 Research
 Open Access
 Published:
Chaotic behavior in a class of delay difference equations
Advances in Difference Equations volume 2013, Article number: 99 (2013)
Abstract
In this paper, we rigorously prove the existence of chaos in a class of delay difference equations, which can be viewed as a discrete analogue of a onedimensional delay differential equation by using the Euler discretization. We first transform this class of delay difference equations into a highdimensional discrete dynamical system. Then we prove that the map of the system is chaotic in the sense of both Devaney and LiYorke under some conditions, by employing the snapback repeller theory. Finally, we give some computer simulations to illustrate the theoretical result.
MSC:34C28, 37D45, 74H65.
1 Introduction
Time delay arises in many realistic systems with feedback in science and engineering. It is well known that time delays in feedback systems can have a considerable influence on the qualitative behavior of these systems. The existence of time delay may lead to some complex dynamic behaviors such as oscillation, divergence, bifurcation and chaos; see [1–4], and many references therein.
Recently, some researchers used the Euler discretization to explore the complex dynamical behaviors of nonlinear differential systems, such as determining the bifurcation diagrams with Hopf bifurcation, observing stable or unstable orbits, and chaotic behavior; see [3–6]etc. However, some complicated behaviors such as chaos which they observed were obtained only by numerical simulations, and have not been proved rigorously. It is noted that there could exist some false phenomena only by virtue of numerical simulations. Therefore, the existence of chaotic behavior of these systems needs to be studied rigorously.
In this paper, we study the chaotic behavior of the following delay difference equation:
where α, β are nonzero real parameters, and k is a positive integer.
Equation (1.1) can be viewed as a discrete analogue of the following onedimensional delay differential equation by using the forward Euler scheme
where $\mu >0$ and γ are real parameters, $\tau >0$ is the delay. Equation (1.2) has been studied by [7, 8] for the periodic solutions, and it is a special case of the following equation:
where $\mu >0$, $\tau >0$ is the delay, and f is a onedimensional nonlinear function. Equation (1.3) is known as the MackeyGlass equation and has found many applications in physics [9], population dynamics [10], physiology [11], medicine [12], neural control [13], and economy [14].
The chaotic behaviors of equation (1.3) for some special forms of f have been studied by many researchers. For example, Heiden and Mackey [15], and Walther [16], proved the existence of chaos in case of either piecewise constant or smooth but ‘nearly’ piecewise constant f. LaniWayda and Walther [17, 18] succeeded to prove chaos for certain smooth, nowhere constant f (and assuming $\mu =0$ in (1.3)).
To the best of our knowledge, the research works on the chaotic behavior of equation (1.2) or its discrete analogue (1.1) with rigorously mathematical proof up to now are still few. The main purpose of this paper is to study the existence of chaos of equation (1.1) by using the snapback repeller theory.
The rest of the paper is organized as follows. In Section 2, some basic concepts and lemmas are introduced. The transformation of the chaos problem is given in Section 3. In Section 4, it is rigorously proved that there exists chaotic behavior in the delay difference equation by using the snapback repeller theory. Finally, an illustrative example is provided with computer simulations.
2 Preliminaries
In this section, some basic concepts and lemmas are introduced.
In 1975, Li and Yorke [19] considered a onedimensional continuous map and obtained the wellknown result, that is, ‘period 3 implies chaos’. They first used the precise word ‘chaos’. However, to the best of our knowledge, there is no unified definition of chaos in mathematics up to now. Depending on the requirements in different problems, there appeared several different definitions of chaos, some are stronger and some are weaker. For convenience, we present two definitions of chaos in the sense of LiYorke and Devaney.
Definition 2.1 Let $(X,d)$ be a metric space, $f:X\to X$ be a map, and 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 distinct points $x,y\in S$,

(i)
${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}d({f}^{n}(x),{f}^{n}(y))=0$;

(ii)
${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}d({f}^{n}(x),{f}^{n}(y))>0$.
