The dynamical behavior of a system of max-type difference equations model

Abstract

Difference equation as a discrete model has been widely used in the fields of algorithm analysis,control theory, computer science, biology, economics and so on. In this paper, we mainly study the dynamical behavior of the system of high order max-type difference equations model,

$x_{n}=\max\left\{{\frac{A}{y_{n-1}},x_{n-k}}\right\},\quad y_{n}=\max\left\{{% \frac{B}{x_{n-1}},y_{n-k}}\right\},\quad n=1,2,\ldots,$

where $A$ and $B$ are non-negative real numbers, initial values are positive real numbers. We use mathematical induction to study the periodic properties of the solutions when $AB=0$ and $AB\neq 0$ . Our conclusion is that: (1) Every positive solution of the difference equations model is periodic with period $k$ if $AB=0$ ; (2) Every positive solution of the difference equations model is eventually periodic with period $k$ if $AB\neq 0$ . At last we give some numberical examples about the system to verify our theoretical results. The conclusions of this paper complement and generalize the existing results.

Keywords

Max-type difference equation positive solution periodicity discrete model

1. Introduction

Equation is an important modeling tool [1, 2], difference equations (systems) is a recurrence relation which contains the unknown function and its difference but does not contain the derivative. It is a powerful modeling tool to describe discrete-time systems in the real world. For example, difference equation model has been widely used in the fields of algorithm analysis, control theory, computer science, biology, economics and so on (see [3, 4, 5, 6, 7, 8, 9]). Furthermore, many continuous mathematical models can be converted to their corresponding discrete versions so as to benefit numerical simulations, which can simplify the process and results of the study. So, difference equation models can be established to depict a variety of natural and social phenomena.

Recently, there has been a great interest in studying properties of higher-order, non-autonomous, max-type difference equations and systems, for example [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. These results are not only valuable in their own right, but they can provide insight into their differential counterparts. In this paper, we study a high order max-type difference equations model and deal with the questions of whether every positive solution of the system of difference equations model is periodic.

In [8], Li made a thorough research on the stability of particle’s trajectory in particle swarm through difference equation and Z transform, discuss the influences of pBest, gBest and randomicity on particle’ s trajectory. In [9], through the study on nonlinear effect off digital image pixel grid, Zhu established one- dimensional and two-dimensional time evolution equations model of nonlinear effect between pixels.

In [10], Elsayed studied the periodic property of following nonlinear difference equation

$\displaystyle x_{n+1}=\max\left\{{\frac{A_{n}}{x_{n-k}},x_{n-s}}\right\},\mbox% { }n=0,1,\ldots,$

where $k$ , $s$ , $\in N$ , $A_{n}\in N$ and initial values are positive real numbers.

In [11], Iričanin studied the property of solutions of following second order max-type system of difference equation as follows

$\displaystyle x_{n+1}=\max\left\{{\frac{A_{n}}{y_{n}},x_{n-1}}\right\},y_{n+1}% =\max\left\{{\frac{B_{n}}{x_{n}},y_{n-1}}\right\},\quad n=0,1,\ldots,$

In [12], Ibrahim studied the periodicity of following max-type system of difference equations with positive two-periodic sequences

$\displaystyle x_{n+1}=\max\left\{\frac{A_{n}}{y_{n-1}},x_{n-1}\right\},y_{n+1}% =\max\left\{\frac{B_{n}}{x_{n-1}},y_{n-1}\right\},\quad n=0,1,\ldots,$

where $A_{n}$ , $B_{n}\in N$ are two-periodic positive sequences, and initial values $x_{i},y_{i}\in(0,\infty)$ , $i=0$ , $-1$ . Motivated by above studies, in this paper, we mainly study the eventually periodicity of following system of high order max-type difference equations model,

$\displaystyle\left\{{{\begin{array}[]{*{20}c}{x_{n}=\max\left\{{\frac{A}{y_{n-% 1}},x_{n-k}}\right\},}\\ {y_{n}=\max\left\{{\frac{B}{x_{n-1}},y_{n-k}}\right\},}\\ \end{array}}}\right.n=1,2,\ldots,$ (1)

where $A$ , $B\in[0,+\infty)$ and $x_{i},y_{i}\in(0,+\infty)$ , $i=0$ , $-$ 1, $\ldots$ , $1-k$ , initial values are positive real numbers. We will show that the solution of this system has following properties: (1) Every positive solution of the system of difference equations model is periodic with period $k$ if $AB=0$ ; (2) Every positive solution of the system of difference equations model is eventually periodic with period $k$ if $AB\neq 0$ .

2. Some definitions

In this section,we will introduce some definitions (see [13, 14]) which will be needed.

