MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 10.1155/2015/917137 917137 Research Article Chaotification for a Class of Delay Difference Equations Based on Snap-Back Repellers http://orcid.org/0000-0002-6986-2886 Li Zongcheng 1, 2 Liu Shutang 1 Li Wei 3 Zhao Qingli 2 Zhang Xinguang 1 College of Control Science and Engineering, Shandong University, Jinan, Shandong 250061 China sdu.edu.cn 2 School of Science, Shandong Jianzhu University, Jinan, Shandong 250101 China sdjzu.edu.cn 3 Department of Public Foundation, Shandong Radio and TV University, Jinan, Shandong 250010 China 2015 12102015 2015 29 04 2015 12 07 2015 12102015 2015 Copyright © 2015 Zongcheng Li 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.

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  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 . 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 , 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  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  and a class of delay difference equations . 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  and the references therein). In the chaotification theorem of , 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 [<xref ref-type="bibr" rid="B11">11</xref>]).

Let (X,d) be a metric space, let F:XX be a map, and let S be a set of X with at least two distinct points. Then S is called a scrambled set of F if, for any two different points x, yS, (1)liminfndFnx,Fny=0,limsupndFnx,Fny>0.The map F is said to be chaotic in the sense of Li-Yorke if there exists an uncountable scrambled set S of F.

Remark 2.

The term “chaos” was first used by Li and Yorke  for a map on a compact interval. Following the work of Li and Yorke, Zhou  gave the above definition of chaos for a topological dynamical system on a general metric space.

Definition 3 (see [<xref ref-type="bibr" rid="B13">13</xref>]).

Let (X,d) be a metric space. A map F:VXV is said to be chaotic on V in the sense of Devaney if

F  is topologically transitive in V;

the periodic points of F are dense in V;

F has sensitive dependence on initial conditions in V.

Remark 4.

In , 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 [<xref ref-type="bibr" rid="B15">15</xref>, Theorem 2.1]; [<xref ref-type="bibr" rid="B16">16</xref>, Theorem 4.4]).

Let F:RnRn be a map with a fixed point zRn. Assume that

F is continuously differentiable in a neighborhood of z and all the eigenvalues of DF(z) have absolute values larger than 1, which implies that there exist a positive constant r and a norm · in Rn such that F is expanding in B¯r(z) in ·, where B¯r(z) is the closed ball of radius r centered at z in (Rn,·);

z is a snap-back repeller of F with Fm(x0)=z, x0z, for some x0Br(z) and some positive integer m, where Br(z) is the open ball of radius r centered at z in (Rn,·). Furthermore, F is continuously differentiable in some neighborhoods of x0,x1,,xm-1, respectively, and detDF(xj)0 for 0jm-1, where xj=F(xj-1) for 1jm-1.

Then for each neighborhood U of z, there exist a positive integer k>m and a Cantor set ΛU such that Fk:ΛΛ is topologically conjugate to the symbolic dynamical system σ:2+2+. Consequently, there exists a compact and perfect invariant set VRn, containing the Cantor set Λ, such that F is chaotic on V in the sense of Devaney as well as in the sense of Li-Yorke and has a dense orbit in V.

Remark 6.

In 1978, Marotto  first gave the concept of snap-back repeller for maps in Rn. Later, in 2004, Shi and Chen  extended this concept to general metric spaces. According to the classifications of snap-back repellers for maps in metric spaces in , the snap-back repeller given by Marotto  is regular and nondegenerate. For more details on snap-back repeller, we refer to  and the references therein. We can easily conclude that the point z 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 Rn 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 (2)xn+1=fxn-k,xn,n0,where k1 is a fixed integer and f:DR2R 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 vn such that the output of the controlled system (3)xn+1=fxn-k,xn+vn,n0,is chaotic in the sense of both Devaney and Li-Yorke. In our earlier paper , by using the result that heteroclinic cycles connecting repellers imply chaos established in , 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  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: (4)vn=αsawɛβxn-k,where ɛ>0 is any given constant, α and β are two undetermined parameters, and sawɛ(·) is the classical sawtooth function; that is, (5)sawɛx=-1mx-2mɛ,2m-1ɛx<2m+1ɛ,mZ,while Z 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).