The map f is said to be chaotic in the sense of LiYorke if there exists an uncountable scrambled set S of f.
There are three conditions in the original characterization of chaos in LiYorke’s theorem [19]. Besides the above conditions (i) and (ii), the third one is that for all $x\in S$ and for all periodic points p of f,
But conditions (i) and (ii) together imply that the scrambled set S contains at most one point x that does not satisfy the above condition. So, the third condition is not essential and can be removed.
Definition 2.2 [20]
Let $(X,d)$ be a metric space and V be a set of X. A map $f:V\to V$ is said to be chaotic on V in the sense of Devaney if

(i)
the set of the periodic points of f is dense in V;

(ii)
f is topologically transitive in V;

(iii)
f has sensitive dependence on initial conditions in V.
In 1992, Banks et al. [21] proved that conditions (i) and (ii) together imply condition (iii) if f is continuous in V and V is infinite. So, condition (iii) is redundant in the above definition. It has been proved that under some conditions, chaos in the sense of Devaney is stronger than that in the sense of LiYorke [22].
Remark 2.1 Some researchers consider that condition (i) in Definition 2.2 is not essential in the chaotic behavior. In 1990, Wiggins [[23], Definition 4.11.2] called the map f chaotic on V if it satisfies conditions (ii) and (iii) in Definition 2.2. So, some researchers use the term ‘chaos in the sense of Wiggins’. It is evident that chaos in the sense of Devaney is stronger than that in the sense of Wiggins.
In the following, by ${B}_{r}(z)$ and ${\overline{B}}_{r}(z)$ denote the open and closed balls of radius r centered at $z\in X$, respectively. For convenience, we present some definitions in [24].
Definition 2.3 [[24], Definitions 2.12.4]
Let $(X,d)$ be a metric space and $f:X\to X$ be a map.

(i)
A point $z\in X$ is called an expanding fixed point (or a repeller) of f in ${\overline{B}}_{r}(z)$ for some constant $r>0$, if $f(z)=z$ and there exists a constant $\lambda >1$ such that
$$d(f(x),f(y))\ge \lambda d(x,y)\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in {\overline{B}}_{r}(z).$$
The constant λ is called an expanding coefficient of f in ${\overline{B}}_{r}(z)$. Furthermore, z is called a regular expanding fixed point of f in ${\overline{B}}_{r}(z)$ if z is an interior point of $f({B}_{r}(z))$. Otherwise, z is called a singular expanding fixed point of f in ${\overline{B}}_{r}(z)$.

(ii)
Assume that z is an expanding fixed point of f in ${\overline{B}}_{r}(z)$ for some $r>0$. Then z is said to be a snapback repeller of f if there exists a point ${x}_{0}\in {B}_{r}(z)$ with ${x}_{0}\ne z$ and ${f}^{m}({x}_{0})=z$ for some positive integer m. Furthermore, z is said to be a nondegenerate snapback repeller of f if there exist positive constants μ and ${r}_{0}<r$ such that ${B}_{{r}_{0}}({x}_{0})\subset {B}_{r}(z)$ and
$$d({f}^{m}(x),{f}^{m}(y))\ge \mu d(x,y)\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in {B}_{{r}_{0}}({x}_{0}).$$
z is called a regular snapback repeller of f if $f({B}_{r}(z))$ is open and there exists a positive constant ${\delta}_{0}$ such that ${B}_{{\delta}_{0}}({x}_{0})\subset {B}_{r}(z)$ and for each positive constant $\delta \le {\delta}_{0}$, z is an interior point of ${f}^{m}({B}_{\delta}({x}_{0}))$. Otherwise, z is called a singular snapback repeller of f.
Remark 2.2 In 1978, Marotto [25] introduced the concept of snapback repeller for maps in the Euclidean space ${\mathbf{R}}^{n}$. It is obvious that Definition 2.3 extends the concept of snapback repeller to maps in metric spaces. According to the above classifications of snapback repellers for maps in metric spaces, the snapback repeller in the Marotto paper [25] is regular and nondegenerate.
