arXiv:1701.00103v1 [math.DS] 31 Dec 2016

On the solutions of a second-order difference
equations in terms of generalized Padovan
sequences
Yacine Halim1 and Julius Fergy T. Rabago2 1 Department of Mathematics and computer sceince
Mila University Center, Mila, Algeria Email: halyacine@yahoo.fr
2 Department of Mathematics and computer sciences, College of Science, University of the Philippines,
Gov. Pack Road, Baguio City 2600, Benguet, Philippines. Email: jfrabago@gmail.com

Abstract

This paper deals with the solution, stability character and asymptotic behavior of the rational difference equation

xn+1

=

xn-1 +  ,  xn xn-1

n  N0,

where N0 = N  {0}, , ,   R+, and the initial conditions x-1 and x0 are non zero real numbers such that their solutions are associated to generalized Padovan numbers. Also, we investigate the two-dimensional case of the this equation given by

xn+1

=

xn-1 +  ,  yn xn-1

yn+1

=

yn-1 +  ,  xn yn-1

n  N0,

and this generalizes the results presented in [34]. Keywords: Difference equations, general solution, stability, generalized Padovan numbers. Mathematics Subject Classification: 39A10, 40A05.

1 Introduction and preliminaries
The term difference equation refers to a specific type of recurrence relation ­ a mathematical relationship expressing xn as some combination of xi with i < n. These equations usually appear as discrete mathematical models of many biological and environmental phenomena such as population growth and predatorprey interactions (see, e.g., [8] and [18]), and so these equations are studied

1

because of their rich and complex dynamics. Recently, the problem of finding

closed-form solutions of rational difference equations and systems of rational of

difference equations have gained considerable interest from many mathemati-

cians. In fact, countless papers have been published previously focusing on the

aforementioned topic, see for example [5, 6, 7, 16, 20] and [21]. Interestingly,

some of the solution forms of these equations are even expressible in terms of

well-known integer sequences such as the Fibonacci numbers, Horadam numbers

and Padovan numbers (see, e.g., [9, 11, 12, 14, 22, 24, 25, 26, 27, 29, 34]).

It is well-known that linear recurrences with constant coefficients, such as

the recurrence relation Fn+1 = Fn + Fn-1 defining the Fibonacci numbers, can be solved through various techniques (see, e.g., [17]). Finding the closed-

form solutions of nonlinear types of difference equations, however, are far more

interesting and challenging compared to those of linear types. In fact, as far

as we know, there has no known general method to deal with different classes

of difference equations solvable in closed-forms. Nevertheless, numerous studies

have recently dealt with finding appropriate techniques in solving closed-form

solutions of some systems of difference equations (see, e.g., [2, 5, 6, 7, 15, 23]).

Motivated by these aforementioned works, we investigate the rational differ-

ence equation

xn+1

=

xn-1 +   xn xn-1

,

n  N0, .

(1)

Particularly, we seek to find its closed-form solution and examine the global

stability of its positive solutions. We establish the solution form of equation

(1) using appropriate transformation reducing the equation into a linear type

difference equation. Also, we examine the solution form of the two-dimensional

analogue of equation (1) given in the following more general form

xn+1

=

xn-1 +   yn xn-1

,

yn+1

=

yn-1 +   xn yn-1

,

n  N0.

(2)

The case  =  =  = 1 has been studied by Tollu, Yazlik and Taskara in [34]. The authors in [34] established the solution form of system (2) (in the case  =  =  = 1) through induction principle.
The paper is organized as follows. In the next section (Section 2), we review some definitions and important results necessary for the success of our study, and this includes a brief discussion about generalized Padovan numbers. In section 3 and 4, we established the respective solution forms of equations (1) and the system (2), and examine their respective stability properties. Finally, we end our paper with a short summary in Section 5.

2 Preliminaries
2.1 Linearized stability of an equation
Let I be an interval of real numbers and let F : Ik+1 - I

2

be a continuously differentiable function. Consider the difference equation

xn+1 = F (xn, xn-1, . . . , xn-k)

(3)

with initial values x0, x-1, . . . x-k  I..

Definition 1. A point x  I is called an equilibrium point of equation(3) if

x = F (x, x, . . . , x).