Set(6)ujnxn+j-k-1,1jk+1,n0.Then (2) and the controlled system (3) with controller (4) can be transformed into the following k+1-dimensional discrete systems on Rk+1: (7)un+1=Fun,(8)un+1=Gun,respectively, where u=(u1,u2,,uk+1)TRk+1 and the maps F, G:Rk+1Rk+1.

As defined in , the maps F and G are called the maps induced by f and g, respectively, where g(x,y)f(x,y)+αsawε(βx). Systems (7) and (8) are called the systems induced by (2) and (3) in the Euclidean space Rk+1, respectively. System (3) is said to be chaotic in the sense of Devaney (or Li-Yorke) on VRk+1 if its induced system (8) is chaotic in the sense of Devaney (or Li-Yorke) on VRk+1.

In the following, without loss of generality and for simplicity, we can suppose that the origin O(0,,0)TRk+1 is always a fixed point of the induced system (7). Otherwise, if none of the fixed points is the origin O, then we can choose a transformation of coordinates such that one of the fixed points becomes the origin O in a new coordinate system. Then the map f in (2) satisfies f(0,0)=0, 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 f is differentiable at (0,0); then the induced map F is differentiable at O. Let fx(x,y) and fy(x,y) denote the first partial derivatives of f with respect to the first and the second variables at the point (x,y), respectively. Then we can get the following results on stability and expansion of the fixed point O of the induced system (7).

Theorem 7.

Assume that k<. Denote afy(0,0), bfx(0,0).

If f is differentiable at (0,0), then, for a=0, the fixed point O of system (7) is asymptotically stable if and only if |b|<1; and for a0, the fixed point O of system (7) is asymptotically stable if and only if |a|<(k+1)/k, and (9)a-1<-b<a2+1-2acosϕ1/2,fork  odd,a+b<1,b<a2+1-2acosϕ1/2,for  k  even,

where ϕ is the solution in (0,π/(k+1)) of equation sin(kθ)/sin[(k+1)θ]=1/|a|.

If f is continuously differentiable in a neighborhood of (0,0) and |b|-|a|>1, then the fixed point O of system (7) is a regular expanding fixed point in some norm in Rk+1.

Proof.

When a=0, it is easy to obtain that all the eigenvalues of DF(O) have absolute values less than 1 if and only if |b|<1. So, the result in (i) holds. When a0, the result in (i) can be directly derived by using Theorem 3 in . Result (ii) can be derived from Lemma 2.1 of . This completes the proof.

Now, we establish a criterion of chaos for the induced system (7).

Theorem 8.

Let f:DR2R be a map and let it be continuously differentiable in a neighborhood of (0,0) with f(0,0)=0. Assume that

|fx(0,0)|-|fy(0,0)|>1, which implies that there exist a positive constant r and a norm · in Rk+1 such that F is continuously differentiable in B¯r(O) and O is a regular expanding fixed point of F in B¯r(O) in the norm ·, where B¯r(O) is the closed ball of radius r centered at O in (Rk+1,·);

there exists a point uD with u0, such that f is continuously differentiable in a neighborhood of (u,0) with f(u,0)=0, fx(u,0)0,

when k=1, there exist x1,x2(-r,r) such that x12+x220, (x1,x2)TBr(O), f is continuously differentiable in a neighborhood of (x2,u), and (10)fx2,u=0,fx1,x2=u,fxx2,u0,fxx1,x20;

when k>1, there exist x1,x2(-r,r) such that x12+x220, (x1,x2,0,,0)TBr(O), f is continuously differentiable in a neighborhood of (x2,u), and (11)fx2,u=0,fx1,0=u,fxx2,u0,fxx1,00.

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 F is continuously differentiable in B¯r(O), all the eigenvalues of DF(O) have absolute values larger than 1, and O is a regular expanding fixed point of F in B¯r(O) in some norm · of Rk+1. Therefore, condition (i) in Lemma 5 is satisfied.

Next, we will show that O is a snap-back repeller of F in the norm ·. In the following, we will show that there exists a point O0W with O0O satisfying (12)Fk+2O0=O,which implies that O is a snap-back repeller of F.

For the case where k=1, it follows from condition (iia) that there exists a point O0=(x1,x2)TBr(O), O0O, such that O1=F(O0)=(x2,u)T, O2=F2(O0)=(u,0)T, and F3(O0)=O.