We now present two lemmas which will be used in Section 4. The following Lemma 2.1 is similar to that Lemma 3.5 in [26].
Lemma 2.1 Let $h:[r,r]\subset \mathbf{R}\to \mathbf{R}$ be a continuously differentiable map. Assume that $h(0)=0$, ${h}^{\mathrm{\prime}}(0)\ne 0$, then for a sufficiently small neighborhood N of $0\in \mathbf{R}$ and any bounded interval I of R, there exists a positive constant ${\beta}^{\ast}:={\beta}^{\ast}(N,I)$ such that the equation $\beta h(x)=y$ has a solution $x\in N$ for any $\beta >{\beta}^{\ast}$ and $y\in I$.
Proof Since h is continuously differentiable on $[r,r]$, and $h(0)=0$, ${h}^{\mathrm{\prime}}(0)\ne 0$, for any sufficiently small neighborhood $N\subset [r,r]$ of 0, there exist two neighborhoods U and V of 0 such that $U,V\subset N$ and $h:U\to V$ is a homeomorphism. In addition, for any bounded interval $I\subset \mathbf{R}$, there exists a positive constant ${\beta}_{0}$ such that
Take ${\beta}^{\ast}:={\beta}^{\ast}(N,I)={\beta}_{0}$, then $\frac{1}{\beta}I\subset V$ for any $\beta >{\beta}^{\ast}$. Hence, for any $y\in I$ and $\beta >{\beta}^{\ast}$, the equation $h(x)=\frac{1}{\beta}y$, i.e., $\beta h(x)=y$ has a solution $x\in U\subset N$. This completes the proof. □
Lemma 2.2 [[27], Theorem 4.4]
Let $f:{\mathbf{R}}^{n}\to {\mathbf{R}}^{n}$ be a map with a fixed point $z\in {\mathbf{R}}^{n}$. Assume that

(1)
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 $\parallel \cdot \parallel $ in ${\mathbf{R}}^{n}$ such that f is expanding in ${\overline{B}}_{r}(z)$ in $\parallel \cdot \parallel $;

(2)
z is a snapback repeller of f with ${f}^{m}({x}_{0})=z$, ${x}_{0}\ne z$, for some ${x}_{0}\in {B}_{r}(z)$ and some positive integer m. Furthermore, f is continuously differentiable in some neighborhoods of ${x}_{0},{x}_{1},\dots ,{x}_{m1}$, respectively, and $detDf({x}_{j})\ne 0$ for $0\le j\le m1$, where ${x}_{j}=f({x}_{j1})$ for $1\le j\le m1$.
Then for each neighborhood U of z, there exist a positive integer $k>m$ and a Cantor set $\mathrm{\Lambda}\subset U$ such that ${f}^{k}:\mathrm{\Lambda}\to \mathrm{\Lambda}$ is topologically conjugate to the symbolic system $\sigma :{\sum}_{2}^{+}\to {\sum}_{2}^{+}$. Consequently, ${f}^{k}$ is chaotic on Λ in the sense of Devaney and f is chaotic in the sense of LiYorke. Further, there exists an uncountable, perfect and compact set $V\subset {\mathbf{R}}^{n}$, containing the Cantor set Λ, such that $f(V)=V$ and f is chaotic on V in the sense of Devaney.
