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This paper is concerned with anticontrol of chaos for a class of delay difference equations via the feedback control technique. The controlled system is first reformulated into a high-dimensional discrete dynamical system. Then, a chaotification theorem based on the heteroclinic cycles connecting repellers for maps is established. The controlled system is proved to be chaotic in the sense of both Devaney and Li-Yorke. An illustrative example is provided with computer simulations.

Anticontrol of chaos (or called chaotification) is a process that makes a nonchaotic system chaotic or enhances a chaotic system to produce a stronger or different type of chaos. In recent years, it has been found that chaos can actually be useful under some circumstances, for example, in human brain analysis [

In the pursuit of chaotifying discrete dynamical systems, a simple yet mathematically rigorous chaotification method was first developed by Chen and Lai [

It is well known that the time delay appears in many realistic systems with feedback in science and engineering. Meanwhile, it has been shown that introducing delays to an undelayed system can be beneficial, especially for chaotic systems. This is the delayed feedback control method, which is widely used in chaos control. For continuous-time control systems, we refer to [

To the best of our knowledge, there are few results on chaotification of delay difference equations. Motivated by the delayed feedback control method, we studied the chaotification problem for a class of delay difference equations with at least two fixed points. Since the sawtooth function and the sine function have some favourable properties, some of which are similar, they are often used as controllers; see [

The rest of the paper is organized as follows. In Section

In this section, we describe the chaotification problem, give a reformulation of the delay difference equation, and introduce some fundamental concepts and lemmas, which will be used in the next section.

In this paper, we consider chaotification of the following delay difference equation:

From the above discussion, we see that the delay difference equation (

For convenience, define

Here, we reformulate (

By setting

The map

Since Li and Yorke [

Let

There are three conditions in the original characterization of chaos in Li-Yorke’s theorem [

Let

the set of the periodic points of

By the result of Banks et al. [

For convenience, some definitions of relevant concepts given in [

Let

Now, we introduce some relative concepts for system (

Consider the following.

A point

The concepts of density of periodic points, topological transitivity, sensitive dependence on initial conditions, and the invariant set for system (

System (

The following two lemmas will be used in the next section.

Assume that the map

It follows from

Since the result in the following lemma is related to the one-sided symbolic dynamical system

Let a map

for each

Under the conditions in Lemma

In this section, a chaotification scheme for the controlled system (

Consider the controlled system (

there exists a point

In order to prove that system (

For convenience, let

By assumption (i), we find that

Next, we need to show that

Take

Take

Finally, we will show that

Therefore, all the assumptions in Lemma

From the proof of Theorem

There are many delay discrete dynamical systems which have more than two fixed points. As all or some of the fixed points satisfy assumptions in Theorem

In the last section, we present an example of chaotification for the delay difference equation (

It is obvious that

In fact, it is obvious that the solutions of the uncontrolled system (

2D computer simulation result shows simple dynamical behaviors of the uncontrolled system (

In order to help better visualize the theoretical result of Theorem

2D computer simulation result shows complex dynamical behaviors of the controlled system (

In summary, the simulated results show that the uncontrolled system (

3D computer simulation result shows simple dynamical behaviors of the uncontrolled system (

3D computer simulation result shows complex dynamical behaviors of the controlled system (

In this paper, we consider anticontrol of chaos for a class of delay difference equations via the feedback control technique. Based on the result that a regular and nondegenerate heteroclinic cycle connecting repellers for maps implies chaos, we establish a chaotification theorem. The controlled system is proved to be chaotic in the sense of both Devaney and Li-Yorke. It is noted that there are many delay discrete dynamical systems which have more than two fixed points. As all or some of the fixed points satisfy assumptions in Theorem

The author declares that there is no conflict of interests regarding the publication of this paper.

This research was supported by the National Natural Science Foundation of China (Grant nos. 11101246 and 11101247). The author would like to thank the editor and the anonymous referees for their valuable comments and suggestions, which have led to an improvement of this paper.