R E S E A R C H
Open Access
Existence of chaos for a simple delay
difference equation
Zongcheng Li
1,2*and Xiaoying Zhu
1*Correspondence:
lizongcheng_0905@163.com
1School of Science, Shandong
Jianzhu University, Jinan, Shandong 250101, China
2College of Control Science and
Engineering, Shandong University, Jinan, Shandong 250061, China
Abstract
In this paper, we study the existence of chaos for a simple delay difference equation. By using the snap-back repeller theory, we prove that the 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 two computer
simulations.
Keywords: delay; snap-back repeller; chaos in the sense of Devaney; chaos in the sense of Li-Yorke
1 Introduction
Time delay appears in many realistic systems with feedback in science and engineering. It affects the behaviors of a dynamical system significantly. The main reason is that it causes a system to be infinite dimensional. For example, Uçar [] proposed and analyzed a very simple first-order delay differential equation that would admit decaying, oscillatory, and even chaotic solutions. Recently, many researchers studied the following delay differential equation:
˙
x(t) = –x(t) +afx(t) –bx(t–τ), ()
wherea,bare real parameters,τ> is the delay, andf : R→Ris a real function. Equation () is known as a generalization of a neural network. When the functionf(x) is taken as
tanh(x), () becomes a neural network, which was studied in [] for the stability character-istics. Particularly, in [–], the authors studied the stability and chaos whenf(x) and the system parametersa,bsatisfy some specific conditions.
It is well known that we can use the discrete analog to study the complex dynamical be-haviors of nonlinear differential systems. Some qualitative properties of the corresponding difference equations can provide much useful information for analyzing the properties of the original differential equations. Therefore, it is very important to study properties of difference equations. Many researchers had studied the following delay difference equa-tion:
x(n+ ) =cx(n) +gx(n) –x(n–k), ()
wherecis a parameter,kis a positive integer, andg: R→Ris a map. Equation () can be viewed as the Euler discretization of () forb= . Such an equation arises from some of
the earliest mathematical models of the macroeconomic ‘trade cycle’, and has attracted a great deal of attention; see [–], and many references therein. Most of them studied the permanence, boundedness, stability, attractivity, oscillations, and bifurcations.
For the research on chaotic behavior of () or discretization of (), even for some spe-cial forms of them, there are few results which are studied with rigorously mathematical proofs. To the best of our knowledge, most of the existing results are studied with the aid of computer simulations. The main goal of this paper is to study the existence of chaos in the discrete analog of (). However, it is very difficult to study chaos for a general functionf. In [], Sprott proposed a simple prototype model with a sinusoidal nonlinearity to study chaos for delay differential equations, which is regarded as one of the simplest chaotic delay differential equations. This motivates us to use the sine function as an example to explore the chaotic behavior in discrete analog of (). That is, we study the existence of chaos of the following delay difference equation:
x(n+ ) =αx(n) +βsinx(n) –γx(n–k), () whereα,β,γ are real parameters, andkis a positive integer.
This paper is organized as follows. In Section , we give some basic concepts and one lemma. In Section , we prove that the delay difference equation is chaotic in the sense of both Devaney and Li-Yorke under some conditions, by using the snap-back repeller theory. Then we give two computer simulations to illustrate the theoretical result. Finally, we conclude this paper in Section .
2 Preliminaries
Up to now, there is no unified definition of chaos in mathematics. For convenience, we list two definitions of chaos which will be used in this paper.
Definition [] Let (X,d) be a metric space,F:X→Xbe a map, andSbe a set ofX with at least two distinct points. ThenSis called a scrambled set ofFif for any two distinct pointsx,y∈S,
(i) lim infn→∞d(Fn(x),Fn(y)) = ;
(ii) lim supn→∞d(Fn(x),Fn(y)) > .
The mapF is said to be chaotic in the sense of Li-Yorke if there exists an uncountable scrambled setSofF.
Remark 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 [] Let (X,d) be a metric space. A mapF:V⊂X→Vis said to be chaotic onVin the sense of Devaney if
(i) the set of the periodic points ofFis dense inV; (ii) Fis topologically transitive inV;
(iii) Fhas sensitive dependence on initial conditions inV.
in Definition ifFis continuous inV. 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.
In the rest of the paper, we useBr(z) andB¯r(z) to denote the open and closed balls of
radiusrcentered atz∈X, respectively. The following definitions in [] are used in this paper.
Definition [, Definitions .-.] Let (X,d) be a metric space andF:X→X be a map.
