Existence of chaos for a simple delay difference equation
- Zongcheng Li^{1, 2}Email author and
- Xiaoying Zhu^{1}
https://doi.org/10.1186/s13662-015-0374-1
© Li and Zhu; licensee Springer. 2015
Received: 29 October 2014
Accepted: 13 January 2015
Published: 11 February 2015
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
1 Introduction
This paper is organized as follows. In Section 2, we give some basic concepts and one lemma. In Section 3, 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 4.
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 1
[13]
- (i)
\(\liminf_{n\to\infty}d(F^{n}(x),F^{n}(y))=0\);
- (ii)
\(\limsup_{n\to\infty}d(F^{n}(x),F^{n}(y))>0\).
Remark 1
The term ‘chaos’ was first used by Li and Yorke [14] for a map on a compact interval. Following the work of Li and Yorke, Zhou [13] gave the above definition of chaos for a topological dynamical system on a general metric space.
Definition 2
[15]
- (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.
Remark 2
If F is continuous in V, then condition (iii) can be concluded by conditions (i) and (ii), which was shown by Banks et al. [16]. Consequently, condition (iii) is unnecessary in Definition 2 if F is continuous in V. In [17], 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 use \(B_{r}(z)\) and \(\bar{B}_{r}(z)\) to denote the open and closed balls of radius r centered at \(z\in X\), respectively. The following definitions in [18] are used in this paper.
Definition 3
[18, Definitions 2.1-2.4]
- (i)A point \(z\in X\) is called an expanding fixed point (or a repeller) of F in \(\bar{B}_{r}(z)\) for some constant \(r>0\), if \(F(z)=z\) and there exists a constant \(\lambda>1\) such thatThe constant λ is called an expanding coefficient of F in \(\bar{B}_{r}(z)\). Furthermore, z is called a regular expanding fixed point of F in \(\bar{B}_{r}(z)\) if z is an interior point of \(F(B_{r}(z))\). Otherwise, z is called a singular expanding fixed point of F in \(\bar{B}_{r}(z)\).$$d\bigl(F(x),F(y)\bigr)\geq\lambda\,d(x,y) \quad \forall x, y\in\bar{B}_{r}(z). $$
- (ii)Assume that z is an expanding fixed point of F in \(\bar{B}_{r}(z)\) for some \(r>0\). Then z is said to be a snap-back repeller of F if there exists a point \(x_{0}\in B_{r}(z)\) with \(x_{0}\neq z\) and \(F^{m}(x_{0})=z\) for some positive integer m. Furthermore, z is said to be a nondegenerate snap-back repeller of F if there exist positive constants μ and \(r_{0}< r\) such that \(B_{r_{0}}(x_{0})\subset B_{r}(z)\) andz is called a regular snap-back 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 \(\delta\leq \delta_{0}\), z is an interior point of \(F^{m}(B_{\delta}(x_{0}))\). Otherwise, z is called a singular snap-back repeller of F.$$d\bigl(F^{m}(x),F^{m}(y)\bigr)\geq \mu \, d(x,y) \quad \forall x,y\in B_{r_{0}}(x_{0}). $$
Remark 3
The concept of snap-back repeller for maps in \({\mathbf{R}}^{n}\) was introduced by Marotto [19] in 1978. Obviously, Definition 3 is given in general metric spaces, which is an extension of Marotto’s definition. In terms of Definition 3, the snap-back repeller given by Marotto [19] is regular and nondegenerate.
