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We study the chaotification problem for a class of delay difference equations by using the snap-back repeller theory and the feedback control approach. We first study the stability and expansion of fixed points and establish a criterion of chaos. Then, based on this criterion of chaos and the feedback control approach, we establish a chaotification scheme such that the controlled 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 some computer simulations.

Research on chaos control has attracted a lot of interest from many scientists and mathematicians. There are two directions in chaos control, that is, control of chaos and anticontrol of chaos (or called chaotification). The former regarded chaos as harmful. So many earlier works focused on stabilizing a chaotic system, which was regarded as the traditional control. The reader is referred to the monographs [

In research on chaotification for discrete dynamical systems, a mathematically rigorous and effective chaotification method was first proposed by Chen and Lai [

To the best of our knowledge, although there already exist many works on chaotification of discrete dynamical systems, there are few results on chaotification of delay difference equations. Motivated by the feedback control approach, we have succeeded in studying the chaotification problems on linear delay difference equations [

This paper is organized as follows. In Section

Up to now, there is no unified definition of chaos in mathematics. For convenience, we present two definitions of chaos, which will be used in this paper.

Let

The term “chaos” was first used by Li and Yorke [

Let

the periodic points of

In [

The following criterion of chaos is established by Shi et al., which plays an important role in the present paper.

Let

Then for each neighborhood

In 1978, Marotto [

In this paper, we will study the chaotification problem of a delay difference equation, chaotic or not, in the form of

The objective here is to design a control input sequence

Set

As defined in [

In the following, without loss of generality and for simplicity, we can suppose that the origin

It is well known that the stability and expansion of a map at a fixed point has a close relationship with the modulus of the eigenvalues of its derivative operator when the map is differentiable at the fixed point. Suppose that

Assume that

If

where

If

When

Now, we establish a criterion of chaos for the induced system (

Let

there exists a point

when

when

We will apply Lemma

It follows from assumption (i) and the second conclusion of Theorem

Next, we will show that

For the case where

For the case where

It is obvious that

Therefore, all the assumptions in Lemma

Since

Based on Theorem

Consider the controlled system (

there exists a point

Then there exist two positive constants

We will use Theorem

For convenience, let

It is obvious that the function

Next, we need to show that

For

For

Finally, let

It is clear that the classical sinusoidal function

In [

Since the point

In the last part of this section, we give an example to illustrate the theoretical result of Theorem

We take the map

In fact, there is only one fixed point

Here, we take

It should be pointed out that the relative existing chaotification scheme in [

Simple dynamical behaviors of uncontrolled system (

Complex dynamical behaviors of controlled system (

Simple dynamical behaviors of uncontrolled system (

Complex dynamical behaviors of controlled system (

In this paper, we study the chaotification problem for a class of delay difference equations with at least one fixed point. We first establish a criterion of chaos by using the snap-back repeller theory. Then, based on this criterion of chaos and the feedback control approach, we establish a chaotification scheme. We have proved that the controlled system is chaotic in the sense of both Devaney and Li-Yorke when the parameters of the system satisfy some mild conditions. Numerical simulations confirm the theoretical analysis. The chaotification problem for more general maps in the original system will be our further research.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (Grants 11101246, 61273088, and 10971120), the Nature Science Foundation of Shandong Province (Grant ZR2010FM010), and the Postdoctoral Science Foundation of China (Grant 2014M561908).