Definition 2. Let x be an equilibrium point of equation(3).

i) The equilibrium x is called locally stable if for every  > 0, there exist  > 0 such that for allx-k, x-k+1, . . . x0  I with

|x-k - x| + |x-k+1 - x| + . . . + |x0 - x| < ,

we have |xn - x| < , for all n  -k.

ii) The equilibrium x is called locally asymptotically stable if it is locally stable, and if there exists  > 0 such that if x-1, x0  I and

|x-k - x| + |x-k+1 - x| + . . . + |x0 - x| < ,

then

lim
n+

xn

=

x.

iii) The equilibrium x is called global attractor if for all x-k, x-k+1, . . . x0  I, we have

lim
n+

xn

=

x.

iv) The equilibrium x is called global asymptotically stable if it is locally stable and a global attractor.

v) The equilibrium x is called unstable if it is not stable.

vi)

Let

pi =

f ui

(x,

x,

.

.

.

,

x),

i = 0, 1, . . . , k.

Then,

the

equation

yn+1 = p0yn + p1yn-1 + . . . + pkyn-k,

(4)

is called the linearized equation of equation (3) about the equilibrium point x.

The next result, which was given by Clark [3], provides a sufficient condition for the locally asymptotically stability of equation (3).

Theorem 1 ([3]). Consider the difference equation (4). Let pi  R, then,

|p0| + |p1| + . . . + |pk| < 1

is a sufficient condition for the locally asymptotically stability of equation (3).

3

2.2 Linearized stability of the second-order systems

Let f and g be two continuously differentiable functions:

f : I2 × J 2 - I, g : I2 × J 2 - J, I, J  R

and for n  N0, consider the system of difference equations

xn+1 = f (xn, xn-1, yn, yn-1) yn+1 = g (xn, xn-1, yn, yn-1)

(5)

where (x-1, x0)  I2 and (y-1, y0)  J 2. Define the map H : I2 ×J 2 - I2 ×J 2 by
H(W ) = (f0(W ), f1(W ), g0(W ), g1(W ))
where W = (u0, u1, v0, v1)T , f0(W ) = f (W ), f1(W ) = u0, g0(W ) = g(W ), g1(W ) = v0. Let Wn = [xn, xn-1, yn, yn-1]T . Then, we can easily see that system (5) is equivalent to the following system written in vector form

Wn+1 = H(Wn), n = 0, 1, . . . ,

(6)

that is

 

xn+1

=

f (xn, xn-1, yn, yn-1)

 

xn

 yn+1

= =

xn g (xn, xn-1, yn, yn-1)

.

 

yn = yn

Definition 3 (Equilibrium point). An equilibrium point (x, y)  I ×J of system (5) is a solution of the system

x = f (x, x, y, y) , y = g (x, x, y, y) .

Furthermore, an equilibrium point W  I2 × J2 of system (6) is a solution of the system
W = H(W ).

Definition 4 (Stability). Let W be an equilibrium point of system (6) and . be any norm (e.g. the Euclidean norm).

1. The equilibrium point W is called stable (or locally stable) if for every  > 0 exist  such that W0 - W <  implies Wn - W <  for n  0.
2. The equilibrium point W is called asymptotically stable (or locally asymptotically stable) if it is stable and there exist  > 0 such that W0-W <  implies Wn - W  0, n  +.

3. The equilibrium point W is said to be global attractor (respectively global attractor with basin of attraction a set G  I2 × J2, if for every W0 (respectively for every W0  G)

Wn - W  0, n  +.

4

4. The equilibrium point W is called globally asymptotically stable (respectively globally asymptotically stable relative to G) if it is asymptotically stable, and if for every W0 (respectively for every W0  G),
Wn - W  0, n  +.
5. The equilibrium point W is called unstable if it is not stable.
Remark 1. Clearly, (x, y)  I × J is an equilibrium point for system (5) if and only if W = (x, x, , y, y, )  I2 × J2 is an equilibrium point of system (6).
From here on, by the stability of the equilibrium points of system (5), we mean the stability of the corresponding equilibrium points of the equivalent system (6).

2.3 Generalized Padovan sequence
The integer sequence defined by the recurrence relation

Pn+1 = Pn-1 + Pn-2, n  N,

(7)

with the initial conditions P-2 = 0, P-1 = 0, P0 = 1 (so P0 = P1 = P2 = 1), is known as the Padovan numbers and was named after Richard Padovan. This is the same recurrence relation as for the Perrin sequence, but with different initial conditions (P0 = 3, P1 = 0, P2 = 2). The first few terms of the recurrence sequence are 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, . . .. The Binet's formula for this recurrence sequence can easily be obtained and is given by

Pn

=

( (

- -

1)( )(

- -

1) )

n

+

( - 1)( ( - )(

- -

1) )

n

+

( (

- -

1)( )(

- -

1) )

n.

where



=

r2+12
6r 

(the

so-called

plastic

number),



=

-

 2



+i

3 2

r 6

-

2 r

and

r = 3 108 + 12 69. The plastic number corresponds to the golden number

1+ 2

5

associated

with

the

equiangular

spiral

related

to

the

conjoined

squares

in

Fibonacci numbers, that is,

lim
n

Pn+1 Pn

= .