Lemma 1
([20, Theorem 4.4], [21, Theorem 2.1])
- (i)
F is continuously differentiable in a neighborhood of z and all the eigenvalues of \(\operatorname {DF}(z)\) have absolute values larger than 1, which implies that there exist a positive constant r and a norm \(\Vert \cdot \Vert \) in \({\mathbf{R}}^{n}\) such that F is expanding in \(\bar{B}_{r}(z)\) in \(\Vert \cdot \Vert \);
- (ii)
z is a snap-back repeller of F with \(F^{m}(x_{0})=z\), \(x_{0}\neq z\), for some \(x_{0}\in B_{r}(z)\) and some positive integer m. Furthermore, F is continuously differentiable in some neighborhoods of \(x_{0},x_{1},\ldots,x_{m-1}\), respectively, and \(\det \operatorname {DF}(x_{j})\neq0\) for \(0\leq j\leq m-1\), where \(x_{j}=F(x_{j-1})\) for \(1\leq j\leq m-1\).
Remark 4
We can easily conclude that z is a regular and nondegenerate snap-back repeller from Lemma 1. Hence, Lemma 1 can be summed as a single word: ‘a regular and nondegenerate snap-back repeller in \({\mathbf{R}}^{n}\) implies chaos in the sense of both Devaney and Li-Yorke’. For more details, one can see [20, 21].
3 Existence of chaos
System (4) is called the system induced by (3) in \({\mathbf{R}}^{k+1}\). It is clear that a solution \(\{x(n-k),\ldots,x(n)\}_{n=1}^{\infty}\) of (3) corresponds to a solution \(\{u(n)\}_{n=1}^{\infty}\) of system (4), where the initial condition \(\{x(-k),\ldots,x(0)\}\) of (3) corresponds to an initial condition \(u(0)=(u_{1}(0),\ldots,u_{k+1}(0))^{T}\in{\mathbf{R}}^{k+1}\) of system (4). Hence, we can study the dynamical behavior of (3) by studying that of its induced system (4) in \({\mathbf{R}}^{k+1}\). So, we call (3) is chaotic in the sense of Devaney (or Li-Yorke) on \(V\subset{\mathbf{R}}^{k+1}\) if its induced system (4) is chaotic in the sense of Devaney (or Li-Yorke) on \(V\subset {\mathbf{R}}^{k+1}\).
Now, we state the main result of this paper as the following theorem.
Theorem 1
Proof
Lemma 1 will be used to prove this theorem. Therefore, we only need to show that the map F of system (4) satisfies all the assumptions in Lemma 1.
For simplifying the proof and convenience, γ is taken as an integer and satisfies condition (5) throughout the proof.
Set \(\beta _{0}:=\max\{\beta _{1},\beta _{2}\}\). From the above discussion, it follows that for arbitrary β satisfying \(\vert \beta \vert >\beta _{0}\), there exists a point \(O_{0}\in W\) satisfying \(O_{0}\neq O\) and \(F^{k+2}{(O_{0})}=O\). Therefore, O is a snap-back repeller of F for arbitrary β satisfying \(\vert \beta \vert >\beta _{0}\).
In summary, the map F satisfies all the assumptions in Lemma 1. Consequently, system (4), i.e., (3), is chaotic in the sense of both Devaney and Li-Yorke. This completes the proof. □
Remark 5
For simplifying the proof of Theorem 1, the parameter γ is taken as an integer. It should be pointed out that γ may be taken as other values such that system (4) is chaotic. In addition, it follows from the above proof that there exists a constant \(\beta _{0}>0\) such that for arbitrary \(\vert \beta \vert >\beta _{0}\), system (4) 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 value \(\beta _{0}\). In practical problems, we can take the parameter \(\vert \beta \vert \) large enough such that (6) or (7) in the proof of Theorem 1 are satisfied.
4 Conclusion
In this paper, we rigorously show the existence of chaos in a simple delay difference equation, which illustrates that the discrete analog of system (1) 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 satisfy some mild conditions. Numerical simulations confirm the theoretical analysis. However, the map f of system (1) is taken as a special function. Therefore, it is very interesting to study the chaotic behavior of system (1) or its discrete analog for a more general form of f, which will be our further research.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant 11101246) and the Postdoctoral Science Foundation of China (Grant 2014M561908).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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