- Open Access
Chaotic behavior in a class of delay difference equations
© Li et al.; licensee Springer. 2013
- Received: 4 December 2012
- Accepted: 23 March 2013
- Published: 10 April 2013
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 one-dimensional delay differential equation by using the Euler discretization. We first transform this class of delay difference equations into a high-dimensional discrete dynamical system. Then we prove that the map of the system is chaotic in the sense of both Devaney and Li-Yorke under some conditions, by employing the snap-back repeller theory. Finally, we give some computer simulations to illustrate the theoretical result.
MSC:34C28, 37D45, 74H65.
- delay difference equation
- snap-back repeller
- chaos in the sense of Devaney
- chaos in the sense of Li-Yorke
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.
where α, β are nonzero real parameters, and k is a positive integer.
where , is the delay, and f is a one-dimensional nonlinear function. Equation (1.3) is known as the Mackey-Glass equation and has found many applications in physics , population dynamics , physiology , medicine , neural control , and economy .
The chaotic behaviors of equation (1.3) for some special forms of f have been studied by many researchers. For example, Heiden and Mackey , and Walther , proved the existence of chaos in case of either piecewise constant or smooth but ‘nearly’ piecewise constant f. Lani-Wayda and Walther [17, 18] succeeded to prove chaos for certain smooth, nowhere constant f (and assuming 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 snap-back 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 snap-back repeller theory. Finally, an illustrative example is provided with computer simulations.
In this section, some basic concepts and lemmas are introduced.
In 1975, Li and Yorke  considered a one-dimensional continuous map and obtained the well-known 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 Li-Yorke and Devaney.
The map f is said to be chaotic in the sense of Li-Yorke if there exists an uncountable scrambled set S 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 
the set of the periodic points of f is dense in V;
f is topologically transitive in V;
f has sensitive dependence on initial conditions in V.
In 1992, Banks et al.  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 Li-Yorke .
Remark 2.1 Some researchers consider that condition (i) in Definition 2.2 is not essential in the chaotic behavior. In 1990, Wiggins [, 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 and denote the open and closed balls of radius r centered at , respectively. For convenience, we present some definitions in .
Definition 2.3 [, Definitions 2.1-2.4]
- (i)A point is called an expanding fixed point (or a repeller) of f in for some constant , if and there exists a constant such that
- (ii)Assume that z is an expanding fixed point of f in for some . Then z is said to be a snap-back repeller of f if there exists a point with and 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 such that and
z is called a regular snap-back repeller of f if is open and there exists a positive constant such that and for each positive constant , z is an interior point of . Otherwise, z is called a singular snap-back repeller of f.
Remark 2.2 In 1978, Marotto  introduced the concept of snap-back repeller for maps in the Euclidean space . It is obvious that Definition 2.3 extends the concept of snap-back repeller to maps in metric spaces. According to the above classifications of snap-back repellers for maps in metric spaces, the snap-back repeller in the Marotto paper  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 .
Lemma 2.1 Let be a continuously differentiable map. Assume that , , then for a sufficiently small neighborhood N of and any bounded interval I of R, there exists a positive constant such that the equation has a solution for any and .
Take , then for any . Hence, for any and , the equation , i.e., has a solution . This completes the proof. □
Lemma 2.2 [, Theorem 4.4]
f is continuously differentiable in a neighborhood of z and all the eigenvalues of have absolute values larger than 1, which implies that there exist a positive constant r and a norm in such that f is expanding in in ;
z is a snap-back repeller of f with , , for some and some positive integer m. Furthermore, f is continuously differentiable in some neighborhoods of , respectively, and for , where for .
Then for each neighborhood U of z, there exist a positive integer and a Cantor set such that is topologically conjugate to the symbolic system . Consequently, is chaotic on Λ in the sense of Devaney and f is chaotic in the sense of Li-Yorke. Further, there exists an uncountable, perfect and compact set , containing the Cantor set Λ, such that 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 (). 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 . From [, Theorem 4.4], we get that is chaotic on Λ in the sense of Devaney. By [, Lemma 2.4], we obtain that is chaotic in the sense of Li-Yorke. Consequently, f is chaotic in the sense of Li-Yorke. The construction of V and f is chaotic on V in the sense of Devaney can be conferred to [, Theorem 4.2]. Under the conditions of Lemma 2.2, z is a regular and nondegenerate snap-back repeller. Therefore, Lemma 2.2 can be briefly stated as the following: ‘a regular and nondegenerate snap-back repeller in implies chaos in the sense of both Devaney and Li-Yorke’. We refer to [24, 30] for details.
In this section, we will transform the delay difference equation (1.1) into a -dimensional discrete dynamical system, and give some definitions about chaos of the two systems.
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 of system (1.1) with an initial condition corresponds to a solution of system (3.1) with an initial condition . We call the solution of (3.1) is induced by the solution of (1.1). Therefore, the dynamical behavior of system (1.1) is the same as that of its induced system (3.1) in . So, we introduce some relative concepts for system (1.1), which are motivated from some works in [, Definitions 5.1 and 5.2].
A point is called an m-periodic point of system (1.1) if is an m-periodic point of its induced system (3.1), that is, and for . In the special case of , z is called a fixed point or steady state of system (1.1).
A point is called a snap-back repeller of system (1.1) if is a snap-back repeller of its induced system (3.1).
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 .
Definition 3.2 System (1.1) is said to be chaotic in the sense of Devaney (or Li-Yorke) on if its induced system (3.1) is chaotic in the sense of Devaney (or Li-Yorke) on .
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 snap-back repeller under some conditions.
where , . We also get that O is the unique fixed point of system (3.1) when . 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 snap-back repeller of system (3.1) under some conditions.
Theorem 4.1 There exists a positive constant such that for any , the fixed point O is a regular and nondegenerate snap-back repeller of system (3.1). Then system (3.1) and consequently, system (1.1), is chaotic in the sense of both Devaney and Li-Yorke.
Proof The idea in the proof is motivated by the proof of [, 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 . So, F satisfies the differentiability in Lemma 2.2.
where is an expanding coefficient of F in , and is the closed ball centered at of radius r with respect to the norm .
which implies that O is a snap-back repeller of F.
Set , then we get that with for any . It follows from (4.2) that , , .
Set , then we get that with for any . It follows from (4.3) that , for , and .
Take . Then for any , there exists a point with satisfying , in the two cases. Fix . Then we get that O is a snap-back repeller of F for .
Therefore, all the assumptions in Lemma 2.2 are satisfied and O is a regular and nondegenerate snap-back repeller of system (3.1). Therefore, system (3.1), i.e., system (1.1), is chaotic in the sense of both Devaney and Li-Yorke. The proof is complete. □
Remark 4.1 From the proof of Theorem 4.1, we see there exists some positive constant such that for any , system (3.1) is chaotic in the sense of both Devaney and Li-Yorke. However, it is very difficult to determine the concrete value 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 k-dimensional linear subsystem and one-dimensional nonlinear subsystem. That is, the folding and stretching only occurs in the variable , and all the other variables are replaced by for . So, system (3.1) can be viewed as one of the simplest systems that can show higher-dimensional chaos. Consequently, system (1.1) can be viewed as one of the simplest delay difference systems that show chaos.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11101246 and 11101247).
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