Remark 2.3 In Lemma 2.2, it is only required that f is continuously differentiable near z and ${x}_{j}$ ($0\le j\le m1$). So, it is easy to be satisfied by many maps in practical models. The conclusions of Lemma 2.2 are slightly different from the original Theorem 4.4 in [27]. From [[27], Theorem 4.4], we get that ${f}^{k}$ is chaotic on Λ in the sense of Devaney. By [[28], Lemma 2.4], we obtain that ${f}^{k}$ is chaotic in the sense of LiYorke. Consequently, f is chaotic in the sense of LiYorke. The construction of V and f is chaotic on V in the sense of Devaney can be conferred to [[29], Theorem 4.2]. Under the conditions of Lemma 2.2, z is a regular and nondegenerate snapback repeller. Therefore, Lemma 2.2 can be briefly stated as the following: ‘a regular and nondegenerate snapback repeller in ${\mathbf{R}}^{n}$ implies chaos in the sense of both Devaney and LiYorke’. We refer to [24, 30] for details.
3 Transformation of the chaos problem
In this section, we will transform the delay difference equation (1.1) into a $(k+1)$dimensional discrete dynamical system, and give some definitions about chaos of the two systems.
Let ${u}_{j}(n):=x(n+jk1)$ for $1\le j\le k+1$, then system (1.1) can be transformed into the following $(k+1)$dimensional discrete system
where $u={({u}_{1},{u}_{2},\dots ,{u}_{k+1})}^{T}\in {\mathbf{R}}^{k+1}$.
The map F is said to be induced by f, and system (3.1) is said to be induced by system (1.1). It is evident that a solution ${\{x(nk),\dots ,x(n)\}}_{n=1}^{\mathrm{\infty}}$ of system (1.1) with an initial condition $\{x(k),\dots ,x(0)\}$ corresponds to a solution ${\{u(n)\}}_{n=1}^{\mathrm{\infty}}$ of system (3.1) with an initial condition $u(0)={({u}_{1}(0),\dots ,{u}_{k+1}(0))}^{T}\in {\mathbf{R}}^{k+1}$. We call the solution ${\{u(n)\}}_{n=1}^{\mathrm{\infty}}$ of (3.1) is induced by the solution ${\{x(nk),\dots ,x(n)\}}_{n=1}^{\mathrm{\infty}}$ of (1.1). Therefore, the dynamical behavior of system (1.1) is the same as that of its induced system (3.1) in ${\mathbf{R}}^{k+1}$. So, we introduce some relative concepts for system (1.1), which are motivated from some works in [[30], Definitions 5.1 and 5.2].
Definition 3.1

(i)
A point $z\in {\mathbf{R}}^{k+1}$ is called an mperiodic point of system (1.1) if $z\in {\mathbf{R}}^{k+1}$ is an mperiodic point of its induced system (3.1), that is, ${F}^{m}(z)=z$ and ${F}^{j}(z)\ne z$ for $1\le j\le m1$. In the special case of $m=1$, z is called a fixed point or steady state of system (1.1).

(ii)
A point $z\in {\mathbf{R}}^{k+1}$ is called a snapback repeller of system (1.1) if $z\in {\mathbf{R}}^{k+1}$ is a snapback repeller of its induced system (3.1).

(iii)
The concepts of density of periodic points, topological transitivity, sensitive dependence on initial conditions, and the invariant set for system (1.1) are defined similarly to those for its induced system (3.1) in ${\mathbf{R}}^{k+1}$.
Definition 3.2 System (1.1) is said to be chaotic in the sense of Devaney (or LiYorke) on $V\subset {\mathbf{R}}^{k+1}$ if its induced system (3.1) is chaotic in the sense of Devaney (or LiYorke) on $V\subset {\mathbf{R}}^{k+1}$.
4 Chaos in the system
In this section, we will investigate the chaotic behavior of system (3.1), i.e., system (1.1), by showing that there exists a regular and nondegenerate snapback repeller under some conditions.
Consider the fixed points of system (3.1). From $F(\overline{u})=\overline{u}$, we obtain that system (3.1) has the following three fixed points for $\frac{1\alpha}{\beta}<1$:
where $p:=\sqrt{1+\frac{\alpha 1}{\beta}}<0$, $q:=\sqrt{1+\frac{\alpha 1}{\beta}}>0$. We also get that O is the unique fixed point of system (3.1) when $\frac{1\alpha}{\beta}\ge 1$. In the two cases, O is always a fixed point of system (3.1). So, in the following, we only show the fixed point O can be a regular and nondegenerate snapback repeller of system (3.1) under some conditions.