Definition 1. [13] Let $I$ be some interval of numbers and let $f:I\times I\to I$ be a continuously difference function. A difference equation of order ( $k+1$ ) is an equation of the form

$\displaystyle x_{n+1}=f(x_{n},x_{n-1},\cdots,x_{n-k}),\mbox{ }n=0,1,\ldots,$

A point $\bar{x}\in I$ is called equilibrium solution of the difference equation if $\bar{x}=f(\bar{x},\bar{x},\ldots,\bar{x})$ , that is $x_{n}=\bar{x}$ for all $n\geqslant-k$ .

Definition 2. [14] A solution $\{x_{n}\}_{n=-k}^{+\infty}$ of difference equation is called periodic with period $p$ (or a period- $p$ solution) if there exists an integer $p\geqslant 1$ such that

$\displaystyle x_{n+p}=x_{n}\ \text{for all}\ n\geqslant-k.$

We say that the solution is periodic with prime period $p$ if the smallest positive integer for $x_{n+p}=x_{n}$ holds for all $n\geqslant-k$ . In this case, a $p$ -tuple $(x_{n},x_{n-1},\ldots,x_{n-k})$ of any $p$ consecutive values of the solutions called a p-cycle of the difference equation.

Definition 3. [14] A solution $\{x_{n}\}_{n=-k}^{+\infty}$ of difference equation is called eventually periodic with period $p$ if there exists an integer $N\geqslant 1$ such that $\{x_{n}\}_{n=-k}^{+\infty}$ is periodic with period $p$ , that is

$\displaystyle x_{n+p}=x_{n}\ \text{for all}\ n\geqslant N.$

3. Main results

In this section, we formulate and prove some lemmas and main theorems.

First, we will study the system (1) in the case of $AB=0$ .

Theorem 1. If $AB=0$ , then every positive solution $\{x_{n},y_{n}\}(n\geqslant 1)$ of system (1) is periodic with period $k$ .

Proof: Let’s assume $A=0$ , so $AB=0$ . According to the system (1), we can easily obtain that $x_{n}=x_{n-k}$ for any $n\geqslant 1$ , at the same time we know

$\displaystyle y_{1}=\max\left\{{\frac{B}{x_{0}},y_{1-k}}\right\},$

$\displaystyle y_{2}=\max\left\{{\frac{B}{x_{1}},y_{2-k}}\right\}=\max\left\{{% \frac{B}{x_{1-k}},y_{2-k}}\right\},$

$\displaystyle y_{3}=\max\left\{{\frac{B}{x_{2}},y_{3-k}}\right\}=\max\left\{{% \frac{B}{x_{2-k}},y_{3-k}}\right\},$

$\displaystyle\ldots$

$\displaystyle y_{1+k}=\max\left\{{\frac{B}{x_{k}},y_{1}}\right\}=\max\left\{{% \frac{B}{x_{0}},y_{1}}\right\}=\max\left\{{\frac{B}{x_{0}},\max\left\{{\frac{B% }{x_{0}},y_{1-k}}\right\}}\right\}=y_{1},$

$\displaystyle y_{2+k}=\max\left\{{\frac{B}{x_{1+k}},y_{2}}\right\}=\max\left\{% {\frac{B}{x_{1}},y_{2}}\right\}=\max\left\{{\frac{B}{x_{1-k}},y_{2}}\right\}=% \max\left\{{\frac{B}{x_{1-k}},\max\left\{{\frac{B}{x_{1-k}},y_{2-k}}\right\}}% \right\}=y_{2},$

$\displaystyle y_{3+k}=\max\left\{{\frac{B}{x_{2+k}},y_{3}}\right\}=\max\left\{% {\frac{B}{x_{2}},y_{3}}\right\}=\max\left\{{\frac{B}{x_{2-k}},y_{3}}\right\}=% \max\left\{{\frac{B}{x_{2-k}},\max\left\{{\frac{B}{x_{2-k}},y_{3-k}}\right\}}% \right\}=y_{3},$

by induction, we obtain

$\displaystyle y_{n+k}=\max\left\{{\frac{B}{x_{n+k-1}},y_{n}}\right\}=\max\left% \{{\frac{B}{x_{n-1}},y_{n}}\right\}=\max\left\{{\frac{B}{x_{n-1}},\max\left\{{% \frac{B}{x_{n-1}},y_{n-k}}\right\}}\right\}=y_{n}.$

So, the positive solution of system (1) is $k$ periodic.

When $B=0$ , the proof is similar. $\Box$

Next, we will study system (1) in the case of $AB\neq 0$ .

Lemma 1. If $AB\neq 0$ , then there exists an integer $t\geqslant k$ , such that $x_{t}=x_{t-k}$ .

