EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 3254-3267
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On the Periodicity Solutions of Five Systems of
Rational Systems of Difference Equations of Order Five

Jawharah Ghuwayzi AL-Juaid

Department of Mathematics and Statistics, Collage of Science, Taif University, P.O. Box
11099, Taif 21944, Saudi Arabia

Abstract. The primary aim of this paper is to explore the behavior of several nonlinear systems
of difference equations following

Tn+1 =
EnEn−4

Tn−3(±1± EnEn−4)
, En+1 =

TnTn−4

En−3(±1± TnTn−4)
,

and obtain solution expressions for them. Moreover, we utilize MATLAB programming to simulate
the dynamics and validate our results.

2020 Mathematics Subject Classifications: 39A10

Key Words and Phrases: Solutions of difference equations, periodic solution, recursive
sequences

1. Introduction

The final 25 years of the 20th century saw significant advancements in the theory
of discrete dynamical systems (DDSs) and difference equations (DEs). Recently, a wide
variety of fields, including biology, economics, physics, resource management, and others,
have seen the use of DDSs and DEs.A key role in practical analysis is played by the theory
of DIFEs. It is improbable that the theory of DEs won’t keep playing a significant part
in mathematics as a whole. In applications, nonlinear difference equations (NDEs) of
order greater than one are crucial. These equations also naturally arise as discrete analogs
and numerical solutions of differential and delay differential equations, which model a
wide range of diverse phenomena in biology, ecology, psychology, engineering, physics,
probability theory, economics, genetics. Finding out how a system of higher-order rational
difference equations (RDEs) behaves and talking about how stable its equilibrium points
are locally asymptotically is quite interesting. Many articles cover the DEs system [1–22]
. For example, In [7] El-Metwally discussed how various systems of third order RDEs
with initial conditions involving non-zero real numbers should be solved. Additionally,

DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5425

Email address: jo.gh@tu.edu.sa (J. G. AL-Juaid)

https://www.ejpam.com 3254 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



J. G. AL-Juaid / Eur. J. Pure Appl. Math, 17 (4) (2024), 3254-3267 3255

he showed additional numerical examples and checked into some of the properties of the
obtained solutions

Tn+1 =
Tn−1Rn

±Tn−1 ±Rn−2
, Rn+1 =

TnRn−1

±Rn−1 ± Tn−2
.

El-Dessoky [6] studied the presence of solutions in the case of four dimensions for a
class of rational systems of differential equations (SDEs) of order four.

Xn+1 =
Xn−3

±1± TnZn−1Yn−2Xn−3
, Yn+1 =

Yn−3

±1±XnTn−1Zn−2Yn−3
.

Zn+1 =
Zn−3

±1± YnXn−1Tn−2Zn−3
, Tn+1 =

Tn−3

±1± ZnYn−1Xn−2Tn−3
,

Elsayed and Alof [11] obtained formulas expressions for solutions of the following frac-
tional SDEs

Xn+1 =
Xn−3Yn−4

Yn(±1−Xn−3Yn−4Rn−1Tn−2)
, Yn+1 =

Yn−3Rn−4

Rn(±1− Yn−3Rn−4Tn−1Xn−2)
,

Rn+1 =
Rn−3Tn−4

Tn(±1±Rn−3Tn−4Xn−1Yn−2)
, Tn+1 =

Tn−3Xn−4

Xn(±1± Tn−3Xn−4Yn−1Rn−2)
.

Mansour et al. [20] checked the behavior of solutions of the SDEs

Wn+1 =
Wn−5

−1 +Wn−5Rn−2
, Rn+1 =

Rn−5

±1±Rn−5Wn−2
.

DEs are also suitable models to describe circumstances where overlapping generations and
seasonal population growth occur. The generalized Beverton-Holt stock recruitment mode
has been studied by researchers in [3]

Tn+1 = αTn +
βTn−1

1 + γTn−1 + ηTn
.

The following system of discrete-time two-predators and the one-prey Lot was explored
dynamically by Khaliq et al.[14]

Xn+1 =
aXn − cXnYn − eXnZn

1 + dXn
, Yn+1 =

bYn + tXnYn − pYnZn

1 + wYn
, Zn+1 =

fZn + rXnZn − hYnZn

1 + kZn
.

Din and Elsayed [5] evaluated the two-directional interacting and invasive species
model’s boundedness nature, persistence, local and global behavior

Rn+1 = η + αRn + βRn−1e
−Tn , Tn+1 = γ + cTn + wTn−1e

−Rn .

