Publishing Company, New York, NY, USA, 1987. View at: MathSciNet - M. Martelli, M. Dang, and T. Seph, “Defining chaos,”
*Mathematics Magazine*, vol. 71, no. 2, pp. 112–122, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Robinson,
*Dynamical Systems: Stability, Symbolic Dynamics and Chaos*, CRC Press, Boca Raton, Fla, USA, 1995. View at: MathSciNet - S. Wiggins,
*Global Bifurcations and Chaos*, Springer, New York, NY, USA, 1988. View at: Publisher Site | MathSciNet - J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, “On Devaney's definition of chaos,”
*The American Mathematical Monthly*, vol. 99, no. 4, pp. 332–334, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. Huang and X. D. Ye, “Devaney's chaos or 2-scattering implies Li-Yorke's chaos,”
*Topology and its Applications*, vol. 117, no. 3, pp. 259–272, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. M. Shi and G. R. Chen, “Chaos of discrete dynamical systems in complete metric spaces,”
*Chaos, Solitons and Fractals*, vol. 22, no. 3, pp. 555–571, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. M. Shi and G. R. Chen, “Discrete chaos in Banach spaces,”
*Science in China A. Mathematics*, vol. 48, no. 2, pp. 222–238, 2005, Chinese version: vol. 34, pp. 595–609, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2014 Zongcheng Li. 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.