On the System of Three Order Rational Difference Equation

This paper is concerned with the local and global asymptotic behavior of positive solution for a system of three order rational difference equations xn+1 = xn α+ xn−1yn−1 , yn+1 = yn β + xn−1yn−1 n = 0, 1, · · · , where α, β ∈ (0,∞), and the initial values x−1, x0 ∈ (0,∞), y−1, y0 ∈ (0,∞). Finally, some numerical examples are provided to illustrate theoretical results obtained.


INTRODUCTION
Difference equations or discrete dynamical systems are diverse fields which impact almost every branch of pure and applied mathematics. Every dynamical system xn+1 = f (xn, xn−1) determines a difference equation and vise versa.
Kurbanli [3] studied a three-dimensional system of rational difference equations where the initial conditions are arbitrary real numbers. Papaschinopoulos et al. [8] investigated the global behavior for a system of the following two nonlinear difference equations.
where A is a positive real number, p, q are positive integers, and x−p, · · · , x0, y−q, · · · , y0 are positive real numbers.
In 2012, Zhang, Yang and Liu [9] investigated the global behavior for a system of the following third order nonlinear difference equations.
Motivated by above discussion, our goal, in this paper is to investigate the solutions of the two-dimensional system of three order rational nonlinear difference equations in the form where α, β ∈ (0, ∞) and the initial values x−1, x0, y−1 and y0 ∈ (0, ∞). Moreover, we have studied the local stability, global stability, boundedness of solutions. We will consider some special cases of (1.1) as applications. Finally, we give some numerical examples.

MAIN RESULTS
Let Ix, Iy be some intervals of real number and f : (iii) (x,ȳ) is called asymptotically stable relative to Ix × Iy if it is stable and an attractor. (iv) Unstable if it is not stable. Theorem 2.1. [25] Assume that X(n + 1) = F (X(n)), n = 0, 1, · · · , is a system of difference equations and X is the equilibrium point of this system i.e., F (X) = X. If all eigenvalues of the Jacobian matrix JF , evaluated at X lie inside the open unit disk |λ| < 1, then X is locally asymptotically stable. If one of them has modulus greater than one, then X is unstable. Proof. (i) We can easily obtain that the linearized system of (1.1) about the equilibrium (0, 0) is where Φn = (xn, xn−1, yn, yn−1) T , The characteristic equation of (2. 2) is This shows that the roots of characteristic equation λ = 1 α and λ = 1 β lie outside unit disk. So the unique equilibrium (0, 0) is locally unstable.

RATE OF CONVERGENCE
In order to study the rate of convergence of positive solutions of (1.1) which converge to equilibrium point (0, 0) of this system, first we consider the following results that gives the rate of convergence of solution of a system of difference equations.
where Xn is m dimensional vector, A ∈ C m×m is a constant matrix. B : Z + → C m×m is a matrix function satisfying as n → ∞, where ∥ · ∥ is any matrix norm which is associated with the vector norm Let E 1 n = xn − 0, E 2 n = yn − 0, then we have where Now the limiting system of error terms can be written as which is similar to linearized system of (1.1) about the equilibrium point (0, 0).
Using Proposition 3.1 and Proposition 3.2, we have following result.