Proof: Proof by contradiction. Assume $x_{n}\neq x_{n-k}$ for all $n\geqslant 1$ , that is $x_{n}=\frac{A}{y_{n-1}}>x_{n-k}$ . Therefore

$\displaystyle x_{1}=\frac{A}{y_{0}}>x_{1-k},x_{2}=\frac{A}{y_{1}}>x_{2-k},% \ldots,x_{1+k}=\max\left\{{\frac{A}{y_{k}},x_{1}}\right\}=\frac{A}{y_{k}}>x_{1% }=\frac{A}{y_{0}},$

so $y_{k}<y_{0}$ , but $y_{k}=\max\left\{{\frac{B}{x_{k-1}},y_{0}}\right\}\geqslant y_{0}(n\geqslant 1)$ , which is contradictory. So there exists an integer $t\geqslant k$ , such that $x_{t}=x_{t-k}$ . $\Box$

Lemma 2. For all $n\geqslant 0$ , followings are true:

(i)
If $k$ is even, then there exists $t\geqslant k$ , such that $\{x_{t+2n}\}$ and $\{y_{t+2n+1}\}$ are periodic with period $k$ ;
(ii)
If $k$ is odd, then there exists $\tau>k$ , such that $\{x_{\tau+2n}\}$ and $\{y_{\tau+2n+1}\}$ are periodic with period $k$ .

Proof: (i) According to Lemma 1, we have a number of $t\in{\rm N}$ , such that $x_{t}=x_{t-k}$ , because

$\displaystyle y_{t+1}=\max\left\{{\frac{B}{x_{t}},y_{t+1-k}}\right\}=\max\left% \{{\frac{B}{x_{t-k}},y_{t+1-k}}\right\},y_{t+1-k}=\max\left\{{\frac{B}{x_{t-k}% },y_{t+1-2k}}\right\}\geqslant\frac{B}{x_{t-k}},$

so $y_{t+1}=y_{t+1-k}$ .

According to Lemma 1, there is

$\displaystyle x_{t+k}=\max\left\{{\frac{A}{y_{t+k-1}},x_{t}}\right\}=\max\left% \{{\frac{A}{y_{t+k-1}},x_{t-k}}\right\},$

so

$\displaystyle x_{t+2}=\max\left\{{\frac{A}{y_{t+1}},x_{t+2-k}}\right\}=\max% \left\{{\frac{A}{y_{t+1-k}},x_{t+2-k}}\right\}=\max\left\{{\frac{A}{y_{t+1-k}}% ,\max\left\{{\frac{A}{y_{t+1-k}},x_{t+2-2k}}\right\}}\right\}=x_{t+2-k},$

$\displaystyle y_{t+3}=\max\left\{{\frac{B}{x_{t+2}},y_{t+3-k}}\right\}=\max% \left\{{\frac{B}{x_{t+2-k}},y_{t+3-k}}\right\}=\max\left\{{\frac{B}{x_{t+2-k}}% ,\max\left\{{\frac{B}{x_{t+2-k}},y_{t+3-2k}}\right\}}\right\}=y_{t+3-k}.$

Similarly, by induction, there is

$\displaystyle x_{t}=x_{t-k},\quad x_{t+2}=x_{t+2-k},\quad x_{t+4}=x_{t+4-k},\ldots,$ $\displaystyle y_{t+1}=y_{t+1-k},\quad y_{t+3}=y_{t+3-k},\quad y_{t+5}=y_{t+5-k% },\ldots.$

Therefore, if $k$ is even, then

$\displaystyle x_{t+k}=x_{t}=x_{t-k},\quad x_{t+2+k}=x_{t+2}=x_{t+2-k},\ldots,$ $\displaystyle y_{t+1+k}=y_{t+1}=y_{t+1-k},\quad y_{t+3+k}=y_{t+3}=y_{t+3-k},\ldots.$

By mathematical induction, for every $n\geqslant 0$ , $\{x_{t+2n}\}$ and $\{y_{t+2n+1}\}$ are periodic with period $k$ .

(ii) If $k$ is odd, $k(k+1)$ is even. According to (i) we know

$\displaystyle x_{t+k(k+1)}=x_{t+k^{2}}=\max\left\{{\frac{A}{y_{t+k^{2}-1}},x_{% t+k^{2}-k}}\right\}.$

Since

$\displaystyle y_{t+k^{2}-1}=\max\left\{{\frac{B}{x_{t+k^{2}-2}},y_{t+k^{2}-1-k% }}\right\}\geqslant y_{t+k^{2}-1-k},$

so

$\displaystyle\frac{A}{y_{t+k^{2}-1}}\leqslant\frac{A}{y_{t+k^{2}-1-k}},$

but

$\displaystyle x_{t+k^{2}-k}=\max\left\{{\frac{A}{y_{t+k^{2}-k-1}},x_{t+k^{2}-2% k}}\right\}\geqslant\frac{A}{y_{t+k^{2}-k-1}},$

so

$\displaystyle x_{t+k(k+1)}=x_{t+k^{2}}=\max\left\{{\frac{A}{y_{t+k^{2}-1}},x_{% t+k^{2}-k}}\right\}=x_{t+k^{2}-k}.$

