Dynamics of some higher order rational difference equations

In this paper we discuss the solution of rational difference equation of the form 𝑧 𝑛+1 = 𝑍 𝑛−20 ±1±𝑍 𝑛−6 𝑍 𝑛−13 𝑍 𝑛−20 , 𝑛 = 0,1,… where the initial values are arbitrary real numbers. To confirm the obtained solutions we consider some numerical examples by assigning different initial values with Matlab.


Introduction
*In this paper we obtain solutions of rational difference Eq. 1 +1 = −20 ±1± −6 −13 −20 , = 0,1 … (1) where the initial values are arbitrary real numbers. Difference equation is a vast field which impact almost found in every branch of pure as well as applied mathematics. In this paper we study the local stability, global attractivity of equilibrium point of Eq. 1 and boundedness of solutions of the Eq. 1). Moreover we obtain solutions of some special cases of this equation. The study and solution of nonlinear higher order difference equation is very challenging. However we have still no suitable generalized method to deal with the global behavior of rational difference equations of higher order so far. Therefore the study of rational difference equations of higher order is worth for consideration. Recently great interest is developed in studying difference equation systems. The reason is that there is need of some techniques whose can be used in investigating problems in different fields. Recently a great effort has been made in studying the qualitative analysis of rational difference equations. Difference equations are very simple in form, but it is very difficult to understand thoroughly the behaviors of their solutions (Cinar 2004a;2004b;2004c). Karatas et al. (2006)  . Van Khuong and Phong (2011) investigated the difference equation −3 = (1 + −1 −2 ), = 0,1, . ... Khaliq and Elsayed (2016)  Suppose that I is some interval of real numbers and F a continuous function defined on I k+1 (k+1 copies of I), where k is some natural number. Throughout this thesis, we consider the following difference equation +1 = ( , −1 , … , − ), = 0,1, . ..
is a sufficient condition for the asymptotic stability of Eq. 2.

First equation
In this section we give a specific form of the first equation in the form with non-zero real numbers initial values.  Suppose that > 0 and that our assumption is true for − 1. That is now it follows from Eq. 3 Similarly, other relations can be proved in same manner. Thus solution is of period 21.

Numerical examples
For confirming the results, suppose some numerical examples which shows different types of solutions of Eq. 4.  = 0,1,2,3,4,5,6) Proof: The proof is similar to the proof of theorem 2.1.

Remark 4.1:
The equilibrium point of Eq. 5 is zero which is not asymptotically stable.

Numerical examples
For confirming the results, take numerical examples which show different types of solutions of Eq. 5.

Conclusion
In this paper we studied solutions, equilibrium points and periodicity of four types of difference equations of Eq. 1. Eq. 3 has zero as equilibrium point and every positive solution is bounded. Eq. 4 has two equilibrium points 0 and √2 3 which are not locally asymptotically stable and has a periodic solution of period 42. The equilibrium point of Eq. 5 is zero which is not asymptotically stable .Every solution of Eq. 6 is periodic with period 21 and has two equilibrium points 0 which is not locally asymptotically stable. To confirm the obtained result, we gave numerical examples of each case by using Matlab.