On the Solutions of Difference Equation Systems with Padovan Numbers ()
1. Introduction
Nonlinear difference equations have long interested researchers in the field of mathematics as well as in other sciences. They play a key role in many applications such as the natural model of a discrete process. There are many recent investigations and interest in the field of nonlinear difference equations from several authors [1-15]. For example, Tollu et al. [14] investigated the solutions of two special types of Riccati difference equations
such that their solutions are associated with Fibonacci numbers. In [2], Aloqeili investigated the stability properties and semi-cycle behavior of the solutions and the form of solutions of the difference equation
In [4], author obtained the formulae of solutions of the difference equations
Also, he studied the global asymptotic stability of the equilibrium points of these equations via the formulae. In [5], Elabbasy et al obtained Fibonacci sequence in solutions of some special cases of the following difference equation
In [6], author deals with the behavior of the solution of the following nonlinear difference equation
Also, he gives specific forms of the solutions of four special cases of this equation. These specific forms also contain Fibonacci numbers. In [7], Cinar studied the positive solutions of the following difference equation system
In [10], Elsayed obtained the form of the solutions of the following rational difference system
In [12], Stevic examined the solutions of the following system of difference equations
Now, we give information about Padovan numbers that establish a large part of our study. The Padovan sequence
, named after Richard Padovan, is defined by
(1.1)
It can be easily obtained that the characteristic equation of (1.1) has the form
(1.2)
having the roots
where 
Furthermore, the unique real root 
is named as plastic number. Also there exists the following limit
where 
kth Padovan number. One can find more information associated with this sequence in [16,17].
We will need the following definition in the sequel.
Definition 1.1 [18] Let 
be an equilibrium point of a map
, where 
and 
are continuously differentiable functions at
. The Jacobian matrix of 
at 
is the matrix
Also, suppose that 
is continuously differentiable on an open set 
in
. Equilibrium point 
is called a saddle point if one of the eigenvalues of 
is larger and another is less than 1 in absolute value.
In this study, we consider the solutions of the following two difference equation systems
(1.3)
and
(1.4)
such that their solutions are associated with Padovan numbers. We also establish a relationship between Padovan numbers and the solutions of systems (1.3) and (1.4).
2. Main Results
In this section, we prove our main results. The following theorem studies the formulae of the solutions of systems (1.3) and (1.4) with initial conditions not making the denominator zero.
Teorem 2.1 Let 
denote the solutions of systems (1.3) and (1.4). Then, the forms of solutions
are given by
(2.1)
and
(2.2)
where 
be the nth Padovan number.
The following lemma is necessary for determining the initial conditions of the well-defined solutions of systems (1.3) and (1.4).
Lemma 2.2 (Forbidden Set) Forbidden sets of systems (1.3) and (1.4) are given by
and
where
respectively.
Proof of Theorem 2.1 We will just prove for system (1.3) since the other part can be proved in the same manner. We use the method of induction on k. For k = 0, we have
For k = 1, we obtain
Now, suppose that our assumption holds for 2k - 1. That is;
From Equation (1.3), we can write for 2k,
and
Similarly, from Equation (1.3), we obtain for 2k + 1,
and
which completes the proof ■.
Theorem 2.3 The following statements hold:
1) System (1.3) has unique real equilibrium point 
and 
is a saddle point2) System (1.4) has unique real equilibrium point 
and 
is a saddle pointwhere p is the plastic number.
Proof
1) Equilibrium point 
of system (1.3) satisfy the system of equations
(2.3)
In (2.3), by subtracting the second equation from the first equation and after some operations, we have
For
, the equations of (2.3) cannot be satisfied and so
. Consequently, we obtain the following cubic equation
The above cubic equation is the characteristic equation of the recurrence relation of the Padovan numbers in (1.2) having the unique real root 
Hence the unique equilibrium point of system (1.3) is point
. Now, we show that the equilibrium point is a saddle point. Firstly, system (1.3) is a special case of the general system of the form
where 
and
. Then, we calculate the Jacobian of the corresponding map
We get
By taking into consideration (1.2), we obtain the characteristic equation of the Jacobian Matrix 
as
Hence, it is clearly seen that 
and 
as desired ■.
2) It can be proved in a similar manner.
Theorem 2.4 Let the initial conditions of the systems (1.3) and (1.4) be 
and
, respectively. Then the following statements hold:
1) The every solution of the system (1.3) converges to point
.
2) The every solution of the system (1.4) converges to point
.
Proof We will only prove for even-subscripted terms of
. Since the other parts of the proof are quite similar, they will be omited.
1) Let us take n = 2k in (2.1). Then, we can write
Also, by taking into account 
we obtain the following equality
as desired ■.
3. Numerical Examples
In order to illustrate and support theoretical results of the previous section, we consider several examples in this section. These examples represent the qualitative behavior of solutions of the mentioned nonlinear difference equation systems.
Example 3.1 Consider system (1.3) with the inital conditions 
(See Figure 1).
Example 3.2 Consider system (1.4) with the inital conditions 
(See Figure 2).
4. Conclusion
In this study, we formulated the solutions of equation systems (1.3) and (1.4) and determined their forbidden sets. Obtained formulae are given by means of Padovan numbers. Also, for 
and
, all the solutions of (1.3) and (1.4) interestingly tend to
their equilibrium points 
and
, respectively, where 
is the plastic number.