Since

$\displaystyle x_{t+k^{2}}=\max\left\{{\frac{A}{y_{t+k^{2}-1}},x_{t+k^{2}-k}}% \right\}\geqslant x_{t+k^{2}-k},$

so

$\displaystyle\frac{B}{x_{t+k^{2}}}\leqslant\frac{B}{x_{t+k^{2}-k}},$

but

$\displaystyle y_{t+k^{2}+1-k}=\max\left\{{\frac{B}{x_{t+k^{2}-k}},y_{t+k^{2}+1% -2k}}\right\}\geqslant\frac{B}{x_{t+k^{2}-k}},$

so

$\displaystyle y_{t+k(k+1)+1}=y_{t+k^{2}+1}=\max\left\{{\frac{B}{x_{t+k^{2}}},y% _{t+k^{2}+1-k}}\right\}=y_{t+k^{2}+1-k}.$

In the same way, for any $i=0,2,4,\ldots$ , we have

$\displaystyle x_{t+k^{2}+k+2}=x_{t+k^{2}+2}=x_{t+k^{2}+2-k},y_{t+k^{2}+k+3}=y_% {t+k^{2}+3}=y_{t+k^{2}+3-k},$ $\displaystyle\ldots$ $\displaystyle x_{t+k^{2}+k+i}=x_{t+k^{2}+i}=x_{t+k^{2}+i-k},y_{t+k^{2}+k+i+1}=% y_{t+k^{2}+i+1}=y_{t+k^{2}+i+1-k}.$

Let $\tau=t+k^{2}-k$ , by mathematical induction, for every $n\geqslant 0$ , $\{x_{\tau+2n}\}$ and $\{y_{\tau+2n+1}\}$ are periodic with period $k$ . $\Box$

Lemma 3. There exists $n_{t}\in N$ , such that $x_{n_{t}}=x_{n_{t}-k}$ and $y_{n_{t}}=y_{n_{t}-k}$ .

Proof: According to Lemma 1 and Lemma 2, when $k$ is even, there exists an integer $t\geqslant k$ , such that $\{x_{t+2n}\}$ and $\{y_{t+2n+1}\}$ are periodic with period $k$ for all $n\geqslant 0$ . We will proof Lemma 3 by contradiction.

Assume if $x_{n_{t}}=x_{n_{t}-k}$ then $y_{n_{t}}\neq y_{n_{t}-k}$ for all $n_{t}\geqslant t$ . Therefore, according to Lemma 2 we know

$\displaystyle x_{t}=x_{t-k},x_{t+1}=\frac{A}{y_{t}}>x_{t+1-k},x_{t+2}=x_{t+2-k% },\ldots,x_{t+k-1}=\frac{A}{y_{t+k-2}},x_{t+k}=x_{t},x_{t+k+1}=\frac{A}{y_{t+k% }}>x_{t+1};$

$\displaystyle y_{t}=\frac{B}{x_{t-1}}>y_{t-k},y_{t+1}=y_{t+1-k},y_{t+2}=\frac{% B}{x_{t+1}},\ldots,y_{t+k-1}=y_{t-1},y_{t+k}=\frac{B}{x_{t+k-1}}>y_{t}.$

$\displaystyle y_{t+k}=\max\left\{{\frac{B}{x_{t+k-1}},y_{t}}\right\}=\frac{B}{% x_{t+k-1}}=\frac{B}{\frac{A}{y_{t+k-2}}}=\frac{B}{A}y_{t+k-2}=\frac{B}{A}\frac% {B}{x_{t+k-3}}=\frac{B}{A}\frac{B}{\frac{A}{y_{t+k-4}}}$

$\displaystyle\qquad=\frac{B^{2}}{A^{2}}y_{t+k-4}=\ldots=\left({\frac{B}{A}}% \right)^{\frac{k}{2}}y_{t}>y_{t},$

it shows $B>A$ . By the same method, we obtain

$\displaystyle x_{t+k+1}=\max\left\{\frac{A}{y_{t+k}},x_{t+1}\right\}=\frac{A}{% y_{t+k}}=\left(\frac{A}{B}\right)^{\frac{k}{2}}x_{t+1}>x_{t+1},$

which means $A>B$ , that is contradictory. So, when $k$ is even, there exists $n_{t}\in N$ , such that $x_{n_{t}}=x_{n_{t}-k}$ and $y_{n_{t}}=y_{n_{t}-k}$ .

When $k$ is odd, the proof is similar. In summary, there exists $n_{t}\in N$ , such that $x_{n_{t}}=x_{n_{t}-k}$ and $y_{n_{t}}=y_{n_{t}-k}$ . $\Box$

Theorem 2. For any $k\in{\rm N}$ , $\{x_{n}\}$ and $\{y_{n}\}$ is eventually periodic with period $k$ .