In a discrete-time COVID-19 epidemic model,The authors [18] used chaos management,
bifurcation analysis, and topological classifications to study local dynamics. See also
[19, 21, 22].



J. G. AL-Juaid / Eur. J. Pure Appl. Math, 17 (4) (2024), 3254-3267 3256

Motivated by the aforementioned researches, the objective of this work is to ascertain
whether solutions to the SDEs exist in the two-dimensional instances

Tn+1 =
EnEn−4

Tn−3(±1± EnEn−4)
, En+1 =

TnTn−4

En−3(±1± TnTn−4)
,

the initial conditions are arbitrary nonzero real numbers.

2. The System Tn+1 =
EnEn−4

Tn−3(1+EnEn−4)
, En+1 =

TnTn−4

En−3(−1+TnTn−4)

In this section, we give a specific form the solutions of the SDE in the form:

Tn+1 =
EnEn−4

Tn−3(1 + EnEn−4)
, En+1 =

TnTn−4

En−3(−1 + TnTn−4)
. (1)

Theorem 1. Assume {Tn, En} are solutions of Eq.(1). Then, for all n = 0, 1, 2, ...,
the following formulas yield periodic solutions of Eq. (1) with period eight.

T8n−4 = s, T8n−3 = d,

T8n−2 = c, T8n−1 = b,

T8n = a, T8n+1 =
fw

d(1 + fw)
,

T8n+2 =
as

c(−1 + 2as)
, T8n+3 =

−fw

b(1− fw)
,

E8n−4 = w, E8n−3 = m,

E8n−2 = g, E8n−1 = k,

E8n = f, E8n+1 =
as

m(−1 + as)
,

E8n+2 =
−fw

g
, E8n+3 =

as

k(1− as)
,

where T0 = a, T−1 = b, T−2 = c, T−3 = d, T−4 = s, E0 = f, E−1 = k, E−2 = g, E−3 =
m, and E−4 = w. Also, where E0E−4 ̸= ±1 and T0T−4 ̸= ±1.
Proof. For n = 0, the outcome is valid. Let us now assume that n > 0 and that n − 1
agrees with our assumption. That’s

T8n−12 = s, T8n−11 = d,

T8n−10 = c, T8n−9 = b,

T8n−8 = a, T8n−7 =
fw

d(1 + fw)
,



J. G. AL-Juaid / Eur. J. Pure Appl. Math, 17 (4) (2024), 3254-3267 3257

T8n−6 =
as

c(−1 + 2as)
, T8n−5 =

−fw

b(1− fw)
,

E8n−12 = w, E8n−11 = m,

E8n−10 = g, E8n−9 = k,

E8n−8 = f, E8n−7 =
as

m(−1 + as)
,

E8n−6 =
−fw

g
, E8n−5 =

as

k(1− as)
.

Next, from system (1) we have

T8n =
E8n−1E8n−5

T8n−4(1 + E8n−1E8n−5)
=

kas
k(1−as)

s(1 + kas
k(1−as))

= a.

E8n =
T8n−1T8n−5

E8n−4(−1 + T8n−1T8n−5)
=

−bfw
b(1−fw)

w(−1− fwb
b(1−fw))

= f.

T8n−4 =
E8n−5E8n−9

T8n−8(1 + E8n−5E8n−9)
=

ask
k(1−as)

a(1 + kas
k(1−as))

= s.

E8n−4 =
T8n−5T8n−9

E8n−8(−1 + T8n−5T8n−9)
=

−bfw
b(1−fw)

f(−1− fwb
b(1−fw))

= w.

Similarly, we can prove the remaining relations. The proof is complete.
Example 1. Figure (1) demonstrates the behavior of the solutions of the SDE Eq.(1)
with T0 = 1, T−1 = 0.5, T−2 = 0.2, T−3 = 2, T−4 = 3, E0 = 0.1, E−1 = 0.5, E−2 =
1, E−3 = 0.2, and E−4 = 0.3.

3. The System Tn+1 =
EnEn−4

Tn−3(1+EnEn−4)
, En+1 =

TnTn−4

En−3(1−TnTn−4)

In this section, For the aforementioned system, we provide the period eight peri-
odic solutions and solution expression.

