Abstract

This paper presents the solutions to the following nonlinear systems of rational difference equations: where initial conditions are nonnegative real numbers. Finally some numerical simulations are presented to verify obtained theoretical results.

1. Introduction

The purpose of present study is to solve and deal with the following difference equations systems:where are arbitrary nonnegative real numbers.

It is anticipated that discrete dynamical systems can be seen as discrete analogous of differential as well as delay differential equations. Moreover, these systems designate certain natural phenomena in economy, physics, biology, and many more. Many scholars and researchers have studied various dynamical properties of difference equations along their systems in recent years. For example, Asiri et al. [1] have investigated the periodicity nature of the following system:

Cinar et al. [2] have obtained the solution of the subsequent recursive system:

Metwally and Elsayed [3] have explored the periodicity nature of the following system:

Touafek et al. [4] have explored the periodicity nature of the following system:

Elsayed [5] has obtained the solution of the following rational system:

For more interesting results regarding dynamical properties of difference and differential equations along their systems, we refer the readers to [2, 6–28] and the references cited therein.

2. On 1st System

This section is about the investigation of the solution form of the following system:

The forms of solutions to (8) are given as Theorem 1.

Theorem 1. Let be a solution to (8), and also let , respectively, be . Then one has

Proof. Obviously results true if . Assuming that for and results hold, that isNow from (8), one hasAgain, from system (8), we obtainHence, the proof of other relations is obvious.

3. On 2nd System

In this section, we explore the solutions of the recursive system:

Theorem 2. Let be a solution to (14), and also let , respectively, be ; then

Proof. Obviously results true if . Assuming that for and results hold, that isNext, one can obtain from system (14) thatSimilarly, system (14) givesObviously, the subsequent unassertive relations follows, and this complete the proof.

4. On 3rd System

In this section, we will explore the solutions of following rational systems:

Theorem 3. Let be a solution of (20), and also let , respectively, are . Then one has

Proof. Obviously results true if . Assuming that for and results hold, that isNext, it can be seen from system (20) thatNow, hereafter, we prove an extra result. It is noted that system (20) leads toIn a similar way, one can establish other formulas.

5. On 4th System

This section determines and studies the formulas for solutions of the following nonlinear system:

Theorem 4. Let be a solution to (26), and also let , respectively, are . Then one has

Proof. Obviously results true if . Assuming that for and results hold, that isNext, one can obtain from system (26) thatAlso from system (26) one gets:Subsequent results follow from the above assertion.

6. Numerical Simulation

In order to verify our theoretical results, we consider four interesting numerical examples by fixing suitable initial values. These simulation shows the solutions of under consideration discrete-time systems, which are depicted in (8), (14), (20), and (26).

Example 1. This example presents that the solutions of discrete-time system (8) with , respectively, are as shown in Figure 1.


Figure 1 
Plot of the solutions of system.

Example 2. This example shows the behavior of system (14). The relevant plot is given in Figure 2 under the initial conditions: , respectively, are .


Figure 2 
Plot of the solutions of system.

Example 3. Figure 3 depicts the dynamics of solutions to (20) when we randomly consider the values as follows: , respectively, are .


Figure 3 
Plot of the solutions of system.

Example 4. The behavior of system (26) are plotted in Figure 4 with , respectively, are .


Figure 4 
Plot of the solutions of system.

7. Conclusion

In this paper, we deal with the form of the solutions of four cases of the nonlinear systems of difference equations . Finally some numerical examples are giving by fixing suitable initial values to show the qualitative behavior of under consideration systems.

Data Availability

All the data used in this article have been included, and the sources were they where adopted were cited accordingly.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

H. S. Alayachi and M. S. M. Noorani would like to acknowledge UKM Grant DIP-2017-011 and Ministry of Education Malaysia Grant FRGS/1/2017/STG06/UKM/01/1 for financial support while the research by A. Q. Khan and A. Khaliq was partially supported by the Higher Education Commission of Pakistan.