Proof: When $k$ is even, according to Lemma 3, we know there exists $n_{t}\in N$ , such tha $x_{n_{t}}=x_{n_{t}-k}\geqslant\frac{A}{y_{n_{t}-1}}$ and $y_{n_{t}}=y_{n_{t}-k}\geqslant\frac{B}{x_{n_{t}-1}}$ , so

$\displaystyle x_{n_{t}+1}=\max\left\{{\frac{A}{y_{n_{t}}},x_{n_{t}+1-k}}\right% \}=\max\left\{{\frac{A}{y_{n_{t}-k}},x_{n_{t}+1-k}}\right\},$

$\displaystyle y_{n_{t}+1}=\max\left\{{\frac{B}{x_{n_{t}}},y_{n_{t}+1-k}}\right% \}=\max\left\{{\frac{B}{x_{n_{t}-k}},y_{n_{t}+1-k}}\right\}.$

Since

$\displaystyle x_{n_{t}+1-k}=\max\left\{{\frac{A}{y_{n_{t}-k}},x_{n_{t}+1-2k}}% \right\}\geqslant\frac{A}{y_{n_{t}-k}},$

$\displaystyle y_{n_{t}+1-k}=\max\left\{{\frac{B}{x_{n_{t}-k}},y_{n_{t}+1-2k}}% \right\}\geqslant\frac{B}{x_{n_{t}-k}},$

so $x_{n_{t}+1}=x_{n_{t}+1-k}$ , $y_{n_{t}+1}=y_{n_{t}+1-k}$ .

$\displaystyle x_{n_{t}+2}=\max\left\{{\frac{A}{y_{n_{t}+1}},x_{n_{t}+2-k}}% \right\}=\max\left\{{\frac{A}{y_{n_{t}+1-k}},x_{n_{t}+2-k}}\right\}=x_{n_{t}+2% -k},$ $\displaystyle y_{n_{t}+2}=\max\left\{{\frac{B}{x_{n_{t}+1}},y_{n_{t}+2-k}}% \right\}=\max\left\{{\frac{B}{x_{n_{t}+1-k}},y_{n_{t}+2-k}}\right\}=y_{n_{t}+2% -k},\ldots,$ $\displaystyle x_{n_{t}+k-1}=x_{n_{t}-1},y_{n_{t}+k-1}=y_{n_{t}-1}.$ $\displaystyle x_{n_{t}+k}=\max\left\{{\frac{A}{y_{n_{t}+k-1}},x_{n_{t}}}\right% \}=\max\left\{{\frac{A}{y_{n_{t}-1}},x_{n_{t}}}\right\}=x_{n_{t}}=x_{n_{t}-k},$ $\displaystyle y_{n_{t}+k}=\max\left\{{\frac{B}{x_{n_{t}+k-1}},y_{n_{t}}}\right% \}=\max\left\{{\frac{B}{x_{n_{t}-1}},y_{n_{t}}}\right\}=y_{n_{t}}=y_{n_{t}-k}.$

By mathematical induction, for every $i=0,1,\ldots,k-1$ , there is

$\displaystyle x_{n_{t}+i}=x_{n_{t}+i+k}=x_{n_{t}+i+2k}=\ldots,y_{n_{t}+i}=y_{n% _{t}+i+k}=y_{n_{t}+i+2k}=\ldots.$

So, $\{x_{n}\}$ and $\{y_{n}\}$ is eventually periodic with period $k$ . $\Box$

Corollary 1. When $x_{n}=y_{n}$ and $A=B$ , every positive solution of following difference equation

$x_{n}=\max\left\{{\frac{A}{x_{n-1}},x_{n-k}}\right\},n=0,1\ldots,$ (2)

is periodic with period $k$ , where $A=B\in[0,+\infty)$ and $x_{i}$ , $y_{i}\in(0,+\infty)$ , $i=0,1,\ldots,1-k$ , initial values are positive real numbers.

Proof: When $x_{n}=y_{n}$ and $A=B$ , system (1) is to be (2), that is

$\displaystyle x_{n}=\max\left\{{\frac{A}{x_{n-1}},x_{n-k}}\right\},n=0,1\ldots.$

When $A=0$ , According to Theorem 1, the conclusion is established, obviously.

When $A\neq 0$ , According to Theorem 2, we know every positive solution of the system of difference equations model is eventually periodic with period $k$ . That is there exists an integer $N\geqslant 1$ such that $x_{n+k}=x_{n}$ for all $n\geqslant N$ .