Tn+1 =
EnEn−4

Tn−3(1 + EnEn−4)
, En+1 =

TnTn−4

En−3(1− TnTn−4)
. (2)

Theorem 2. Suppose that {Tn, En} be solutions of the system of SDEs (2). Then for
n = 0, 1, 2, .., the solutions of Eq.(2) can be formed as follows

T8n−4 = s, T8n−3 = d,

T8n−2 = c, T8n−1 = b,



J. G. AL-Juaid / Eur. J. Pure Appl. Math, 17 (4) (2024), 3254-3267 3258

Figure 1: Periodic solutions of Eq.(1)

T8n = a, T8n+1 =
fw

d(1 + fw)
,

T8n+2 =
as

c
, T8n+3 =

fw

b(1 + fw)
,

E8n−4 = w, E8n−3 = m,

E8n−2 = g, E8n−1 = k,

E8n = f, E8n+1 =
as

m(1− as)
,

E8n+2 =
fw

g
, E8n+3 =

as

k(1− as)
,

where E0E−4 ̸= −1 and T0T−4 ̸= 1.
Proof. For n = 0, the outcome is valid. Assume for the moment that n > 0 and that
n− 1 falls under our hypothesis. That’s

T8n−12 = s, T8n−11 = d,

T8n−10 = c, T8n−9 = b,

T8n−8 = a, T8n−7 =
fw

d(1 + fw)
,

T8n−6 =
as

c
, T8n−5 =

fw

b(1 + fw)
,

E8n−12 = w, E8n−11 = m,



J. G. AL-Juaid / Eur. J. Pure Appl. Math, 17 (4) (2024), 3254-3267 3259

E8n−10 = g, E8n−9 = k,

E8n−8 = f, E8n−7 =
as

m(−1 + as)
,

E8n−6 =
fw

g
, E8n−5 =

as

k(1− as)
.

Next, from system (2) we have

T8n+1 =
E8nE8n−4

T8n−3(1 + E8nE8n−4)
=

fw

d(1 + fw)
.

E8n+1 =
T8nT8n−4

E8n−3(1− T8nT8n−4)
=

as

m(1− as)
.

T8n =
E8n−1E8n−5

T8n−4(1 + E8n−1E8n−5)
=

ask
k(1−as)

s(1 + kas
k(1−as))

= a.

E8n =
T8n−1T8n−5

E8n−4(−1 + T8n−1T8n−5)
=

bfw
b(1+fw)

w(1− fwb
b(1+fw))

= f.

We can confirm the other forms using the same method. The evidence is finished.
Example 2. Figure (2) illustrates the behavior of the solutions of the SDEs (2) with T0 =
2, T−1 = 1.5, T−2 = 0.2, T−3 = 0.4, T−4 = 1, E0 = 0.1, E−1 = 0.6, E−2 = 3, E−3 = 1,
and E−4 = 0.3.

Figure 2: Chart the behavior of solutions of the SDEs(2)



J. G. AL-Juaid / Eur. J. Pure Appl. Math, 17 (4) (2024), 3254-3267 3260

4. The System Tn+1 =
EnEn−4

Tn−3(1+EnEn−4)
, En+1 =

TnTn−4

En−3(−1−TnTn−4)

In this section, we find form the solutions of the SDEs in the form:

Tn+1 =
EnEn−4

Tn−3(1 + EnEn−4)
, En+1 =

TnTn−4

En−3(−1− TnTn−4)
. (3)

Theorem 3. Suppose that {Tn, En} are solutions of Eq.(3). After that, all of solutions
of Eq.(3) are periodic with period eight and given by the following formulas for n=0,1,2,...
,

T8n−4 = s, T8n−3 = d,

T8n−2 = c, T8n−1 = b,

T8n = a, T8n+1 =
fw

d(1 + fw)
,

T8n+2 =
−as

c
, T8n+3 =

fw

b(−1− fw)
,

E8n−4 = w, E8n−3 = m,

E8n−2 = g, E8n−1 = k,

E8n = f, E8n+1 =
as

m(−1− as)
,

E8n+2 =
fw

g(−1− 2fw)
, E8n+3 =

−as

k(−1 + as)
,

where E0E−4 ̸= ±1 and T0T−4 ̸= ±1.
Proof. The outcome is valid for n = 0. Let us now assume that n > 0 and that n− 1 is
consistent with our assumption. That is

T8n−12 = s, T8n−11 = d,

T8n−10 = c, T8n−9 = b,

T8n−8 = a, T8n−7 =
fw

d(1 + fw)
,

T8n−6 =
−as

c
, T8n−5 =

fw

b(−1− fw)
,

E8n−12 = w, E8n−11 = m,

E8n−10 = g, E8n−9 = k,

E8n−8 = f, E8n−7 =
as

m(−1− as)
,

E8n−6 =
fw

g(−1− 2fw)
, E8n−5 =

−as

k(−1 + as)
.



