Global Dynamics of a Higher Order Difference Equation with Quadratic Term

In this paper, we investigate the dynamics of following higher order difference equation xn+1 = A+B xn xn−m with A,B and initial conditions are positive numbers. Especially we study the boundedness, periodicity, oscillation behaviours, global asymptotically stability and rate of convergence of related higher order difference equations.


Introduction
Di¤erence equations and their systems have captured the attention of the researchers over the last two decades. This attention result from area of usage of di¤erence equations. In particular they which arise in mathematical models that describe problems in ecology, probability and engineering, etc. Since we know very little about such equations, it is very important to study higher order di¤erence equations.
In [23], Jafar et al handled the following higher order rational di¤erence equation x n + x n k A + Bx n + Cx n k where the parameters , , A, B and C and the initial conditions are nonnegative real numbers, k = f1; 2; :::g. They studied the boundedness, invariant intervals, semi-cycles and global stability of related equation.
In [24], Saleh et al analyzed the invariant intervals, periodic character, the character of semi-cycles and global asymptotic stability of all positive solutions of di¤erence equation where the parameters , , , B and C and the initial conditions are nonnegative real numbers, k = f1; 2; :::g.
In [12], Devault et al investigated the boundedness, global stability and periodic character of solutions of the di¤erence equation where p and the initial conditions are arbitrary positive numbers.
In [14], Saleh et al studied global asymptotic stability, periodicity and semicycle analysis of the unique positive equilibrium of following di¤erence equation where A is positive and k 2 f2; 3; g. In [8], Abu-Saris et al examined the global stability of unique positive equilibrium point of following higher order di¤erence equations where A is positive and k 2 f2; 3; g. Additionally, in [17], Saleh et al dealt with the global asymptotic stability of the negative equilibrium of the di¤erence equation (1) where A < 0 and k 2 f1; 2; g. In [11], Hassan dealt with dynamics of following di¤erence equation where p and q lie in (0; 1).
In [10], Bešo et al investigated boundedness, global attractivity and Neimark-Sacker bifurcation of following di¤erence equation where ; are positive real numbers and the initial conditions are positive real numbers. Motivated by the above studies, we consider the dynamics of following higher order di¤erence equation where A; B are positive real numbers, and m 2 f2; 3; g, and the initial conditions are positive numbers. Additionally, we investigate the boundedness, periodicity, global asymptotic stability and rate of convergence of related higher order di¤erence equations. Now, we present some important theorems which used by us during this study.
Theorem 1 (See [4]) Assume that q i 2 R, i = 1; 2; ; and k 2 f0; 1; g. Then is a su¢ cient condition for the asymptotic stability of the di¤ erence equation x n+k + q 1 x n+k 1 + ::: + q k x n = 0; n = 0; 1; : where k is a positive integer, and where [a; b] is an interval of real numbers and consider the following di¤ erence equation x n+1 = f (x n ; ; x n k ) ; n = 0; 1; : Suppose that f satis…es the following conditions: ii. If ; k + 1, we set Then there exists exactly one equilibrium point x of the di¤ erence equation (3), and every solution of (3) converges to x.
Consider the scalar kth-order linear di¤erence equation where k is a positive integer and p i : exist in C. Consider the limiting equation of (4): x (n + k) + q 1 x (n + k 1) + + q k x (n) = 0: Theorem 3 (Poincaré' s Theorem) Consider (4) subject to condition (5). Let 1 ; ; k be the roots of the characteristic equation of the limiting equation (6) and suppose that j i j 6 = j j j for i 6 = j. If x (n) is a solution of (4), then either x (n) = 0 for all large n or there exists an index j 2 f1; ; kg such that lim n!1 x (n + 1) The following results were obtained by Perron, and one of Perron's results was improved by Pituk, see [19].