Theorem 4.1 There exists a positive constant ${\beta}_{0}$ such that for any $\beta >{\beta}_{0}$, the fixed point O is a regular and nondegenerate snapback repeller of system (3.1). Then system (3.1) and consequently, system (1.1), is chaotic in the sense of both Devaney and LiYorke.
Proof The idea in the proof is motivated by the proof of [[26], Theorem 3.2]. We will apply Lemma 2.2 to prove this theorem. So, it suffices to show that all the assumptions in Lemma 2.2 are satisfied. It is clear that the map F in system (3.1) is continuously differentiable in ${\mathbf{R}}^{k+1}$. So, F satisfies the differentiability in Lemma 2.2.
First, we show that there exists a positive constant ${\beta}_{1}$ such that O is an expanding fixed point of the map F in some norm in ${\mathbf{R}}^{k+1}$ for any $\beta >{\beta}_{1}$. In fact, the Jacobian matrix of F at O is
Its eigenvalues are determined by
Let ${\beta}_{1}:=1+\alpha $. From (4.1), we get that all the eigenvalues of $DF(O)$ have absolute values larger than 1 for any $\beta >{\beta}_{1}$. Otherwise, suppose that there exists an eigenvalue ${\lambda}_{0}$ of $DF(O)$ with ${\lambda}_{0}\le 1$, then we get the following inequality:
which is a contradiction. Hence, it follows from the first condition of Lemma 2.2 that there exists a positive constant r and a norm ${\parallel \cdot \parallel}^{\ast}$ in ${\mathbf{R}}^{k+1}$ such that O is an expanding fixed point of F in ${\overline{B}}_{r}(O)$ in the norm ${\parallel \cdot \parallel}^{\ast}$, that is,
where $\gamma >1$ is an expanding coefficient of F in ${\overline{B}}_{r}(O)$, and ${\overline{B}}_{r}(O)$ is the closed ball centered at $O\in {\mathbf{R}}^{k+1}$ of radius r with respect to the norm ${\parallel \cdot \parallel}^{\ast}$.
Next, we show that O is a snapback repeller of F in the norm ${\parallel \cdot \parallel}^{\ast}$. Suppose that $W\subset {\overline{B}}_{r}(O)$ is an arbitrary neighborhood of O in ${\mathbf{R}}^{k+1}$, then there exists a small interval $U\subset \mathbf{R}$ containing 0 such that $\underset{k+1}{\underset{\u23df}{U\times U\times \cdots \times U}}\subset W$. In the following, we will show that there exists a positive constant ${\beta}_{2}$ such that for any $\beta >{\beta}_{2}$, there exists a point ${O}_{0}\in W$ with ${O}_{0}\ne O$ satisfying
which implies that O is a snapback repeller of F.
For convenience, let $h(x):=x(1{x}^{2})$. It is clear that $h(x)$ is continuously differentiable on R and satisfies
For $k=1$. From Lemma 2.1, it follows that there exists a positive constant ${\beta}_{2}^{\mathrm{\prime}}$ such that for any $\beta >{\beta}_{2}^{\mathrm{\prime}}$, there exist two points ${x}_{1},{x}_{2}\in U$ satisfying the following equations:
Set ${O}_{0}={({x}_{1},{x}_{2})}^{T}\in {\mathbf{R}}^{2}$, then we get that ${O}_{0}\in U\times U\subset W$ with ${O}_{0}\ne O$ for any $\beta >{\beta}_{2}^{\mathrm{\prime}}$. It follows from (4.2) that $F({O}_{0})={({x}_{2},1)}^{T}$, ${F}^{2}({O}_{0})={(1,0)}^{T}$, ${F}^{3}({O}_{0})=O$.