J. G. AL-Juaid / Eur. J. Pure Appl. Math, 17 (4) (2024), 3254-3267 3261

Next, from system (3) we have

T8n+2 =
E8n+1E8n−3

T8n−2(1 + E8n+1E8n−3)
=

mas
m(−1−as)

c(1 + mas
m(−1−as))

=
−as

c
.

E8n+2 =
T8n+1T8n−3

E8n−2(−1− T8n+1T8n−3)
=

fwd
d(1+fw)

g(−1− fwd
d(1+fw))

=
fw

g(−1− 2fw)
.

T8n+3 =
E8n+2E8n−2

T8n−1(1 + E8n+2E8n−2)
=

fwg
g(−1−2fw)

b(1 + fwg
g(−1−2fw))

=
fw

b(−1− fw)
.

E8n+3 =
T8n+2T8n−2

E8n−1(−1− T8n+2T8n−2)
=

−asc
c

k(−1 + asc
c

=
−as

k(−1 + as)
.

We can validate the other forms by using the same procedure. The proof is complete.
Example 3. The solution is periodic of period eight when Theorem 3 is met and the
initial values are T0 = 3, T−1 = 1, T−2 = 0.1, T−3 = 2, T−4 = 0.2, E0 = 1, E−1 =
0.5, E−2 = 0.3, E−3 = 2, and E−4 = 0.9, as shown in Figure (3).

Figure 3: Draw a graph displaying periodicity of the solutions of SDEs (3)

5. The System Tn+1 =
EnEn−4

Tn−3(−1+EnEn−4)
, En+1 =

TnTn−4

En−3(−1+TnTn−4)

In this section, we investigate the solutions of the SDEs in the form:

Tn+1 =
EnEn−4

Tn−3(−1 + EnEn−4)
, En+1 =

TnTn−4

En−3(−1 + TnTn−4)
. (4)



J. G. AL-Juaid / Eur. J. Pure Appl. Math, 17 (4) (2024), 3254-3267 3262

Theorem 4. Consider {Tn, En} are solutions of Eq.(4). Afterwards, for all n = 0, 1,
2,..., the following formulas obtain periodic solutions of Eq. (4) with period eight.

T8n−4 = s, T8n−3 = d,

T8n−2 = c, T8n−1 = b,

T8n = a, T8n+1 =
fw

d(−1 + fw)
,

T8n+2 =
as

c
, T8n+3 =

fw

b(−1 + fw)
,

E8n−4 = w, E8n−3 = m,

E8n−2 = g, E8n−1 = k,

E8n = f, E8n+1 =
as

m(−1 + as)
,

E8n+2 =
fw

g
, E8n+3 =

as

k(−1 + as)
,

where E0E−4 ̸= 1 and T0T−4 ̸= 1.
Proof. The result is valid for n = 0. Let us now assume that n > 0 and that n− 1 agrees
with our assumption. That is

T8n−12 = s, T8n−11 = d,

T8n−10 = c, T8n−9 = b,

T8n−8 = a, T8n−7 =
fw

d(−1 + fw)
,

T8n−6 =
as

c
, T8n−5 =

fw

b(−1 + fw)
,

E8n−12 = w, E8n−11 = m,

E8n−10 = g, E8n−9 = k,

E8n−8 = f, E8n−7 =
as

m(−1 + as)
,

E8n−6 =
fw

g
, E8n−5 =

as

k(−1 + as)
.

Next, from system (4) we have

T8n−4 =
E8n−5E8n−9

T8n−8(−1 + E8n−5E8n−9)
=

ask
k(−1+as)

a(−1 + ask
k(−1+as))

= s.



J. G. AL-Juaid / Eur. J. Pure Appl. Math, 17 (4) (2024), 3254-3267 3263

E8n−4 =
T8n−5T8n−9

E8n−8(−1 + T8n−5T8n−9)
=

fwb
b(−1+fw)

f(−1 + fwb
b(−1+fw))

= w.

T8n =
E8n−1E8n−5

T8n−4(−1 + E8n−1E8n−5)
=

ask
k(−1+as)

s(−1 + kas
k(−1+as))

= a.

E8n =
T8n−1T8n−5

E8n−4(−1 + T8n−1T8n−5)
=

bfw
b(−1+fw)

w(−1 + fwb
b(−1+fw))

= f.