Theorem 4 Suppose that (5) holds. If x (n) is a solution of (4), then either x (n) = 0 eventually or where 1 ; ; k are the (not necessarily distinct) roots of the characteristic equation (7).
2 Analysis of the periodicity, boundedness, semicycles, and global stability of solutions of Eq. (2) In this section, we …rstly investigate the existence of two periodic solutions of Eq.
(2) as m is odd or even. We also study the boundedness and persistence of solutions of Eq.(2). Moreover we …nd out the semi-cycles of solutions of Eq.(2). Then we scrutiny the global asymptotic stability of solutions of Eq.(2). We further handle the rate of convergence of Eq. (2). First of all, we take the change of the variables for Eq.(2) as follows y n = xn A . From this, we obtain the following di¤erence equation From now on, we handle the di¤erence equation (8). The unique positive equilibrium point of Eq. (8) is In this here, we study the periodic solutions of Eq.(8) with period two. Proof. We assume that there exist two periodic solution such that ; ; ; ; ; where and are positive and distinct real numbers. We handle two cases for the proof of Theorem. Firstly we consider a case such that m is even. We have from Eq.(8) = 1 + p ; = 1 + p : Hence we obtain that 2 p = 0: So we get = 1+ p 1+4p 2 = y = which is a trivial solution. Now we deal with the other case such that m is odd. Now we apply Elsayed's new method for two periodic solution, see [18]. We have from Eq.(8) Hence if we take = n for n 2 R f0; 1; 1g. Therefore we obtain that n = 1 + p n 2 ; = 1 + pn : Thus, subtracting (10) from (9) gives the following (n 1) = p 1 n 2 n = p 1 n 3 n 2 : From n 6 = 1, we have 2 = p n 2 + n + 1 n 2 ; = r p (n 2 + n + 1) n 2 : Since is real number, (11) is impossible for all real n and p > 0. This is a contradiction. So the proof is completed. Now, we investigate the boundedness of solutions of Eq.(8).
We know that y n > 1 for all n 1. So we obtain from (12), y n < 1 + p y n 1 y n 2 y n 2 y n 3 y n m y n m 1 : Additinally, we set Eq.(8) such that for i = 1; 2; . Thus we have from (13) and (14) y n < 1 + p 1 y n 2 + p y 2  Proof. Let fy n g be a solution of Eq. (8). We …rstly consider a negative semicycle. The positive semi-cycle is similar and can be omitted. Suppose that a solution fy n g of Eq.(8) has a negative semi-cycle with 2m + 1 terms. Assume that y N is the …rst term in this negative semi-cycle. Thus we have y N ; y N +1 ; ; y N +2m < y: Additionally, we get the followings for i = 1; 2; ; m y N +m+i = 1+p y N +m+i 1 y 2 N +i 1 > 1+p y N +m+i 1 y 2 > y N +m+i 1 1 + p y 2 > y N +m+i 1 : Hence we obtain that y N +m < y N +m+1 < < y N +2m 1 < y N +2m : Therefore we get y N +m < y N +2m . So we have the followings Now we handle the proof of (b). Assume that Eq.(8) has a semi-cycle of length at least k. We again consider a negative semi-cycle. The positive semicycle is similar. Let y N be the …rst term of the following positive semi-cycle. Hence y N m ; y N m+1 ; ; y N 1 < y < y N : Therefore we obtain that ; k 1. Thus negative semi-cycle occurs at least m + 1 terms. So, every semi-cycle of following this semi-cycle consist at least m + 1 terms.
Proof. Firstly, we consider the following function

Numerical Simulations of Eq.(2)
This section, we present two numerical examples for verify our theoretical results.

Conclusion and Open Problems
During this paper, we investigate the dynamics of di¤erence equation (8). We …rstly …nd out that Eq. (8) has not periodic solution with period two. Then we reveal the bounded solution of Eq. (8). We further study the semi-cycles of Eq. (8). Moreover, we discover that the equilibrium point y of Eq. (8)  where the initial values are real numbers and r 2 f3; 4; g. Open Problem 2: Investigate the dynamics of following higher order difference equation where the initial values are real numbers and r 2 f2; 3; g, q 2 f2; 3; g.