Abstract
We study the chaotification problem for a class of delay difference equations by using the snap-back repeller theory and the feedback control approach. We first study the stability and expansion of fixed points and establish a criterion of chaos. Then, based on this criterion of chaos and the feedback control approach, we establish a chaotification scheme such that the controlled system is chaotic in the sense of both Devaney and Li-Yorke when the parameters of this system satisfy some mild conditions. For illustrating the theoretical result, we give some computer simulations.
1. Introduction
Research on chaos control has attracted a lot of interest from many scientists and mathematicians. There are two directions in chaos control, that is, control of chaos and anticontrol of chaos (or called chaotification). The former regarded chaos as harmful. So many earlier works focused on stabilizing a chaotic system, which was regarded as the traditional control. The reader is referred to the monographs [1–3] for more details. However, in recent years, it has been found that chaos can actually be very useful in some applications; a typical example is chaos-based cryptography [4]. Hence, sometimes it is useful and even important to make a nonchaotic system chaotic, or to make a chaotic system produce a stronger or different type of chaos. This progress is called chaotification or anticontrol of chaos.
In research on chaotification for discrete dynamical systems, a mathematically rigorous and effective chaotification method was first proposed by Chen and Lai [5–7], where they first used the feedback control technique. This method plays an important role in studying chaotification problems of discrete dynamical systems. For a survey on chaotification of discrete dynamical systems, one can see [8] and some references therein.
To the best of our knowledge, although there already exist many works on chaotification of discrete dynamical systems, there are few results on chaotification of delay difference equations. Motivated by the feedback control approach, we have succeeded in studying the chaotification problems on linear delay difference equations [9] and a class of delay difference equations [10]. In the two papers, we use the sine functions as controllers to establish some chaotification schemes. The reason of using this type of controllers is that the sine function has some favorable properties and this designed controller is also simple, cheap, and implementable in real engineering applications (see [8–10] and the references therein). In the chaotification theorem of [10], the delay difference equations need to have at least two fixed points. However, there are also many delay difference equations with only one fixed point, which cannot satisfy the above condition. This motivates us to study this case. In this paper, we will apply the feedback control approach and the snap-back repeller theory to study chaotification for a class of delay difference equations with at least one fixed point.
This paper is organized as follows. In Section 2, we give some basic concepts and one lemma. In Section 3, we study the stability and expansion of fixed points and establish a criterion of chaos. Based on this criterion of chaos, we establish a chaotification scheme for a class of delay difference equations with at least one fixed point. Then, we give some computer simulations to illustrate the theoretical result. Finally, we conclude this paper in Section 4.
2. Preliminaries
Up to now, there is no unified definition of chaos in mathematics. For convenience, we present two definitions of chaos, which will be used in this paper.
Definition 1 (see [11]). Let be a metric space, let be a map, and let be a set of with at least two distinct points. Then 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 .
Remark 2. The term “chaos” was first used by Li and Yorke [12] for a map on a compact interval. Following the work of Li and Yorke, Zhou [11] gave the above definition of chaos for a topological dynamical system on a general metric space.
Definition 3 (see [13]). Let be a metric space. A map is said to be chaotic on in the sense of Devaney if (i)is topologically transitive in ;(ii)the periodic points of are dense in ;(iii) has sensitive dependence on initial conditions in .
Remark 4. In [14], Huang and Ye showed that chaos in the sense of Devaney is stronger than that in the sense of Li-Yorke under some conditions.
The following criterion of chaos is established by Shi et al., which plays an important role in the present paper.
Lemma 5 (see [15, Theorem 2.1]; [16, Theorem 4.4]). Let be a map with a fixed point . Assume that (i) is continuously differentiable in a neighborhood of and all the eigenvalues of have absolute values larger than 1, which implies that there exist a positive constant and a norm in such that is expanding in in , where is the closed ball of radius centered at in ;(ii) is a snap-back repeller of with , , for some and some positive integer , where is the open ball of radius centered at in . Furthermore, is continuously differentiable in some neighborhoods of , respectively, and for , where for .
Then for each neighborhood of , there exist a positive integer and a Cantor set such that is topologically conjugate to the 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 and has a dense orbit in .
Remark 6. In 1978, Marotto [17] first gave the concept of snap-back repeller for maps in . Later, in 2004, Shi and Chen [18] extended this concept to general metric spaces. According to the classifications of snap-back repellers for maps in metric spaces in [18], the snap-back repeller given by Marotto [17] is regular and nondegenerate. For more details on snap-back repeller, we refer to [16–19] and the references therein. We can easily conclude that the point in Lemma 5 is a regular and nondegenerate snap-back repeller. Hence, Lemma 5 can be summed as a single word: “a regular and nondegenerate snap-back repeller in implies chaos in the sense of both Devaney and Li-Yorke.” For more details, one can see [15, 16].
3. Chaotification Based on Snap-Back Repellers
In this paper, we will study the chaotification problem of a delay difference equation, chaotic or not, in the form of where is a fixed integer and is a map. Equation (2) is a discrete analogue of many one-dimensional delay differential equations, such as the well known Mackey-Glass equation.
