Dynamical study of the class of difference equation Xn+1= Xn-2q+1/A+BXn-2qXn-q+1

Article history: Received 25 October 2019 Received in revised form 10 February 2020 Accepted 14 February 2020 This work is devoted to present a study of the class of the difference equations xn+1 = xn−2q+1 A + Bxn−2q+1xn−q+1, q = 1,2, ... ⁄ with arbitrary initial data, where A and B are arbitrary parameters, and q is an arbitrary nonnegative integer. We present a detailed investigation of the behavior of the solution, including their dependence on parameters and initial conditions. Local and global stabilities of the equilibrium points are discussed. The existence of a periodic solution is studied. Numerical simulations are given to assure the correctness of the analytical results. This study improves and surpasses studies of several forms of difference equations that have been investigated earlier by many researchers.


*Mathematics
Difference equations play an important role in describing dynamical systems and presenting many numerical schemes (Agiza and Elsadany, 2004;Ahmed et al., 2015;Ahmed and Hegazi, 2006;Askar, 2014;Elabbasy et al., 2014;El-Metwally et al., 2015;El-Morshedy and Liz, 2005;Elsadany, 2010;Elsadany et al., 2013;Elsadany and Matouk, 2014;Karatas et al., 2006;Matouk et al., 2015). Many applications of difference equations can be found in various fields of science such as game theory, mathematical biology, physics, and engineering Ahmed and Hegazi, 2006;Askar et al., 2016;Elabbasy et al., 2014;El-Morshedy and Liz, 2006;Elsadany et al., 2013;Elsadany and Matouk, 2014;Wang et al., 2017a). Because of these applications, many researchers focused on studying difference equations. For some of these studies, we refer the reader to Cinar (2004aCinar ( , 2004b and Elsayed (2008Elsayed ( , 2009aElsayed ( , 2009bElsayed ( , 2009cElsayed ( , 2009dElsayed ( , 2010aElsayed ( , 2010bElsayed ( , 2010c. In recent years, many researchers investigated the qualitative behavior of nonlinear rational difference equations and systems of difference equations. Elabbasy et al. (2007)  . Motivated by the work of Cinar (2004b) and others, Wang et al. (2010) studied the asymptotic behavior of the solutions for the difference equation where the parameters , , , and are positive real numbers. Their technique is based on a variational iteration method. They introduce the notion of mixed monotone property of functions. Then they use it to obtain an interesting result [54,Theorem 3.2]. This result gives sufficient conditions that grantee that the equation has a unique equilibrium point, and this equilibrium point is globally attractor. Using a similar technique, Wang et al. (2011) investigated the asymptotic behavior of the solutions for the difference equation , respectively. For more recent studies of systems of rational difference equations (Haddad et al. 2017a;. Elsayed (2011a;2011b;2011c) obtained the solutions for the following difference equations . Then later, Elsayed et al. (2017) generalized this work, where they gave a detailed analytical study and behavior of the solutions of x n+1 = −3 + −1 −3 and investigated many of its properties such as local stability and global attractivity of its equilibrium points.  considered the difference equations , where they obtained the solution of this equation, investigated its asymptotic behavior, determined its forbidden set, and discussed the existence of periodic solutions. Wang et al. (2017c) Wang et al. (2018) investigated the boundedness and asymptotic behavior of systems of max-type difference equation. Their work generalizes and improves many results concerning systems of maxtype non-linear difference equations. The sufficient conditions obtained in their paper provide flexibility for applications and analysis of such systems.
The aim of this paper is to give a great generalization to the study of the qualitative behavior of nonlinear rational difference equations. We consider a general class of difference equations of the form: where and are arbitrary constants and with arbitrary initial data −2 +1 = −2 +1 , −2 +2 = −2 +2 , . . . , 0 = 0 . We give a detailed analytical study of this difference equation. Where we obtain the solution of this equation and investigate its convergence. We also investigate its asymptotic behavior, determine its forbidden set, and discuss the existence of its periodic solutions. The order of this difference equation, namely 2 , is kept as an arbitrary parameter. This allows us to make a significant contribution to the study of difference equations. We were able to generalize and improve results about many forms of difference equations such as the ones studied in Aloqeili (2006), Cinar (2004a), Elsayed (2011aElsayed ( , 2011bElsayed ( , 2011c, and Ghazel et al. (2017). For instance, taking = 1, = 1/ , and = / in Eq. 1, yields the second order difference equation that was considered in Cinar (2004b). On the other hand, substituting = 3 and replacing and by / and / respectively produce the sixth order difference equation considered in Ghazel et al. (2017).
In the next section, we introduce some basic definitions and primary results that will be needed in later sections. Section 3, discusses the equilibrium points and their stability. We show that Eq. 1 has either one equilibrium point, namely ̅ = 0, or three equilibrium points, ̅ = 0 , ∓√(1 − )/ . The stability of these equilibrium points is also investigated. Theorem (4.1) gives complete analytical expressions for the solutions of Eq. 1, and the good sets are described in Theorem (4.2). In Section 5, we present a detailed study for the convergence of the solutions of Eq. 1, and the periodicity of these solutions is dealt with in Section 6. In the last section, some numerical simulations are given to support the theoretical results.
4. If ̅ is both locally stable and global attractor, then it is called globally asymptotically stable.