(i) A pointz∈Xis called an expanding fixed point (or a repeller) ofFinB¯r(z)for some
constantr> , ifF(z) =zand there exists a constantλ> such that
dF(x),F(y)≥λd(x,y) ∀x,y∈ ¯Br(z).
The constantλis called an expanding coefficient ofFinB¯r(z). Furthermore,zis
called a regular expanding fixed point ofFinB¯r(z)ifzis an interior point of
F(Br(z)). Otherwise,zis called a singular expanding fixed point ofFinB¯r(z).
(ii) Assume thatzis an expanding fixed point ofFinB¯r(z)for somer> . Thenzis said
to be a snap-back repeller ofFif there exists a pointx∈Br(z)withx=zand
Fm(x
) =zfor some positive integerm. Furthermore,zis said to be a nondegenerate
snap-back repeller ofFif there exist positive constantsμandr<rsuch that
Br(x)⊂Br(z)and
dFm(x),Fm(y)≥μd(x,y) ∀x,y∈Br(x).
zis called a regular snap-back repeller ofFifF(Br(z))is open and there exists a
positive constantδsuch thatBδ(x)⊂Br(z)and for each positive constantδ≤δ, zis an interior point ofFm(Bδ(x)). Otherwise,zis called a singular snap-back
repeller ofF.
Remark The concept of snap-back repeller for maps in Rnwas introduced by Marotto
[] in . Obviously, Definition is given in general metric spaces, which is an extension of Marotto’s definition. In terms of Definition , the snap-back repeller given by Marotto [] is regular and nondegenerate.
Lemma ([, Theorem .], [, Theorem .]) Let F: Rn→Rnbe a map with a fixed
point z∈Rn.Assume that
(i) Fis continuously differentiable in a neighborhood ofzand all the eigenvalues of
DF(z)have absolute values larger than,which implies that there exist a positive
constantrand a norm · inRnsuch thatFis expanding inB¯
r(z)in · ;
(ii) zis a snap-back repeller ofFwithFm(x
) =z,x=z,for somex∈Br(z)and some
positive integerm.Furthermore,Fis continuously differentiable in some
neighborhoods ofx,x, . . . ,xm–,respectively,anddet DF(xj)= for≤j≤m– ,
wherexj=F(xj–)for≤j≤m– .
containing the Cantor set,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 We can easily conclude thatzis a regular and nondegenerate snap-back re-peller from Lemma . Hence, Lemma can be summed as a single word: ‘a regular and nondegenerate snap-back repeller in Rnimplies chaos in the sense of both Devaney and Li-Yorke’. For more details, one can see [, ].
3 Existence of chaos
Now, we state the main result of this paper as the following theorem.
Theorem There exists a constantβ> such that for arbitraryβsatisfying|β|>βand
for someγ satisfying
|γ|> +|α+β|
|β| , ()
system(),and consequently(),is chaotic in the sense of both Devaney and Li-Yorke.
Proof Lemma will be used to prove this theorem. Therefore, we only need to show that the mapFof system () satisfies all the assumptions in Lemma .
It is clear thatO:= (, . . . , )T∈Rk+is always a fixed point of system (), and other fixed
pointsP:= (x, . . . ,x)T∈Rk+satisfy
sin( –γ)x=( –α)x β .
For simplifying the proof and convenience,γ is taken as an integer and satisfies condi-tion () throughout the proof.
Firstly, it is to show thatOis an expanding fixed point ofFin Rk+under condition (). It is obvious thatFis continuously differentiable in Rk+, and its Jacobian matrix atOis
DF(O) =
The characteristic equation ofDF(O) is
λk+– (α+β)λk+βγ= ,
from which we obtain the result that all the eigenvalues of DF(O) have absolute values larger than under condition (). If it is not true, then there will exist at least an eigenvalue λofDF(O) satisfies|λ| ≤. So, we can obtain the following contradiction:
+|α+β| ≥λk++(α+β)λk
≥λk+– (α+β)λk=|–βγ|> +|α+β|.
Therefore, we find that Ois an expanding fixed point ofF from the first condition of Lemma , that is,
F(x) –F(y)∗≥μx–y∗, ∀x,y∈ ¯Br(O),
wherer> is a constant, · ∗is some norm in Rk+, andμ> is an expanding coefficient
ofFinB¯r(O).
Secondly, one is to prove thatOis a snap-back repeller of the mapF. LetW⊂ ¯Br(O)
be an arbitrary neighborhood ofOin Rk+. Then we can obtain a small intervalU⊂R
Whenk> , we can also achieve a positive constantβsuch that for arbitrary|β|>β,
there exist two pointsx,x∈Usatisfying ⎧
⎨ ⎩
βsin(γx) = –π,
βsin(γx) = –απ.