The objective here is to design a control input sequence such that the output of the controlled system is chaotic in the sense of both Devaney and Li-Yorke. In our earlier paper [10], by using the result that heteroclinic cycles connecting repellers imply chaos established in [20], we have studied the chaotification problem of (2) for the case where (2) has at least two fixed points. However, there are also many delay discrete dynamical systems which only have one fixed point. Then, the chaotification scheme established in [10] cannot be used. In this paper, we will study the chaotification problem for the case where (2) has at least one fixed point. We design the controller as follows: where is any given constant, and are two undetermined parameters, and is the classical sawtooth function; that is, while denotes the integer set. Many researchers have succeeded in using the sawtooth function as a controller to chaotify discrete dynamical systems (see [15, 21] and the references therein).
SetThen (2) and the controlled system (3) with controller (4) can be transformed into the following -dimensional discrete systems on : respectively, where and the maps , .
As defined in [10], the maps and are called the maps induced by and , respectively, where . Systems (7) and (8) are called the systems induced by (2) and (3) in the Euclidean space , respectively. System (3) is said to be chaotic in the sense of Devaney (or Li-Yorke) on if its induced system (8) is chaotic in the sense of Devaney (or Li-Yorke) on .
In the following, without loss of generality and for simplicity, we can suppose that the origin is always a fixed point of the induced system (7). Otherwise, if none of the fixed points is the origin , then we can choose a transformation of coordinates such that one of the fixed points becomes the origin in a new coordinate system. Then the map in (2) satisfies , throughout the rest of the paper.
It is well known that the stability and expansion of a map at a fixed point has a close relationship with the modulus of the eigenvalues of its derivative operator when the map is differentiable at the fixed point. Suppose that is differentiable at ; then the induced map is differentiable at . Let and denote the first partial derivatives of with respect to the first and the second variables at the point , respectively. Then we can get the following results on stability and expansion of the fixed point of the induced system (7).
Theorem 7. Assume that . Denote , . (i)If is differentiable at , then, for , the fixed point of system (7) is asymptotically stable if and only if ; and for , the fixed point of system (7) is asymptotically stable if and only if , and where is the solution in of equation .(ii)If is continuously differentiable in a neighborhood of and , then the fixed point of system (7) is a regular expanding fixed point in some norm in .
Proof. When , it is easy to obtain that all the eigenvalues of have absolute values less than 1 if and only if . So, the result in (i) holds. When , the result in (i) can be directly derived by using Theorem 3 in [22]. Result (ii) can be derived from Lemma 2.1 of [10]. This completes the proof.
Now, we establish a criterion of chaos for the induced system (7).
Theorem 8. Let be a map and let it be continuously differentiable in a neighborhood of with . Assume that (i), which implies that there exist a positive constant and a norm in such that is continuously differentiable in and is a regular expanding fixed point of in in the norm , where is the closed ball of radius centered at in ;(ii) there exists a point with , such that is continuously differentiable in a neighborhood of with , ,(iia) when , there exist such that , , is continuously differentiable in a neighborhood of , and (iib) when , there exist such that , , is continuously differentiable in a neighborhood of , and Then the induced system (7), and consequently system (2), is chaotic in the sense of both Devaney and Li-Yorke.
Proof. We will apply Lemma 5 to prove this theorem. So, we only need to show that all the assumptions in Lemma 5 are satisfied.
It follows from assumption (i) and the second conclusion of Theorem 7 that is continuously differentiable in , all the eigenvalues of have absolute values larger than 1, and is a regular expanding fixed point of in in some norm of . Therefore, condition (i) in Lemma 5 is satisfied.
Next, we will show that is a snap-back repeller of in the norm . In the following, we will show that there exists a point with satisfying which implies that is a snap-back repeller of .
For the case where , it follows from condition (iia) that there exists a point , , such that , , and .
For the case where , it follows from condition (iib) that there exists a point , , such that , for , and
It is obvious that is continuously differentiable in some neighborhoods of for . So, we need to show that the following holds: If is differentiable at , then a direct calculation shows that From condition (i), we get that , which together with condition (ii) and (14) implies that conclusion (13) holds for and .
Therefore, all the assumptions in Lemma 5 are satisfied. Then the induced system (7), and consequently system (2), is chaotic in the sense of both Devaney and Li-Yorke. This completes the proof.
Remark 9. Since is a function of two variables, the conditions in (ii) of Theorem 8 are not very strict conditions.
Based on Theorem 8, a chaotification scheme for the controlled system (3) with controller (4) is established in the following.
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 any given constant and is some 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. We will use Theorem 8 to prove this theorem. So, it suffices to show that the map satisfies all the assumptions in Theorem 8.
For convenience, let , throughout the proof, where is an undetermined integer. Let denote the induced map of .
