STABILITY CHAOS AND PERIODIC SOLUTION OF DELAYED RATIONAL RECURSIVE SEQUENCE

A BSTRACT . In this paper, we will investigate a non-linear rational difference equation of higher order. Our con-centration is on invariant intervals, periodic character, the character of semi cycles and global asymptotic stability of all positive solutions of where the parameters A, B, α,β and A 0 , B 0 and the initial conditions s − r , s − r +1 , s − r +2 ,..., s 0 are arbitrary positive real numbers, r = max { l,k } . Finally, we study the global stability of this equation through numerically solved examples and conﬁrm our theoretical discussion through it. difference For positive parameters and initial values, the dynamics of the Eq.(1.1.) is limited to local asymptotic stability of the positive equilibrium and periodic, bounded solutions. It does not show chaotic, aperiodic, fractal-like solutions. It is seen that if we consider the negative values of the controlling parameters A and/or B , the Eq.(1.1) exhibits very complicated dynamics viz. fractal-like and chaotic and periodic.


INTRODUCTION
Our aim is to investigate the global stability character and the periodicity of the solutions of the following rational higher order difference equation (1) s n+1 = As n + Bs n−l + α + βs n−k A 0 + B 0 s n−k , n = 0, 1, ..., where the parameters A 0 , B 0 , α, β and A, B and the initial conditions s −k , s −k+1 , ..., s 0 are positive real numbers, k = {1, 2, 3, ...} is a positive integer and the initial conditions s −k , s −k+1 , ..., s 0 are non-negative real numbers. Discrete dynamical systems or difference equations are varied field because various biological systems naturally leads to their study by means of a discrete variable. Every dynamical system u n+1 = f (u n ) determines a difference equation and vice versa. The theory of discrete dynamical systems and difference equations developed greatly during the last twenty-five years of the twentieth century. Applications of discrete dynamical systems and difference equations have appeared recently in many areas. The theory of difference equations occupies a central position in applied sciences and applicable areas. There is no doubt that the theory of difference equations will continue to play an important role in mathematics as a whole. Nonlinear difference equations of order greater than one are of paramount importance in applications. Such equations also appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations, which model various diverse phenomena in biology, ecology, physiology, physics, engineering, economics, probability theory, genetics, psychology and resource management. It is very interesting to investigate the behaviour of solutions of a higher-order rational difference equation and to discuss the local asymptotic stability of its equilibrium points. Rational difference equations have been studied by several authors. Especially there has been a great interest in the study of the attractivity of the solutions of such equations. For more results for the rational difference equations, we refer the interested reader to [6][7][8][9][10]. "Saleh and Baha et al. [27] investigated the global attractivity of the following rational recursive sequence Several other researchers have studied the behavior of the solution of difference equations, for example, in [15] Elsayed et al. investigated the solution of the following non-linear difference equation.
Elabbasy et al. [16] studied the boundedness, global stability, periodicity character and gave the solution of some special cases of the difference equation.
Keratas et al. [20] gave the solution of the following difference equation Elabbasy et al. [17] investigated the global stability, periodicity character and gave the solution of some special cases of the difference equation Yalçınkaya et al. [18] has studied the following difference equation Wang et. al. [19] existence and uniqueness of the positive solutions and the asymptotic behaviour of the equilibrium points of the fuzzy difference equation where x n is a sequence of positive fuzzy numbers, the parameters A, B, C, D and the initial conditions Elsayed et al. [22] studied the global behaviour of rational recursive sequence where the initial conditions x −r , x −r+1 , x −r+2 , ..., x 0 are arbitrary positive real numbers, r = max{l, k, s,t} is non-negative integer and a, b, c, d, e are positive constants: L.alseda et al. [4] studied the following rational difference equation Q. Wang et al. [31] investigated the local stability, asymptotic behaviour, periodicity and oscillation of solutions for the difference equation with the initial conditions .., b 0 are given k + 1 constants, A, B, C are positive constants. As a matter of fact, numerous papers negotiate with the problem of solving non-linear difference equations in any way possible, see, for instance [7]- [15]. The long-term behaviour and solutions of rational difference equations of order greater than one has been extensively studied during the last decade. For example, various results about periodicity, boundedness, stability, and closed form solution of the second-order rational difference equations, see [5][6][7][8][9][21][22][23][24][25][26][27][28][29].