()
LetO= (x,x, , . . . , )T∈Rk+, it also follows thatO∈U ×U× · · · × U
k+
⊂WwithO=
Ofor arbitrary|β|>β. From (), we also obtain the result thatF(O) = (x, , . . . ,π)T,
Fj(O) = (, . . . , ,π , , . . . ,
j
)Tfor ≤j≤k+ , andFk+(O ) =O.
Setβ:=max{β,β}. From the above discussion, it follows that for arbitraryβsatisfying
|β|>β, there exists a pointO∈W satisfyingO=OandFk+(O) =O. Therefore,Ois
a snap-back repeller ofFfor arbitraryβsatisfying|β|>β.
Thirdly, one is to prove that for arbitrary|β|>β, the following holds:
det DF(Oj)= , ≤j≤k+ ,
where Oj:=F(Oj–) for ≤j≤k+ . The existence of the Jacobian matrices ofF at Oj
(≤j≤k+ ) is becauseFis continuously differentiable in Rk+.
It is easy to conclude that for arbitraryu= (u, . . . ,uk+)T∈Rk+, the following holds:
det DF(u) = (–)k+βγcos(uk+–γu). ()
Whenk= , it follows thatO= (x,x)T,O= (x,π)T,O= (π, )T∈R. From (), we
getsin(γx)= andsin(x–γx)= for arbitrary|β|>β. Together with (), we get the
following for arbitrary|β|>β:
det DF(O) =βγcos(x–γx)= ,
det DF(O) = –βγcos(γx)= ,
det DF(O) =βγcos(γ π) =±βγ= .
When k > , it follows that O = (x,x, , . . . , )T, O = (x, , . . . , ,π)T, and Oj =
(, . . . , ,π, , . . . ,
j
)T∈Rk+for ≤j≤k+ . Similarly, it follows from () thatsin(γx )=
and sin(γx)= for arbitrary |β|>β. Consequently, the following hold for arbitrary
|β|>β:
det DF(O) = (–)k+βγcos(γx)= ,
det DF(O) = (–)kβγcos(γx)= ,
det DF(Oj) = (–)k+βγ= , for ≤j≤k,
det DF(Ok+) = (–)k+βγcos(γ π) =±βγ= .
In summary, the mapF satisfies all the assumptions in Lemma . Consequently, sys-tem (),i.e., (), is chaotic in the sense of both Devaney and Li-Yorke. This completes the
Remark For simplifying the proof of Theorem , the parameterγ is taken as an integer. It should be pointed out thatγmay be taken as other values such that system () is chaotic. In addition, it follows from the above proof that there exists a constantβ> such that
for arbitrary|β|>β, system () is chaotic in the sense of both Devaney and Li-Yorke.
However, there are few methods to determine the concrete expanding area of a fixed point in the literature. So it is not easy to get the particular valueβ. In practical problems, we
can take the parameter|β|large enough such that () or () in the proof of Theorem are satisfied.
For illustrating the theoretical result, we present two computer simulations of system (), from which we can see that system (),i.e., () indeed has complex dynamical behav-iors. The parameters are taken asα= .,β= ,γ= ,k= , . From the proof of Theo-rem , we see thatOis an expanding fixed point of the mapFfor|γ|= > [ +|α+β|]/|β|= .. It is also easy to obtain the result that there exist two pairs of pointsx≈–.,
x ≈–. satisfying () when k= , andx≈–.,x≈–. satisfying ()
whenk> . Therefore,Ois a regular and nondegenerate snap-back repeller of the map F. Two simulation results are given in Figures and fork= , , which exhibit complex dynamical behaviors of the system.
4 Conclusion
In this paper, we rigorously show the existence of chaos in a simple delay difference equa-tion, which illustrates that the discrete analog of system () indeed has very complicated dynamical behaviors. By using the snap-back repeller theory, we prove that the system is chaotic in the sense of both Devaney and Li-Yorke when the parameters of this system sat-isfy some mild conditions. Numerical simulations confirm the theoretical analysis. How-ever, the mapf of system () is taken as a special function. Therefore, it is very interesting to study the chaotic behavior of system () or its discrete analog for a more general form off, which will be our further research.
Figure 2 Computer simulation of system (4) forα= 0.1,β= 200,γ= 6,k= 2, andnfrom 0 to 20,000, with the initial conditionu(0) = (0.01, 0.01, 0.01)T.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant 11101246) and the Postdoctoral Science Foundation of China (Grant 2014M561908).
Received: 29 October 2014 Accepted: 13 January 2015
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