For more informations associated with Padovan sequence, see [4] and [19].

Here we define an extension of the Padovan sequence in the following way

S-2 = 0, S-1 = 0, S0 = 1, Sn+1 = pSn-1 + qSn-2, n  N. (8)

The Binet's formula for this recurrence sequence is given by

Sn

=

( (

- -

1)( )(

- -

1) )

n

+

( (

- -

1)( )(

- -

1) )

n

+

( (

- 1)( - )(

- -

1) )

n.

5

where 

=

R2+12p 6R

,



=



-

 2

+i

3 2

R 6

-

2p R

and R = 3 108q + 12

One can easily verify that

lim
n

Sn+1 Sn

= .

-12p3 + 81q2.

3 Closed-Form solutions and stability of equation (1)
For the rest of our discussion we assume Sn, the n-th generalized Padovan number, to satisfy the recurrence equation
Sn+1 = pSn-1 + qSn-2, n  N0,
with initial conditions S-2 = 0, S-1 = 0, S0 = 1.

3.1 Closed-Form solutions of equation (1)

In this section, we derive the solution form of equation (1) through an analytical

approach.

We

put

q

=

 

and

p

=

 

,

hence

we

have

the

equation

xn+1

=

pxn-1 + xnxn-1

q

;

n  N0.

(9)

Consider the equivalent form of equation (9) given by

xn+1

=

p xn

+

q xn xn-1

which, upon the change of variable xn+1 = zn+1/zn, transforms into

zn+1 = pzn-1 + qzn-2.

(10)

Now, we iterate the right hand side of equation (10) as follows

zn+1 = pzn-1 + qzn-2 = qzn-2 + p2zn-3 + qpzn-4 = p2zn-3 + 2pqzn-4 + q2zn-5 = 2pqzn-4 + (p3 + q2)zn-5 + qp2zn-6 = (p3 + q2)zn-5 + 3p2qzn-6 + 2pq2zn-7 = 3p2qzn-6 + (p4 + 3pq2)zn-7 + (p3 + q3)zn-8 = (p4 + 3pq2)zn-7 + (q3 + 4qp3)zn-8 + 3p2q2zn-9 ...
= Sn+1z0 + Sn+2z-1 + Snqz-2.

6

Hence,

xn+1

=

zn+1 zn

=

Sn+1z0 + Sn+2z-1 + Snqz-2 Snz0 + Sn+1z-1 + Sn-1qz-2.

=

Sn+1

z0 z-1

+ Sn+2

+

Sn-2

q

z-2 z-1

Sn

z0 z-1

+ Sn+1

+

Sn-1 q

z-2 z-1

=

Sn+1x0

+

Sn+2

+

Snq

1 x-1

Sn x0

+

Sn+1

+

Sn-1q

1 x-1

=

Sn+1 x0 x-1 Snx0x-1 +

+ Sn+2x-1 Sn+1x-1 +

+ Snq Sn-1q

.

The above computations prove the following result.

Theorem 2. Let {xn}n-1 be a solution of (9). Then, for n = 1, 2, . . . ,

xn

=

Sn+1x-1 + Snx0x-1 Snx-1 + Sn-1x0x-1

+ +

qSn-1 qSn-2

.

(11)

where the initial conditions x-1, x0  R - F , with F is the Forbidden Set of equation (9) given by



F=

(x-1, x0) : Snx-1 + Sn-1x0x-1 + qSn-2 = 0 .

n=-1

If  =  = , then from (11) we get

xn

=

Pn+1x-1 Pnx-1 +

+ Pnx0x-1 Pn-1 x0 x-1

+ +

qPn-1 qPn-2

.

Hence, for  =  =  we have Sn = Pn, n  N, and consequently we get the solution given in [34].

3.2 Global stability of solutions of equation (1)

In this section we study the global stability character of the solutions of equation
(9). It is easy to show that eqrefeq1 has a unique positive equilibrium point given by x = . Let I = (0, +), and consider the function f : I2 - I defined by

f (x,

y)

=

py + xy

q

.