We claim that the smallest positive integer $N$ is 1. That is $x_{n+k}=x_{n}$ for all $n\geqslant 1$ . Otherwise, $x_{1+k}\neq x_{1}$ , so

$\displaystyle x_{1+k}=\max\left\{{\frac{A}{x_{k}},x_{1}}\right\}=\frac{A}{x_{k% }}=\max\left\{{\frac{A}{\max\left\{{\frac{A}{x_{k-1}},x_{0}}\right\}},x_{1}}% \right\}=\max\left\{{\min\left\{{x_{k-1},\frac{A}{x_{0}}}\right\},x_{1}}\right\},$

since $x_{1}=\max\left\{{\frac{A}{x_{0}},x_{1-k}}\right\}\geqslant\frac{A}{x_{0}}$ , so $x_{1+k}=x_{1}$ . That is contradictory, so $x_{n+k}=x_{n}$ for all $n\geqslant 1$ , that is every positive solution of difference Eq. (2) is periodic with period $k$ . $\Box$
4. Numericai experiments

Computer numerical simulation helps to understand and observe the dynamical behavior of difference equations model (1) betterly. In this section, we will use Matlab to carry out numerical experiments on the main conclusions of Section 3.

We will give some examples about the system (1) of difference equations, and draw two-dimensional graphics using Matlab mathematical software. These graphics are the trajectory of solutions with initial values are given, in order to show the periodicity of the solution.

Model 1. When $A=$ 0, $B=$ 6 and $k=$ 3, system (1) is

$\displaystyle\left\{{{\begin{array}[]{l}x_{n}=x_{n-3},\\ {y_{n}=\max\left\{{\frac{6}{x_{n-1}},y_{n-3}}\right\},}\hfill\\ \end{array}}}\right.n=1,2,\ldots,$ (3)

It satisfies the conditions of Theorem 1, so, every positive solution of system (3) is periodic with period 3. For example, taking initial value $x_{0}=2$ , $x_{-1}=0.4$ , $x_{-2}=10$ , $y_{0}=20$ , $y_{-1}=$ 0.4, $y_{-2}=$ 5, let’s observe the trajectory of the solution (see Fig. 1, Table 1) .

Table 1

The data of solution of system (3)

Initial value $x_{0}=$ 2, $x_{-1}=$ 0.4, $x_{-2}=$ 10, $y_{0}=$ 20, $y_{-1}=$ 0.4, $y_{-2}=$ 5
$n$	1	2	3	4	5	6	7	8	9	…
$x(n)$	10	0.4	2	10	0.4	2	10	0.4	2	…
$y(n)$	5	0.6	20	5	0.6	20	5	0.6	20	…

Figure 1.

The solution of system (3) when initial value $x_{0}=2$ , $x_{-1}=$ 0.4, $x_{-2}=$ 10, $y_{0}=$ 20, $y_{-1}=$ 0.4, $y_{-2}=$ 5.

According to the model, we can see from Fig. 1 and Table 1 that $x_{n+3}=x_{n}$ , $y_{n+3}=y_{n}$ for all $n\geqslant 1$ , when initial value $x_{0}=2$ , $x_{-1}=$ 0.4, $x_{-2}=$ 10, $y_{0}=$ 20, $y_{-1}=$ 0.4, $y_{-2}=$ 5. That is model (1) is periodic with period 3.

Model 2. When $A=1,B=10$ and $k=6$ , system (1) is

$\displaystyle\left\{{{\begin{array}[]{l}{x_{n}=\max\left\{{\frac{1}{y_{n-1}},x% _{n-6}}\right\},}\hfill\\ {y_{n}=\max\left\{{\frac{10}{x_{n-1}},y_{n-6}}\right\},}\hfill\\ \end{array}}}\right.\mbox{ }n=1,2,\ldots,$ (4)

It satisfies the conditions of Theorem 2, so, every positive solution of system (4) is eventually periodic with period 2. For example, taking initial value $x_{0}=$ 40, $x_{-1}=$ 0.1, $y_{0}=$ 2, $y_{-1}=$ 10, let’s observe the trajectory of the solution (see Fig. 2, Table 2) .

Table 2

The data of solution of system (4)

Initial value $x_{0}=$ 40, $x_{-1}=$ 2.5, $x_{-2}=$ 4.8, $x_{-3}=$ 27, $x_{-4}=$ 30, $x_{-5}=$ 0.6, $y_{0}=$ 6.4, $y_{-1}=$ 5, $y_{-2}=$ 10, $y_{-3}=$ 18, $y_{-4}=$ 4.2, $y_{-5}=$ 5.8
$n$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
$x(n)$	0.6	30	27	4.8	2.5	40	0.6	30	27	4.8	2.5	40	0.6	30	27	4.8	2.5	40	0.6	30
$y(n)$	5.8	4.2	18	10	5	6.4	5.8	4.2	18	10	5	16.7	5.8	4.2	18	10	5	16.7	5.8	4.2

Table 3

The data of solution of system (5)

Initial value $x_{0}=$ 4, $x_{-1}=$ 0.5, $x_{-2}=$ 0.6, $x_{-3}=$ 0.8, $x_{-4}=$ 2.4, $y_{0}=$ 2, $y_{-1}=$ 16, $y_{-2}=$ 4.8, $y_{-3}=$ 5, $y_{-4}=$ 0.4
$n$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	$\ldots$
$x(n)$	2.4	1.6	0.8	0.64	4	2.4	2.4	1.2	0.96	4	2.4	2.4	1.2	0.96	4	…
$y(n)$	2.5	5	6.25	12.5	15.625	2.5	5	6.25	12.5	15.625	2.5	5	6.25	12.5	15.625	…

Figure 2.