For $k>1$, it also follows from Lemma 2.1 that there exists a positive constant ${\beta}_{2}^{\ast}$ such that for any $\beta >{\beta}_{2}^{\ast}$, there exist two points ${x}_{1},{x}_{2}\in U$ satisfying the following equations:
Set ${O}_{0}={({x}_{1},{x}_{2},0,\dots ,0)}^{T}\in {\mathbf{R}}^{k+1}$, then we get that ${O}_{0}\in \underset{k+1}{\underset{\u23df}{U\times U\times \cdots \times U}}\subset W$ with ${O}_{0}\ne O$ for any $\beta >{\beta}_{2}^{\ast}$. It follows from (4.3) that $F({O}_{0})={({x}_{2},0,\dots ,1)}^{T}$, ${F}^{j}({O}_{0})={(0,\dots ,0,\underset{j}{\underset{\u23df}{1,0,\dots ,0}})}^{T}$ for $2\le j\le k+1$, and ${F}^{k+2}({O}_{0})=O$.
Take ${\beta}_{2}:=max\{{\beta}_{2}^{\mathrm{\prime}},{\beta}_{2}^{\ast}\}$. Then for any $\beta >{\beta}_{2}$, there exists a point ${O}_{0}\in W$ with ${O}_{0}\ne O$ satisfying ${F}^{k+2}({O}_{0})=O$, in the two cases. Fix ${\beta}_{0}\ge max\{{\beta}_{1},{\beta}_{2}\}$. Then we get that O is a snapback repeller of F for $\beta >{\beta}_{0}$.
Now, we shall show that for $\beta >{\beta}_{0}$,
where ${O}_{j}:=F({O}_{j1})$ for $1\le j\le k+1$. It is obvious that $h(x)$ is continuously differentiable on R and satisfies
Hence, from the second conclusion of (4.4), it follows that there exists a sufficiently small neighborhood ${U}_{1}\subset \mathbf{R}$ containing 0 such that ${h}^{\mathrm{\prime}}(x)\ne 0$ for all $x\in {U}_{1}$. We can take sufficiently large ${\beta}_{0}$ such that ${x}_{1}$, ${x}_{2}$ obtained above also lie in ${U}_{1}$ for $\beta >{\beta}_{0}$ and satisfy equations (4.2) or (4.3). Consequently, we have
A direct calculation shows that for any $u={({u}_{1},\dots ,{u}_{k+1})}^{T}\in {\mathbf{R}}^{k+1}$,
For $k=1$, we get that ${O}_{0}={({x}_{1},{x}_{2})}^{T}$, ${O}_{1}={({x}_{2},1)}^{T}$, ${O}_{2}={(1,0)}^{T}\in {\mathbf{R}}^{2}$. It follows from the second conclusion of (4.4), (4.5) and (4.6) that for $\beta >{\beta}_{0}$,
For $k>1$, we get that ${O}_{0}={({x}_{1},{x}_{2},0,\dots ,0)}^{T}$, ${O}_{1}={({x}_{2},0,\dots ,0,1)}^{T}$, and ${O}_{j}={(0,\dots ,0,\underset{j}{\underset{\u23df}{1,0,\dots ,0}})}^{T}\in {\mathbf{R}}^{k+1}$ for $2\le j\le k+1$. Hence, from the second conclusion of (4.4), (4.5) and (4.6), we get that for $\beta >{\beta}_{0}$,
Therefore, all the assumptions in Lemma 2.2 are satisfied and O is a regular and nondegenerate snapback repeller of system (3.1). Therefore, system (3.1), i.e., system (1.1), is chaotic in the sense of both Devaney and LiYorke. The proof is complete. □
Remark 4.1 From the proof of Theorem 4.1, we see there exists some positive constant ${\beta}_{0}$ such that for any $\beta >{\beta}_{0}$, system (3.1) is chaotic in the sense of both Devaney and LiYorke. However, it is very difficult to determine the concrete value ${\beta}_{0}$ since the expanding area of a fixed point is not easy to obtain. This will be left for our further research.