Theorem 3. The equilibrium point x is locally asymptotically stable.

Proof. The linearized equation of equation (9) about the equilibrium x is

yn+1 = t1yn + t2yn-1

7

where

t1

=

f x

(x,

x)

=

-

R6

pR2 + + pR2

12p2 + + 12p2

6qR

+

48p3 R2

and

t2

=

f y

(x,

x)

=

- R6

+

6qR pR2 + 12p2

+

48p3 R2

and the characteristic polynomial is

2 + t1 + t2 = 0.

Consider the two functions defined by

a() = 2, b() = -(t1 + t2).

We have Then

pR2 + 12p2 + 12qR

R6

+

pR2

+

12p2

+

48p3 R2

< 1.

|b()| < |a()| ,  : || = 1

Thus, by Rouche's theorem, all zeros of P () = a() - b() = 0 lie in || < 1. So, by Theorem (1) we get that x is locally asymptotically stable.

Theorem 4. The equilibrium point x is globally asymptotically stable.

Proof. Let {xn}n-k be a solution of equation (9). By Theorem (3) we need only to prove that E is global attractor, that is

lim
n

xn

=

.

it follows from Theorem (2) that

Then

lim
n

xn

=

lim
n

Sn+1x-1 Snx-1 +

+ Snx0x-1 Sn-1 x0 x-1

+ +

qSn-1 qSn-2

=

Sn lim

x Sn+1
Sn -1

+

x0 x-1

+

q

Sn-1 Sn

n Sn

x-1

+

Sn-1 Sn

x0

x-1

+

q Sn-2
Sn

=

lim
n

x-1

Sn+1 Sn

x-1

+

x0 x-1

+

q

Sn-1 Sn

+

Sn-1 Sn

x0

x-1

1
( ) q

Sn+1

-

p q

Sn-1

+ q Sn

=

lim
n

 x-1 x-1 +

+

1 

x0

x-1

1 

x0 x-1

+

+

q

1 



+

p 

lim
n

xn

=

.

8

Example 1. For confirming results of this section, we consider the following

numerical example. Let  = 2,  = 5 and  = 4 in (1), then we obtain the

equation

xn+1

=

2xn-1 + 5 4xnxn-1

.

(12)

Assume x-1 = 3 and x0 = 0.2, (see Fig. 1).

x(n)

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

10

20

30

40

50

60

70

n

Figure 1: This figure shows that the solution of the equation (12) is global attractor, that is, lim xn = .
n

4 Closed-form and stability of solutions of system (2)

4.1 Closed-form solutions of system (2)

In this section, we derive the respective solution form of system (2). We put

q

=

 

and

p=

 

.

Hence,

we

have

the

system

xn+1

=

pxn-1 + ynxn-1

q

,

yn+1

=

pyn-1 + xn yn-1

q

,

n  N0

(13)

The following theorem describes the form of the solutions of system (13).

Theorem 5. Let {xn, yn}n-1 be a solution of (13). Then for n = 1, 2, . . . ,

9


   xn =
  

Sn+1y-1 Sny-1 +

+ Snx0y-1 Sn-1 x0 y-1

+ +

qSn-1 qSn-2

,

Sn+1x-1 Snx-1 +

+ Sny0x-1 Sn-1y0x-1

+ +

qSn-1 qSn-2

,

if n is even, if n is odd,

(14)


   yn =
  

Sn+1 x-1 Snx-1 +

+ Sny0x-1 Sn-1 y0 x-1

+ +

qSn-1 qSn-2

,

Sn+1 y-1 Sny-1 +

+ Snx0y-1 Sn-1 x0 y-1

+ +

qSn-1 qSn-2

,

if n is even, if n is odd,

(15)

where the initial conditions x-1, x0, y-1 and y0  R \ (F1  F2), with F1 and F2 are the forbidden sets of equation (9) given by



F1 =

(x-1, x0, y-1, y0) : Snx-1 + Sn-1y0x-1 + qSn-2 = 0 ,

n=-1

and



F2 =

(x-1, x0, y-1, y0) : Sny-1 + Sn-1x0y-1 + qSn-2 = 0 .