For the case where k>1, it follows from condition (iib) that there exists a point O0=(x1,x2,0,,0)TBr(O), O0O, such that O1=F(O0)=(x2,0,,0,u)T, Oj=Fj(O0)=(0,,0,u,0,,0j)T for 2jk+1, and Fk+2(O0)=O.

It is obvious that F is continuously differentiable in some neighborhoods of OjF(Oj-1) for 1jk+1. So, we need to show that the following holds: (13)detDFOj0,0jk+1.If F is differentiable at u=(u1,,uk+1)TRk+1, then a direct calculation shows that (14)detDFu=-1kfxu1,uk+1.From condition (i), we get that |fx(0,0)|>1+|fy(0,0)|>0, which together with condition (ii) and (14) implies that conclusion (13) holds for k=1 and k>1.

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 f 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

f is continuously differentiable in [-r,r]2 for some r>0 with f(0,0)=0, which implies that there exist positive constants M and N such that for any (x,y)[-r,r]2(15)fx,yM,fxx,yN,fyx,yN;

there exists a point u(-r,r) with u0 such that f(u,0)=0.

Then there exist two positive constants α0 and β0 satisfying (16)α0>M+uɛ,β02m0ɛu>max1+2Nα0,3ɛr,where ɛ>0 is any given constant and m0 is some integer, such that, for any α>α0 and β=β0, 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 g(x,y)f(x,y)+αsawɛ(βx) satisfies all the assumptions in Theorem 8.

For convenience, let α>α0, β=2mɛ/u>max{(1+2N)/α0,3ɛ/r} throughout the proof, where m is an undetermined integer. Let G denote the induced map of g.

It is obvious that the function sawɛ(βx) is continuously differentiable in (-ɛ/β,ɛ/β). Then, from assumption (i), we obtain that g is continuously differentiable in (-ɛ/β,ɛ/β)2 with g(0,0)=0, O is a fixed point of the map G, and G is continuously differentiable in (-ɛ/β,ɛ/β)k+1. It follows from the last two relations of (15) that (17)gx0,0=fx0,0+αβαβ-fx0,0αβ-N>1+N1+fy0,0=1+gy0,0.So condition (i) in Theorem 8 holds. Consequently, there exist a positive constant r and a norm · in Rk+1 such that G is continuously differentiable in B¯r(O) and O is a regular expanding fixed point of G in B¯r(O) in the norm ·, where B¯r(O)(-ɛ/β,ɛ/β)k+1 is the closed ball of radius r centered at O in (Rk+1,·). Further, suppose that WBr(O) is an arbitrary neighborhood of O in Rk+1. Then there exists a neighborhood U of 0 such that U×U××UW.

Next, we need to show that g satisfies assumption (ii) in Theorem 8. It is obvious that sawɛ(βu)=0 and sawɛ(βx) is continuously differentiable in a neighborhood of u. So, g(x,y) is continuously differentiable in a neighborhood of (u,0). From assumption (ii) and condition (15), it follows that (18)gu,0=fu,0+αsawɛβu=0,gxu,0=fxu,0+-1mαβαβ-fxu,0>1+N>0.

For k=1, let (19)h1xfx,u+αsawɛβx.It follows from assumption (i) and the definition of sawtooth function that h1 is continuous in [-ɛ/β,3ɛ/β]. From the first relation of (15), we get that (20)h1ɛβ=fɛβ,u+αɛαɛ-M>0,h13ɛβ=f3ɛβ,u-αɛM-αɛ<0.Therefore, by the intermediate value theorem, there exists a point x2 with ɛ/β<x2<3ɛ/β, such that h1(x2)=0; that is, g(x2,u)=0. Similarly, let (21)h2xfx,x2+αsawɛβx-u.It is also clear that h2 is continuous in [-ɛ/β,3ɛ/β]. It also follows from the first relation of (15) that (22)h2ɛβ=fɛβ,x2+αɛ-uαɛ-M-u>0,h23ɛβ=f3ɛβ,x2-αɛ-uM+u-αɛ<0.By the intermediate value theorem again, there exists a point x1 with ɛ/β<x1<3ɛ/β, such that h2(x1)=0; that is, g(x1,x2)=u. It is clear that x1 and x2 are both in (ɛ/β,3ɛ/β)=(uɛ/2mr,uɛ/2mr). So we can take a sufficiently large integer m1>0, such that x1, x2U with x12+x220, and (x1,x2)TW for any |m|m1. It can easily be proved that g is continuously differentiable in some neighborhoods of (x2,u) and (x1,x2). Now, we show gx(x2,u)0. Otherwise, if gx(x2,u)=0, then the following equality holds: (23)fxx2,u+-1mαβ=0.Hence, αβ=|fx(x2,u)|N, which is a contradiction. Similarly, we can prove that gx(x1,x2)0. Hence, condition (iia) in Theorem 8 holds.