The solution of system (4) when initial value $x_{0}=$ 40, $x_{-1}=$ 2.5, $x_{-2}=$ 4.8, $x_{-3}=$ 27, $x_{-4}=$ 30, $x_{-5}=$ 0.6, $y_{0}=$ 6.4, $y_{-1}=$ 5, $y_{-2}=$ 10, $y_{-3}=$ 18, $y_{-4}=$ 4.2, $y_{-5}=$ 5.8.

According to the model, we can see from Fig. 2 and Table 2 that $x_{n+6}=x_{n}$ , $y_{n+6}=y_{n}$ for all $n\geqslant 7$ , when initial value $x_{0}=$ 40, $x_{-1}=$ 2.5, $x_{-2}=$ 4.8, $x_{-3}=$ 27, $x_{-4}=$ 30, $x_{-5}=$ 0.6, $y_{0}=$ 6.4, $y_{-1}=$ 5, $y_{-2}=$ 10, $y_{-3}=$ 18, $y_{-4}=$ 4.2, $y_{-5}=$ 5.8. That is model (1) is periodic with period 6.

Model 3. When $A=$ 6, $B=$ 0.4 and $k=5$ , system (1) is

$\displaystyle\left\{{{\begin{array}[]{*{20}c}{x_{n}=\max\left\{{\frac{6}{y_{n-% 1}},x_{n-5}}\right\},}\\ {y_{n}=\max\left\{{\frac{0.4}{x_{n-1}},y_{n-5}}\right\},}\\ \end{array}}}\right.\mbox{ }n=1,2,\ldots,$ (5)

It satisfies the conditions of Theorem 2, so, every positive solution of system (5) is eventually periodic with period 5. For example, taking initial value $x_{0}=$ 4, $x_{-1}=$ 0.5, $x_{-2}=$ 0.6, $x_{-3}=$ 0.8, $x_{-4}=$ 2.4, $y_{0}=$ 2, $y_{-1}=$ 16, $y_{-2}=$ 4.8, $y_{-3}=$ 5, $y_{-4}=$ 0.4, let’s observe the trajectory of the solution (see Fig. 3, Table 3) .

Figure 3.

The Solution of system (5) when initial value $x_{0}=$ 4, $x_{-1}=$ 0.5, $x_{-2}=$ 0.6, $x_{-3}=$ 0.8, $x_{-4}=$ 2.4, $y_{0}=$ 2, $y_{-1}=$ 16, $y_{-2}=$ 4.8, $y_{-3}=$ 5, $y_{-4}=$ 0.4s.

According to the model, we can see from Fig. 3 and Table 3 that $x_{n+5}=x_{n}$ , $y_{n+2}=y_{n}$ , for all $n\geqslant 5$ , when initial value $x_{0}=$ 4, $x_{-1}=$ 0.5, $x_{-2}=$ 0.6, $x_{-3}=$ 0.8, $x_{-4}=$ 2.4, $y_{0}=$ 2, $y_{-1}=$ 16, $y_{-2}=$ 4.8, $y_{-3}=$ 5, $y_{-4}=$ 0.4. That is model (3) is eventually periodic with period 5.

Model 4. When $x_{n}=y_{n}$ , $A=B=1$ and $k=5$ , system (1) is

$x_{n}=\max\left\{{\frac{1}{x_{n-1}},x_{n-5}}\right\},n=1,2,\ldots,$ (6)

It satisfies the conditions of Corollary 1, so, every positive solution of system (6) is eventually periodic with period 4. For example, taking initial value $x_{0}=$ 2, $x_{-1}=$ 4.8, $x_{-2}=$ 12, $x_{-3}=$ 8, $x_{-4}=$ 0.3, let’s observe the trajectory of the solution (see Fig. 4, Table 4) .

Figure 4.

The solution of system (6) when initial value $x_{0}=$ 2, $x_{-1}=$ 4.8, $x_{-2}=$ 12, $x_{-3}=$ 8, $x_{-4}=$ 0.3.