n=-1

Proof. The closed-form solution of (13) can be established through a similar approach we used in proving the one-dimensional case. However, for convenience, we shall prove the theorem by induction. For the basis step, we have

x1

=

px-1 + q y0x-1

and

y1

=

py-1 + x0 y-1

q

,

so the result clearly holds for n = 0. Suppose that n > 0 and that our assumption holds for n - 1. That is,

x2n-2

=

S2n-1y-1 S2n-2y-1

+ S2n-2x0y-1 + S2n-3x0y-1

+ +

qS2n-3 qS2n-4

,

x2n-1

=

S2nx-1 + S2n-1y0x-1 + qS2n-2 S2n-1x-1 + S2n-2y0x-1 + qS2n-3

,

y2n-2

=

S2n-1 x-1 S2n-2x-1

+ +

S2n-2 y0 c-1 S2n-3 y0 x-1

+ +

qS2n-3 qS2n-4

,

y2n-1

=

S2ny-1 + S2n-1x0y-1 + qS2n-2 S2n-1y-1 + S2n-2x0y-1 + qS2n-3

.

10

Now it follows from system (13) that

x2n

=

px2n-2 + q y2n-1x2n-2

=

p

S2n-1 S2n-2

y-1 y-1

+ S2n-2x0y-1 + S2n-3x0y-1

+ qS2n-3 + qS2n-4

+q

S2ny-1 + S2n-1x0y-1 + qS2n-2 S2n-1y-1 + S2n-2x0y-1 + qS2n-3

S2n-1y-1 + S2n-2x0y-1 + qS2n-3 S2n-2y-1 + S2n-3x0y-1 + qS2n-4

=

p(S2n-1 y-1

+ S2n-2x0y-1 + qS2n-3) + q(S2n-2y-1 + S2n-3x0y-1 S2ny-1 + S2n-1x0y-1 + qS2n-2

+ qS2n-4)

So, we have

x2n

=

S2n+1y-1 S2ny-1 +

+ S2nx0y-1 S2n-1 x0 y-1

+ +

qS2n-1 qS2n-2

.

Also it follows from system (13) that

y2n

=

py2n-2 + q x2n-1 y2n-2

=

p

S2n-1x-1 S2n-2x-1

+ S2n-2y0x-1 + S2n-3y0x-1

+ qS2n-3 + qS2n-4

+q

S2nx-1 + S2n-1y0x-1 + qS2n-2 S2n-1x-1 + S2n-2y0c-1 + qS2n-3

S2n-1x-1 + S2n-2y0x-1 + qS2n-3 S2n-2x-1 + S2n-3y0x-1 + qS2n-4

=

p(S2n-1 x-1

+ S2n-2y0x-1 + qS2n-3) + q(S2n-2x-1 + S2n-3y0x-1 S2nx-1 + S2n-1y0x-1 + qS2n-2

+ qS2n-4) .

Hence, we have

y2n

=

S2n+1x-1 S2nx-1 +

+ S2ny0c-1 S2n-1y0x-1

+ +

qS2n-1 qS2n-2

.

Using the same argument it follows from system (13) that

x2n+1

=

px2n-1 + q y2nx2n-1

=

p

S2nx-1 + S2n-1y0x-1 + qS2n-2 S2n-1x-1 + S2n-2y0x-1 + qS2n-3

+

q

S2n+1x-1 + S2ny0x-1 + qS2n-1 S2nx-1 + S2n-1y0x-1 + qS2n-2

S2nx-1 + S2n-1y0x-1 + qS2n-2 S2n-1x-1 + S2n-2y0x-1 + qS2n-3

=

p(S2n x-1

+

S2n-1y0x-1 + qS2n-2) + q(S2n-1x-1 + S2n-2y0x-1 S2n+1x-1 + S2ny0x-1 + qS2n-1

+

qS2n-3) .

This yields

x2n+1

=

S2n+2x-1 S2n+1 x-1

+ +

S2n+1 S2ny0

y0x-1 c-1 +

+ qS2n qS2n-1

.

11

Moreover, we have

y2n+1

=

py2n-1 + q x2n y2n-1

=

p

S2ny-1 + S2n-1x0y-1 + qS2n-2 S2n-1y-1 + S2n-2x0y-1 + qS2n-3

+

q

S2n+1y-1 + S2nx0y-1 + qS2n-1 S2ny-1 + S2n-1x0y-1 + qS2n-2

S2ny-1 + S2n-1x0y-1 + qS2n-2 S2n-1y-1 + S2n-2x0y-1 + qS2n-3

=

p(S2n y-1

+

S2n-1x0y-1 + qS2n-2) + q(S2n-1y-1 + S2n-2x0y-1 S2n+1y-1 + S2nx0y-1 + qS2n-1

+

qS2n-3) ,

and this implies that

y2n+1

=

S2n+2y-1 S2n+1 y-1

+ +

S2n+1 S2n x0

x0 y-1 y-1 +

+ qS2n qS2n-1

.