Remark 4.2 The system (3.1) consists of a kdimensional linear subsystem and onedimensional nonlinear subsystem. That is, the folding and stretching only occurs in the variable ${u}_{k+1}$, and all the other variables ${u}_{j}$ are replaced by ${u}_{j+1}$ for $1\le j\le k$. So, system (3.1) can be viewed as one of the simplest systems that can show higherdimensional chaos. Consequently, system (1.1) can be viewed as one of the simplest delay difference systems that show chaos.
In order to help better visualize the theoretical result, two computer simulations are done, which exhibit complicated dynamical behaviors of the induced system (3.1), i.e., system (1.1). We take $\alpha =0.1$, $\beta =2.7$, $k=1,2$ for computer simulations. It follows from the proof of Theorem 4.1 that O is an expanding fixed point of system (3.1) when $\beta >1+\alpha =1.1$. Furthermore, there exists some positive constant ${\beta}_{0}>1.1$ such that for $\beta >{\beta}_{0}$, we have two couples of points ${x}_{1}\approx 0.4879$, ${x}_{2}\approx 0.0371$ satisfying equation (4.2) for $k=1$, and ${x}_{1}\approx 0.4832$, ${x}_{2}\approx 0.0371$ satisfying equation (4.3) for $k>1$. Consequently, O is a regular and nondegenerate snapback repeller of system (3.1) for $\beta >{\beta}_{0}$. Therefore, system (3.1), i.e., system (1.1), is chaotic in the sense of both Devaney and LiYorke. Some simulation results are shown in Figures 12 for $k=1,2$ and n from 0 to 20,000, respectively. It is a dense orbit in each picture, which clearly shows that the system has very complicated dynamical behaviors.
References
 1.
Busenbrg S, Martelli M: Delay Differential Equations and Dynamical Systems. Springer, Berlin; 1991.
 2.
Jiang M, Shen Y, Jian J, Liao X: Stability, bifurcation and a new chaos in the logistic differential equation with delay. Phys. Lett. A 2006, 350: 221–227. 10.1016/j.physleta.2005.10.019
 3.
Peng MS, Uçar A: The use of the Euler method in identification of multiple bifurcations and chaotic behavior in numerical approximations of delay differential equations. Chaos Solitons Fractals 2004, 21: 883–891. 10.1016/j.chaos.2003.12.044
 4.
Peng MS, Yuan Y: Stability, symmetrybreaking bifurcation and chaos in discrete delayed models. Int. J. Bifurc. Chaos 2008, 18: 1477–1501. 10.1142/S0218127408021117
 5.
He ZM, Lai X: Bifurcation and chaotic behavior of a discretetime predatorprey system. Nonlinear Anal., Real World Appl. 2011, 12: 403–417. 10.1016/j.nonrwa.2010.06.026
 6.
Fan DJ, Wei JJ: Bifurcation analysis of discrete survival red blood cells model. Commun. Nonlinear Sci. Numer. Simul. 2009, 14: 3358–3368. 10.1016/j.cnsns.2009.01.015
 7.
Kaplan JL, Yorke JA: Ordinary differential equations which yield periodic solutions of differential delay equations. J. Math. Anal. Appl. 1974, 48: 317–324. 10.1016/0022247X(74)901620
 8.
Dormayer P:The stability of special symmetric solutions of $\dot{x}=\alpha f(x(t1))$ with small amplitudes. Nonlinear Anal., Theory Methods Appl. 1990, 14: 701–715. 10.1016/0362546X(90)90045I
 9.
Furomochi T: Existence of periodic solutions of onedimensional differentialdelay equations. Tohoku Math. J. 1978, 30: 13–35. 10.2748/tmj/1178230094
 10.