Also, we can prove the other relations. This completes the proof.
Example 4. We assume that T0 = 0.1, T−1 = 2, T−2 = 0.5, T−3 = 1, T−4 = 0.3, E0 =
0.1, E−1 = 1, E−2 = 0.6, E−3 = 1.1, and E−4 = 0.7 for the Eq.(4). (See Fig. 4).

Figure 4: Sketch the periodicity of the solution of Eq.(4)

6. The System Tn+1 =
EnEn−4

Tn−3(−1−EnEn−4)
, En+1 =

TnTn−4

En−3(1−TnTn−4)

In this section, we obtain the form of the solutions of the SDEs

Tn+1 =
EnEn−4

Tn−3(−1− EnEn−4)
, En+1 =

TnTn−4

En−3(1− TnTn−4)
. (5)

Theorem 5. Let that {Tn, En} are solutions of Eq. (5). Next, each of solutions Eq.(5)
are periodic, having a period of eight, and is given by formulas for n = 0, 1, 2,...,

T8n−4 = s, T8n−3 = d,

T8n−2 = c, T8n−1 = b,



J. G. AL-Juaid / Eur. J. Pure Appl. Math, 17 (4) (2024), 3254-3267 3264

T8n = a, T8n+1 =
fw

d(−1− fw)
,

T8n+2 =
−as

c
, T8n+3 =

fw

b(1 + fw)
,

E8n−4 = w, E8n−3 = m,

E8n−2 = g, E8n−1 = k,

E8n = f, E8n+1 =
as

m(1− as)
,

E8n+2 =
fw

g(−1− 2fw)
, E8n+3 =

−as

k(1 + as)
,

where E0E−4 ̸= ±1 and T0T−4 ̸= ±1.
Proof. For n = 0, the result is true. Now, let’s suppose that n > 0 and that n − 1
supports our hypothesis. That’s

T8n−12 = s, T8n−11 = d,

T8n−10 = c, T8n−9 = b,

T8n−8 = a, T8n−7 =
fw

d(−1− fw)
,

T8n−6 =
−as

c
, T8n−5 =

fw

b(1 + fw)
,

E8n−12 = w, E8n−11 = m,

E8n−10 = g, E8n−9 = k,

E8n−8 = f, E8n−7 =
as

m(1− as)
,

E8n−6 =
fw

g(−1− 2fw)
, E8n−5 =

−as

k(1 + as)
.

Next, from system (5) we have

T8n+1 =
E8nE8n−4

T8n−3(−1− E8nE8n−4)
=

fw

d(−1− fw)
.

E8n+1 =
T8nT8n−4

E8n−3(1− T8nT8n−4)
=

as

m(1− as)
.

T8n+3 =
E8n+2E8n−2

T8n−1(−1− E8n+2E8n−2)
=

fwg
g(−1−2fw)

b(−1− fwg
g(−1−2fw))

=
fw

b(1 + fw)
.



J. G. AL-Juaid / Eur. J. Pure Appl. Math, 17 (4) (2024), 3254-3267 3265

E8n+3 =
T8n+2T8n−2

E8n−1(1− T8n+2T8n−2)
=

−asc
c

k(1− −asc
c )

=
−as

k(1 + as)
.

We can prove the other relations as well. This brings the proof to a close.
Example 5. See Figure (5) where we takes system (5) with T0 = 0.5, T−1 = 4, T−2 =
7, T−3 = 0.5, T−4 = 1, E0 = 1.2, E−1 = 0.5, E−2 = 3, E−3 = 4, and E−4 = 0.2.

Figure 5: Chart the behavior of the solutions of system (5)

Remark 1 The solutions of the following systems can be also obtained .

Tn+1 =
EnEn−4

Tn−3(1 + EnEn−4)
, En+1 =

TnTn−4

En−3(1 + TnTn−4)
.

Tn+1 =
EnEn−4

Tn−3(−1 + EnEn−4)
, En+1 =

TnTn−4

En−3(−1− TnTn−4)
.

Tn+1 =
EnEn−4

Tn−3(−1 + EnEn−4)
, En+1 =

TnTn−4

En−3(1− TnTn−4)
.

7. Conclusion

This study investigates the solutions of five SDEs. Across five sections, we derive
periodic solutions with a period of eight for each system. To corroborate our theoretical
findings, numerical examples are presented for each system, with Figures 1-5 providing
visual confirmation of the results.



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