It is obvious that the function is continuously differentiable in . Then, from assumption (i), we obtain that is continuously differentiable in with , is a fixed point of the map , and is continuously differentiable in . It follows from the last two relations of (15) that So condition (i) in Theorem 8 holds. Consequently, there exist a positive constant and a norm in such that is continuously differentiable in and is a regular expanding fixed point of in in the norm , where is the closed ball of radius centered at in . Further, suppose that is an arbitrary neighborhood of in . Then there exists a neighborhood of 0 such that .
Next, we need to show that satisfies assumption (ii) in Theorem 8. It is obvious that and is continuously differentiable in a neighborhood of . So, is continuously differentiable in a neighborhood of . From assumption (ii) and condition (15), it follows that For , let It follows from assumption (i) and the definition of sawtooth function that is continuous in . From the first relation of (15), we get that Therefore, by the intermediate value theorem, there exists a point with , such that ; that is, . Similarly, let It is also clear that is continuous in . It also follows from the first relation of (15) that By the intermediate value theorem again, there exists a point with , such that ; that is, . It is clear that and are both in . So we can take a sufficiently large integer , such that , with , and for any . It can easily be proved that is continuously differentiable in some neighborhoods of and . Now, we show . Otherwise, if , then the following equality holds: Hence, , which is a contradiction. Similarly, we can prove that . Hence, condition (iia) in Theorem 8 holds.
For , the determination of can be derived from the proof of the above paragraph as . That is, there exists a point in , such that ; that is, . Set With a similar method to the above paragraph, we can also get that there exists a point in such that , which implies that . So we can also take a sufficiently large integer , such that , with , and for any . It can also easily be proved that is continuously differentiable in some neighborhoods of and . The proofs of and are similar to the above paragraph. So, the details are omitted.
Finally, let . Then condition (ii) in Theorem 8 is satisfied for . Therefore, for any and , the controlled system (3) with controller (4) is chaotic in the sense of both Devaney and Li-Yorke. The proof is complete.
Remark 11. It is clear that the classical sinusoidal function has similar geometric properties to the sawtooth function. So the following function can also be used as a controller to chaotify system (2), where is some constant to be determined and is the controlled parameter. In fact, with a similar argument to the proof of Theorem 10, one can show that there also exist two positive constants and such that for any constant and the result in Theorem 10 holds.
Remark 12. In [10], a similar result is given for a class of maps with at least two fixed points. In such a case, the two chaotification schemes obtained in [10] and this paper can be used. However, there will be many chaotic invariant sets as pointed out in Lemma 2.2 of [10] when using the chaotification scheme in [10]. It seems that the chaotic behaviors induced by a heteroclinic cycle connecting repellers are more complex than that induced by a single snap-back repeller. The difference between them will be our further research. But when the original system only has one fixed point, the chaotification scheme obtained in [10] cannot be used. Then, we can use the chaotification scheme obtained in this paper to chaotify this system.
Remark 13. Since the point in assumption (ii) of Theorem 10 can be negative, the value of determined in this paper can be a negative integer. In addition, it is very difficult to determine the concrete value since the concrete expanding area of a fixed point is not easy to obtain. To the best of our knowledge, there are few methods to determine the concrete expanding area of a fixed point in the existing literatures. So, in practical problems, we can take large enough such that the chaotification scheme can be effective.
In the last part of this section, we give an example to illustrate the theoretical result of Theorem 10.
Example 14. We take the map in (2) as the following: It is clear that is continuously differentiable in and satisfies . Without loss of generality, we take in Theorem 10. Then, for any , we get that Hence, we take , , and in assumption (i) of Theorem 10. It is also clear that the equation has a nonzero solution , which lies in . Therefore, all the assumptions in Theorem 10 are satisfied. Here, we take the constant in controller (4). Then, 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, there is only one fixed point in the uncontrolled system (7). It is obvious that and , which imply that is asymptotically stable from result (i) in Theorem 7. It is also clear that all the solutions of the uncontrolled system (7) are bounded if the initial values are taken from . Therefore, if we take an initial condition , then the solution of the uncontrolled system (7) should tend to the asymptotically stable fixed point when tends to infinity. This is confirmed in Figures 1 and 3.
Here, we take , , , , , and from 0 to 20000 for computer simulations. The simulated results show that the original system (7) has simple dynamical behaviors, and the controlled system (8) has complex dynamical behaviors; see Figures 1–4.
It should be pointed out that the relative existing chaotification scheme in [10] is not available for this map since there is only one fixed point.
4. Conclusion
In this paper, we study the chaotification problem for a class of delay difference equations with at least one fixed point. We first establish a criterion of chaos by using the snap-back repeller theory. Then, based on this criterion of chaos and the feedback control approach, we establish a chaotification scheme. We have proved that the controlled system is chaotic in the sense of both Devaney and Li-Yorke when the parameters of the system satisfy some mild conditions. Numerical simulations confirm the theoretical analysis. The chaotification problem for more general maps in the original system will be our further research.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grants 11101246, 61273088, and 10971120), the Nature Science Foundation of Shandong Province (Grant ZR2010FM010), and the Postdoctoral Science Foundation of China (Grant 2014M561908).