Gurney WSC, Blythe SP, Nisbeth RM: Nicholson’s blowflies revisited. Nature 1978, 287: 17–21.
 11.
Heiden U: Delays in physiological systems. J. Math. Biol. 1979, 8: 345–364. 10.1007/BF00275831
 12.
Mackey MC, Glass L: Oscillation and chaos in physiological control systems. Science 1977, 197: 287–289. 10.1126/science.267326
 13.
Mackey MC, Heiden U: The dynamics of recurrent inhibition. J. Math. Biol. 1984, 19: 211–225. 10.1007/BF00277747
 14.
Mackey MC: Commodity price fluctuations: price dependent delays and nonlinearities as explanatory factors. J. Econ. Theory 1990., 48: Article ID 497
 15.
Heiden U, Mackey MC: The dynamics of production and destruction: analytic insight into complex behavior. J. Math. Biol. 1982, 16: 75–101. 10.1007/BF00275162
 16.
Heiden U, Walther HO: Existence of chaos in control systems with delayed feedback. J. Differ. Equ. 1983, 47: 273–295. 10.1016/00220396(83)900372
 17.
LaniWayda B, Walther HO: Chaotic motion generated by delayed negative feedback, Part I: a transversality criterion. Differ. Integral Equ. 1995, 8: 1407–1452.
 18.
LaniWayda B, Walther HO: Chaotic motion generated by delayed negative feedback, Part II: construction of nonlinearities. Math. Nachr. 1996, 180: 141–211.
 19.
Li TY, Yorke JA: Period three implies chaos. Am. Math. Mon. 1975, 82: 985–992. 10.2307/2318254
 20.
Devaney RL: An Introduction to Chaotic Dynamical Systems. AddisonWesley, New York; 1987.
 21.
Banks J, Brooks J, Cairns G, Davis G, Stacey P: On Devaney’s definition of chaos. Am. Math. Mon. 1992, 99: 332–334. 10.2307/2324899
 22.
Huang W, Ye XD: Devaney’s chaos or 2scattering implies LiYorke’s chaos. Topol. Appl. 2002, 117: 259–272. 10.1016/S01668641(01)000256
 23.
Wiggins S: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York; 1990.
 24.
Shi YM, Chen GR: Chaos of discrete dynamical systems in complete metric spaces. Chaos Solitons Fractals 2004, 22: 555–571. 10.1016/j.chaos.2004.02.015
 25.
Marotto FR:Snapback repellers imply chaos in ${R}^{n}$. J. Math. Anal. Appl. 1978, 63: 199–223. 10.1016/0022247X(78)901154
 26.
Huang Y, Zou XF: Coexistence of chaos and stable periodic orbits in a simple discrete neural network. J. Nonlinear Sci. 2005, 15: 291–303. 10.1007/s003320050647z
 27.
Shi YM, Chen GR: Discrete chaos in Banach spaces. Sci. China Ser. A 2004, 34: 595–609. (English version 48, 222–238 (2005))
 28.
Shi YM, Yu P: Chaos induced by regular snapback repellers. J. Math. Anal. Appl. 2008, 337: 1480–1494. 10.1016/j.jmaa.2007.05.005
 29.
Shi YM, Yu P: Study on chaos by turbulent maps in noncompact sets. Chaos Solitons Fractals 2006, 28: 1165–1180. 10.1016/j.chaos.2005.08.162
 30.
Shi YM, Yu P, Chen GR: Chaotification of dynamical systems in Banach spaces. Int. J. Bifurc. Chaos 2006, 16: 2615–2636. 10.1142/S021812740601629X
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11101246 and 11101247).
Author information
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
ZL carried out the main part of this manuscript. QZ and DL participated in discussion. All authors read and approved the final manuscript.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Received
Accepted
Published
DOI
Keywords
 chaos
 delay difference equation
 snapback repeller
 chaos in the sense of Devaney
 chaos in the sense of LiYorke