Table 4

The data of solution of system (3)

Initial value $x_{0}=$ 2, $x_{-1}=$ 4.8, $x_{-2}=$ 12, $x_{-3}=$ 8, $x_{-4}=$ 0.3
$n$	1	2	3	4	5	6	7	8	9	10	…
$x(n)$	0.5	8	12	4	2	0.5	8	12	4	2	…

According to the model, we can see from Fig. 4 Table 4 that $x_{n+5}=x_{n}$ , for al $n\geqslant 1$ , when initial value $x_{0}=$ 2, $x_{-1}=$ 4.8, $x_{-2}=$ 12, $x_{-3}=$ 8, $x_{-4}=$ 0.3. That is model (4) is periodic with period 5.

5. Conclusion

In this paper, we investigate the dynamical behavior of the positive solution of the system model of max-type difference Eq. (1).

First, we showed every positive solution of the system of difference equations model is periodic with period $k$ if $AB=$ 0. Then, we proved every positive solution of the system of difference equations model is eventually periodic with period $k$ if $AB\neq 0$ .

At last, we give four models of system (1), draw trajectories of the solutions by giving initial values and assign a value for $k$ , thus intuitively reflect the periodic property of system (1).

Conflict of interest

The authors confirm that this article content has no conflicts of interest.

Footnotes

Acknowledgments

This work is supported by National Natural Science Foundation of China (No.11461007), the promotion project of basic ability for young and middle-aged teachers in colleges and universities of Guangxi in 2016 (No. KY2016YB069). This work is also supported by Program on the High Level Innovation Team and Outstanding Scholars in Universities of Guangxi Province, the project of Guangxi Colleges and Universities Key Laboratory of Mathematical andStatistical Model (No. 2016 GXKLMS010) and the NaturalScience Foundation of Guangxi Normal University.

References

Mez

and Luiggi

A.N.

, Adapting the condensation and evaporation model to the study of kinetics of phase transformations in binary metal systems, Journal of Computational Methods in Sciences & Engineering 14(1–3) (2014), 179–194.

Mohammadi

, A computational approach for solution of boundary layer equations for the free convection along a vertical plate, Journal of Computational Methods in Sciences & Engineering 15(3) (2015), 317–326.

El-Metwally

, Global behavior of an economic model, Chaos Solitons & Fractals 33(2007), 994–1005.

El-Metwally

and El-Afifi

M.M.

, On the behavior of some extension forms of some population Models, Chaos Solitons & Fractals 36 (2008), 104–114.

Zhou

Honghua

H.U.

Liang

et al., Research on difference equation model in traffic flow calculation, Journal of Changchun University of Science & Technology 2 (2014), 117–123.

Huang

C.M.

and Wang

W.P.

, Applications of difference equation in population forecasting model, Advanced Materials Research 1–2 (2015), 1079–1080.

Touafek

and Elsayed

E.M.

, On the solutions of systems of rational difference equations, Mathematical and Computer Modelling 55 (2012), 1987–1997.

Sun

D.B.

Zou

et al., An analysis for a particle’ s trajectory of PSO based on difference equation, Chinese Journal of Computers 29 (2006), 2052–2060.

Zhu

Z.M.

Liu

R.Q.

Liu

F.F.

et al, Analysis of image processing model based on pixels of solitary waves, Journal of Beijing University of Aeronautics and Astronautics 41 (2015), 2335–2339.

10.

Elsayed

E.M.

Iričanin

B.D.

and Stevic

, On the max-type equation xn+1=max⁡{An/xn,xn-1}, Ars Combinatoria 95 (2010), 187–192.

11.

Iričanin

B.D.

and Touafek

, On a second order max-type system of difference equations, Indian Journal of Mathematics 54 (2012), 119–142.

12.

Ibrahim

T.F.

and Touafek

, Max-type system of difference equations with positive two-periodic sequences, Mathe-matical Methods in the Applied Sciences 16 (2014), 2562–2569.

13.

Kulenovic

M.R.S.

and Ladas

, Dynamics of second order rational difference equations: with open problems and conjectures, New York: Chapman Hall/CRC, 2002.

14.

Grove

E.A.

and Ladas

, Periodicities in nonlinear difference equations, NewYork: Chapman Hall/CRC, 2005.

15.

Sun

T.X.

Q.L.

Wua

and Xi

H.J.

, Global behavior of the max-type difference equation, Applied Mathematics and Computation 248 (2014), 687–692.

16.

Ganeriwala

and Zohdi

T.I.

, A coupled discrete element-finite difference model of selective laser sintering, Granular Matter 18 (2016), 1–15.

17.

Abo-Zeid

, Global behavior of a higher order difference equation, Discrete Dynamics in Nature & Society 64 (2014), 830–839.

18.

Yazlik

Tollu

D.T.

and Taskara

, On the solutions of a max-type difference equation system, Mathematical Methods in the Applied Sciences 38 (2015), 4388–4410.