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We consider the higher order nonlinear rational difference equation

Recently, dynamics of nonnegative solutions of higher order rational difference equation has been an area of intense interest. Related to this subject, researches are done by Dehghan et al. [

Our aim in this paper is to study the higher order nonlinear rational difference equation

The periodic character of positive solutions of (

Motivated by the above results, our interest is now to study and generalize the previous results to the general case depicted in (

The change of variable

This paper is organized besides this introduction in three sections. In Section

For the sake of self-containment and convenience, we recall the following definitions and results from [

Let

A solution of (

(i) The equilibrium point

(ii) The equilibrium point

(iii) The equilibrium point

(iv) The equilibrium point

(v) The equilibrium point

(i) A solution

(ii) A solution

Let

(a) If all roots of (

(b) If at least one of the roots of (

The following result from [

If

The aforementioned lemma leads to the following conclusion.

If

In this section, we give a necessary and sufficient condition for (

Equation (

(a) If

(b) If

Assume that there exist distinct nonnegative real numbers

(b) If

Construct the quadratic equation

Suppose (

To investigate the local stability of the two cycles

We have

Now let

However; by Lemma

Hence, the characteristic polynomial is given by

Assume that

With that in mind, it is clear that

Next we will establish inequality (

Now, by applying Theorem

However, by the Descartes’ Rule of Signs

The proof is complete.

The characteristic equation of the linearized equation at the equilibrium solution is given by

In order to illustrate the results of the previous section and to support our theoretical discussion, we consider several numerical examples generated by MATLAB.

Dynamics of

Dynamics of

Dynamics of

In this paper, we showed that the period-two solution of the higher order nonlinear rational difference equation

We consider the aforementioned result as a step forward in investigating bigger classes of difference equations which afford the ELAS property; that is, the existence of a periodic solution implies its local asymptotic stability.

The authors are very grateful to the anonymous referees for carefully reading the paper and for their comments and valuable suggestions that lead to an improvement in the paper.