Definition 2.2:
Let ̅ be an equilibrium point of Eq. 2 and let be the set of all roots of the characteristic Eq. 4 about ̅ . Then: 1. ̅ is said to be hyperbolic if | | ≠ 1 for all ∈ , otherwise it is called nonhyperbolic; 2. ̅ is called saddle if there are two roots 1 , 2 ∈ such that 1 < 1 and 2 > 1; 3. ̅ is called repeller if | | > 1 for all ∈ .
The next theorem, obtained by Kocic and Ladas (1993), is called the Linearized Stability Theorem.
Theorem 2.1: Let ̅ be an equilibrium point of Eq. 2 and let be the set of all roots of the characteristic Eq. 4 about ̅ .

Definition 2.3:
A solution ( ) ≥− +1 of Eq. 2 is said to be periodic with period or a periodic− solution if: A solution is called periodic with prime period if is the smallest positive integer for which Eq. 5 holds.

Stability analysis of the equilibrium points
In this section, we discuss the nature of the equilibrium points of the difference equation given by where is the continuous function defined on ℝ 2 as ( 0 , 1 , ⋯ , 2 −1 ) = u 2q−1 A+Bu 2q−1 u q−1 Theorem 3.1: Let ( ) ≥−2 +1 be a solution of Eq. 6.

Proof:
The result follows directly from the fact that the equilibrium points of Eq. 6 are the real roots of the equation ̅ ( ̅ 2 − + 1) = 0.

Analytical expressions of ( )
In this section, we give some analytical expressions of the sequence ( ) ≥−2 +1 . Theorem 4.1: Let ( ) ≥−2 +1 be a sequence given by Eq. 1. Then for each ∈ {2 − 1, 2 − 2, … , } and all ≥ 0, we have: and . We will do this by induction on . It is evident that the results hold for = 0. Let ≥ 0 be an integer, and suppose that the results hold for all nonnegative integers ≤ . We shall now prove that the identities hold for the step + 1.
. This completes the proof.
The good set of a difference equation is the set of conditions that any initial data must satisfy so that the associated solution of the difference equation is defined for all natural number . The complement of the good set is usually called the forbidden set. Determining good sets is a problem of great importance in the study of difference equations, and the interest in this problem has increased recently. For general information on good sets and forbidden sets of difference equations, we refer the reader to Azizi (2012), Grove and Ladas (2005), Kocic and Ladas (1993), Kulenovic and Ladas (2002), and Rubió-Massegú (2009). The next theorem describes the good set of the difference Eq. 1.

Convergence
In this section, we study the asymptotic behavior of a solution of difference Eq. 1.

Proof:
We divide the proof into two cases.

Conclusion
In this paper, we have presented a complete study of the class of rational difference equations x n+1 = x n−2q+1 A+Bx n−2q+1 x n−q+1 with arbitrary initial data, where A and B are arbitrary parameters, and q is an arbitrary nonnegative integer. Keeping the order 2q as an arbitrary parameter allowed us to make a significant contribution and improves and surpasses studies of several forms of difference equations that have been investigated in the literature. In this study, we have given a detailed analytical investigation of the convergence of the solutions including their dependence on parameters and initial data. We also provided a complete discussion of the local and global stability of the equilibrium points as well as the existence of periodic solutions of this class. At the end, numerical simulations have been done to confirm the correctness of analytical results.