For k>1, the determination of x2 can be derived from the proof of the above paragraph as k=1. That is, there exists a point x2 in (ɛ/β,3ɛ/β), such that h1(x2)=0; that is, g(x2,u)=0. Set (24)h3xfx,0+αsawɛβx-u.With a similar method to the above paragraph, we can also get that there exists a point x1 in (ɛ/β,3ɛ/β) such that h3(x1)=0, which implies that g(x1,0)=u. So we can also take a sufficiently large integer m2>0, such that x1, x2U with x12+x220, and (x1,x2,0,,0)TW for any |m|m2. It can also easily be proved that g is continuously differentiable in some neighborhoods of (x2,u) and (x1,x2). The proofs of gx(x2,u)0 and gx(x1,0)0 are similar to the above paragraph. So, the details are omitted.

Finally, let |m0|=max{m1,m2}. Then condition (ii) in Theorem 8 is satisfied for m=m0. Therefore, for any α>α0 and β=β0, 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 sinx has similar geometric properties to the sawtooth function. So the following function (25)vn=αsinβxn-kcan also be used as a controller to chaotify system (2), where β is some constant to be determined and α>0 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 β0 and α0 such that for any constant α>α0 and β=β0 the result in Theorem 10 holds.

Remark 12.

In , 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  and this paper can be used. However, there will be many chaotic invariant sets as pointed out in Lemma 2.2 of  when using the chaotification scheme in . 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  cannot be used. Then, we can use the chaotification scheme obtained in this paper to chaotify this system.

Remark 13.

Since the point u in assumption (ii) of Theorem 10 can be negative, the value of m0 determined in this paper can be a negative integer. In addition, it is very difficult to determine the concrete value m0 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 |m0| 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 f in (2) as the following: (26)fx,y=0.01xx-1-0.01y2.It is clear that f is continuously differentiable in R2 and satisfies f(0,0)=0. Without loss of generality, we take r=3 in Theorem 10. Then, for any (x,y)[-3,3]2, we get that (27)fx,y0.21,fxx,y0.07,fyx,y0.06.Hence, we take M=0.21, N=0.07, and r=3 in assumption (i) of Theorem 10. It is also clear that the equation f(x,0)=0 has a nonzero solution u=1, which lies in (-3,3). Therefore, all the assumptions in Theorem 10 are satisfied. Here, we take the constant ɛ=1 in controller (4). Then, it follows from Theorem 10 that there exist two positive constants (28)α0>M+uɛ=1.21,β0=2m0ɛu=2m0>max1+2Nα0,3ɛr=max1.14α0,1=1,where m0 is some positive integer, such that, for any α>α0 and β=β0, 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 O(0,,0)TRk+1 in the uncontrolled system (7). It is obvious that fy(0,0)=0 and fx(0,0)=-0.01, which imply that O 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 [-3,3]k+1. Therefore, if we take an initial condition u(0)=(0.1,,0.1)TRk+1, then the solution u(n) of the uncontrolled system (7) should tend to the asymptotically stable fixed point O when n tends to infinity. This is confirmed in Figures 1 and 3.

Here, we take ɛ=1, m0=10, α=30, β=20, k=1,2, and n 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 14.

It should be pointed out that the relative existing chaotification scheme in  is not available for this map since there is only one fixed point.

Simple dynamical behaviors of uncontrolled system (7) for k=1, where the initial value is taken as u(0)=(0.1,0.1)T.

Complex dynamical behaviors of controlled system (8) for α=30, β=20, ɛ=1, and k=1, where the initial value is taken as u(0)=(0.1,0.1)T.

Simple dynamical behaviors of uncontrolled system (7) for k=2, where the initial value is taken as u(0)=(0.1,0.1,0.1)T.

Complex dynamical behaviors of controlled system (8) for α=30, β=20, ɛ=1, and k=2, where the initial value is taken as u(0)=(0.1,0.1,0.1)T.

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).

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