PRELIMINARIES AND DEFINITIONS
Here, we recall some basic definitions and some theorems that we need in the sequel. Let I be some interval of real numbers and let That is, s n = s for n ≥ 0, is a solution of Eq.(2.1), or equivalently, s is a fixed point of F .

Definition 2. (Periodicity)
A Sequence {s n } ∞ n=−k is said to be periodic with period p if s n+p = s n for all n ≥ −k.
The following theorem will be useful for the proof of our results in this paper. is an interval of real numbers. Consider the difference equation (5) s n+1 = f (s n , s n−1 , ..., s n−k ), n = 0, 1, ....
Suppose that f satisfies the following conditions:  In this section we will study the equilibrium points of Eq.(1.1). The equilibrium points of Eq.(1.1) are the positive solutions of the equation Hence, for s = So, the linearized equation about the s = is locally asypmtotically stable if and only if.
Proof. Let f : (0, ∞) 3 → (0, ∞) be a continuous function defined by It follows from Theorem A that, the equilibriums of the Eq.(1.1) is locally asymptotically stable ⇔ Here at the positive equilibriums, On simple algebraic simplification, we get, Here we present an example with different pairs of (l, k) with fixed A = 0.771941 and B = 0.115124 of local stability of the positive equilibrium. In all the above four cases of even and odd pair of (l, k), the trajectories for different choice of parameters except (A, B) are locally asymptotically stable as shown in the Fig. 2.

EXISTENCE OF PERIODIC SOLUTION
In this section, we will investigate positive prime period two solution of Eq.(1.1). The following theorems states the necessary and sufficient conditions that this equation has periodic solution of prime period two.
we will prove just the 1 st case, the remaining cases are the same.
We would like to show that s 1 = q, and s 2 = p It follows from Eq.(1.1) that dividing numerator and denominator by 2AB 0 we get multiply denominator and numerator of right hand side by 2AA 0 B 0 + B 0 (ξ − δ) and by computation we get Similarly Then by induction we get s 2n = p and s 2n+1 = q for all n ≥ − max{l, k}.
Thus Eq.(1.1) has prime period two solution.
6. GLOBAL STABILITY OF EQ.(1.1) In this section we will study the global stability character of the solutions of Eq.(1.1).

Lemma 3.
For any values of the quotient α A0 and β B0 the function f (u, v, w) defined by Eq.(4.2) has monotonicity behaviour in its two arguments.
Proof. The proof follows by easy computations and is omitted.  Proof. Let ζ, η are real numbers and assume that g : [ζ, η] 2 → [ζ, η] be a function defined by Now, two cases must be considered.
By using comparison, the right hand side can be written as follows So, we can write y n = a n y 0 + constant, and this equation is locally asymptotically stable because A+B < 1, and converges to the equilibrium point y = αB0+βA0
the right hand side can be written as follows y n+1 = Ay n ⇒ y n = A n y 0 , and this equation is unbounded because A > 1 and lim n→∞ y n = ∞. Then by using ratio test {s n } ∞ n=− max{l,k} is unbounded from above. Similarly from Eq.(1.1) we see that x n+1 = As n + Bs n−l + α+βs n−k A0+B0s n−k > Bs n−l f or all n ≥ 0.
The right hand side can be written as follows y n+1 = By n−l ⇒ y 2n−l = B n y −l , and y 2n = B n y 0 and this equation is unstable because B > 1, and lim n→∞ y 2n−l = lim n→∞ y n = ∞. Then by using ratio test {s n } ∞ n=− max{l,k} is unbounded from above.          This work is related to the qualitative behaviour of a rational difference equation, which may be considered as generalized equation studied in [36]. Thus our results, considerably extend some previous investigations in literature. Firstly existence and uniqueness of positive equilibrium point is prove. Then it investigated that Eq. (1.1) is bounded and persists. We proved that the Eq. (1.1) has a unique positive equilibrium point, which is locally asymptotically stable. The method of linearization is used to prove the local asymptotic stability of unique equilibrium point. Linear stability analysis shows that the positive steady-state of Eq.