This completes the proof of the theorem.

4.2 Global attractor of solutions of system (2)
Our aim in this section is to study the asymptotic behavior of positive solutions of system (13). Let I = J = (0, +), and consider the functions

f : I2 × J 2 - I and g : I2 × J 2 - J

defined by

f (u0, u1, v0, v1)

=

pu1 + v0u1

q

and

g(u0, u1, v0, v1)

=

pv1 + u0v1

q

,

respectively.

Lemma 1. System (9) has a unique equilibrium point in I × J, namely

E=

R2

+ 12p 6R

,

R2

+ 12p 6R

.

Proof. Clearly the system

x

=

px + xy

q,

y

=

py + yx

q

,

has a unique solution in I2 × J2 which is

E=

R2

+ 12p 6R

,

R2

+ 12p 6R

.

Theorem 6. The equilibrium point E is global attractor. 12

Proof. Let {xn, yn}n0 be a solution of system (9). Let n   in Theorem 5. That is, we have

lim
n

x2n

=

lim
n

Sn+1 y-1 Sny-1 +

+ Snx0y-1 Sn-1 x0 y-1

+ +

qSn-1 qSn-2

Sn = lim
n Sn

Sn+1 Sn

y-1

+

x0y-1

+

q

Sn-1 Sn

y-1

+

Sn-1 Sn

x0

y-1

+

q Sn-2
Sn

=

lim
n

y-1

Sn+1 Sn

y-1

+

x0 y-1

+

q

Sn-1 Sn

+

Sn-1 Sn

x0

y-1

+

1
( ) q

Sn+1

-

p q

Sn-1

q Sn

=

 y-1 y-1 +

+

1 

x0y-1

+

q

1 

1 

x0y-1

+



+

p 

= .

and

lim
n

x2n+1

=

lim
n

Sn+1x-1 Snx-1 +

+ Sny0x-1 Sn-1y0x-1

+ +

qSn-1 qSn-2

Sn = lim
n Sn

Sn+1 Sn

x-1

+

y0x-1

+

q

Sn-1 Sn

x-1

+

Sn-1 Sn

y0

x-1

+

q Sn-2
Sn

=

lim
n

x-1

Sn+1 Sn

x-1

+

y0x-1

+

q

Sn-1 Sn

+

Sn-1 Sn

y0

x-1

+

1
( ) q

Sn+1

-

p q

Sn-1

q Sn

=

 x-1 x-1 +

+

1 

y0

x-1

1 

y0x-1

+

+

q

1 



+

p 

= .

Then

lim
n

xn

=

.

Similarly,

we

obtain

lim
n

yn

=

.

Thus,

we

have

nlim(xn, yn) = E.

Example 2. As an illustration of our results, we consider the following numer-

ical example. Let  = 2,  = 3 and  = 5 in system (2), then we obtain the

system

xn+1

=

2xn-1 + 3 5ynxn-1

,

yn+1

=

2yn-1 + 3 5xnyn-1

,

n  N0

(16)

Assume x-1 = 1.2, x0 = 3.6, y-1 = 2.3 and y0 = 0.8. (See Fig. 2).

13

x(n), y(n)

3.5

3

2.5

2

1.5

1

0.5

0

0

10

20

30

40

50

60

70

n

Figure 2: This figure shows that the solution of the system (16) is global at-

tractor,

that

is

lim
n

xn

=

E.

5 Summary and Recommendations

In this work, we have successfully established the closed-form solution of the rational difference equation

xn+1

=

xn-1 +   xn xn-1

as well as the closed-form solutions of its corresponding two-dimensional case

xn+1

=

xn-1 +   yn xn-1

,

yn+1

=

yn-1 +   xn yn-1

.

Also, we obtained stability results for the positive solutions of these systems.

Particularly, we have shown that the positive solutions of each of these equations

tends to a computable finite number, and is in fact expressible in terms of the

well-known plastic number. Meanwhile, for future investigation, one could also

derive the closed-form solution and examine the stability of solutions of the

system

xn+1

=

xn-1 -   yn xn-1

,

yn+1

=

yn-1 ±   xn yn-1

.

This work we leave to the interested readers.

14

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