ON THE ASYMPTOTIC CHARACTER OF A GENERALIZED RATIONAL DIFFERENCE EQUATION

. We investigate the global asymptotic stability of the solutions of X n +1 = βX n − l + γX n − k A + X n − k for n = 1 , 2 ,... , where l and k are positive integers such that l (cid:54) = k . The parameters are positive real numbers and the initial conditions are arbitrary nonnegative real numbers. We ﬁnd necessary and suﬃcient conditions for the global asymptotic stability of the zero equilibrium. We also investigate the positive equilibrium and ﬁnd the regions of parameters where the positive equilibrium is a global attractor of all positive solutions. Of particular interest for this generalized equation would be the existence of unbounded solutions and the existence of prime period two solutions depending on the combination of delay terms ( l , k ) being (odd, odd), (odd, even), (even, odd) or (even, even). In this manuscript we will investigate these aspects of the solutions for all such combinations of delay terms.

for n = 1, 2, . . ., where l and k are positive integers such that l = k. The parameters are positive real numbers and the initial conditions are arbitrary nonnegative real numbers. We find necessary and sufficient conditions for the global asymptotic stability of the zero equilibrium. We also investigate the positive equilibrium and find the regions of parameters where the positive equilibrium is a global attractor of all positive solutions. Of particular interest for this generalized equation would be the existence of unbounded solutions and the existence of prime period two solutions depending on the combination of delay terms (l, k) being (odd, odd), (odd, even), (even, odd) or (even, even). In this manuscript we will investigate these aspects of the solutions for all such combinations of delay terms.

Introduction. Consider the difference equation
x n+1 = βx n−l + γx n−k A + x n−k , n = 0, 1, . . . , where l and k are positive integers such that l = k, the parameters β, γ, and A are positive real numbers, and the initial conditions are arbitrary non-negative real numbers.
A special case where l = 1 and k = 2 is studied in [7]. The case l = 0 was investigated in [4]. In this case, without loss of generality, it was assumed that γ = 1. It was shown that when A ≥ β + 1 all solutions converge to the zero equilibrium. When A < β + 1, two sufficient conditions for global attractivity of the positive equilibriumx = β + 1 − A are A > β − 1 and A ≥ (k−1)β−1 k . Some other special cases of Eq.(1) has also been investigated. See [3] and [12] for related work in some of these special cases. See also [2].
We consider here the most general case of double long delay recursion. The linear fractional form of the recursion is maintained in the generalization in order 1708 ESHA CHATTERJEE AND SK. SARIF HASSAN to systematically study the effects of the long delays. Although, in some sections, the functional form of the recursion will be further generalized and not just specific to the linear fractional form.
The change of variables x n = γy n reduces Eq.(1) to the difference equation with l and k positive integers such that l = k and with positive parameters and non-negative initial conditions. Our goal is to investigate the global stability of the solutions of Eq.(2). Note that, for initial conditions that are not all zero we have an eventually positive solution of Eq. (2). Without loss of generality, in the sequel we need to consider only the positive solutions of Eq.(2).
In the sequel we will assume l < k without loss of generality. Hence we obtain the recursion in k + 1 space. The proofs for the other case l > k follows in a similar manner.

Equilibria and invariant intervals.
In this section we show that when q > p, the interval [0, 1] is an invariant interval for all solutions of Eq. (2). Furthermore, in this case, every solution of Eq.(2) is eventually bounded from above by the constant q/p. We also show that when q < p, the interval (1, ∞) is an invariant interval for all solutions of Eq.(2) and every solution in this case is eventually bounded from below by the constant q/p.
The equilibrium points of Eq.(2) are the non-negative solutions of the equation Clearly, zero is always an equilibrium point of Eq.(2). However, when q < p + 1, Eq.(2) also has the unique positive equilibrium pointȳ = p + 1 − q. The following lemma, the proof of which is straightforward and will be omitted, exhibits two identities that will be useful in the study of Eq.(2).
Lemma 2.1. Let {y n } ∞ n=−k be a solution of Eq.(2). Then for n ≥ 0 the following two identities hold: The following lemma establishes the existence of invariant intervals for solutions of Eq.(2). (a) Suppose that q > p and that there exists N ≥ 0 such that y N −l , y N −l+1 , . . . , y N ≤ 1. Then y n < 1 for all n > N . (b) Suppose that q < p and that there exists N ≥ 0 such that y N −l , y N −l+1 , . . . , y N ≥ 1. Then y n > 1 for all n > N .
Proof. We will prove (a). The proof of (b) is similar and will be omitted. From Eq.
(3) we see that y N −l ≤ 1 < q p and so y N +1 < 1. The result now follows by induction.
The next result establishes an eventual bound for the solutions of Eq.(2)from above and below for the cases p < q and p > q respectively. Proof. (a) Suppose, for the sake of contradiction, that there exists N sufficiently large, such that Then clearly, and similarly which eventually leads to a contradiction. (b) Suppose for the sake of contradiction that there exists N sufficiently large, such that Then clearly, and similarly This eventually leads to a contradiction and completes the proof.
3. The stability character of the zero equilibrium. Recall that zero is always an equilibrium point of Eq. (2). In this section we investigate the stability of the zero equilibrium of Eq.(2). The linearized equation of Eq.(2) with respect to the zero equilibrium is with associated characteristic equation Lemma 3.1. When q > p+1, the zero equilibrium of Eq.(2) is locally asymptotically stable, while if q < p + 1 it is unstable. In the case q = p + 1 the zero equilibrium is stable.
Proof. It follows by Clark's Theorem, see [5], that the zero equilibrium is locally asymptotically stable when Note that f (1) = q−(p+1) q < 0 when q < p + 1. Thus, the zero equilibrium is unstable when q < p + 1. Now assume q = p + 1. Let > 0, and let {y n } ∞ n=−k be a nonnegative solution of Eq. (2). Letting δ = , if the initial conditions are such that The proof follows by induction. The next result gives the global stability character of the zero equilibrium of Eq.(2).
with associated characteristic equation As a consequence of Clark's Theorem, see [5], it follows that the positive equilibrium of Eq. (2) is locally asymptotically stable when q > p − 1.
We now discuss the global stability of the positive equilibrium of Eq.(2).
Proof. It suffices to show that the positive equilibrium is a global attractor of all solutions of Eq.(2). Let {y n } ∞ n=−k be a solution of Eq. (2). Observe that the function increases in u for all v ∈ (0, ∞), increases in v for all u ∈ (0, q p ] and decreases in v for all u ∈ ( q p , ∞). We divide the proof into the following three cases: Case(i). p < q < p + 1.
By Lemma (2.3) (a) it follows that there exists N ≥ 0, such that y n < q p for all n ≥ N.
Hence eventually, y n+1 = py n−l + y n−k q + y n−k < py n−l + 1 q + 1 and by the comparison principle lim sup n→∞ y n ≤ 1 q + 1 − p .
Therefore the solution {y n } ∞ n=−k is bounded from above and below by positive constants. Set The proof is complete.

5.
Existence of prime period two solutions. In this section we establish the necessary and sufficient conditions for the existence of prime period two solutions of Eq.(2). It is interesting to note that the range of parameters for which such solutions exist, vary according to the parity of the delay terms l and k. That is, the conditions under which prime period two solutions of Eq.(2) exist depends on the combination of (l,k) being (odd, odd), (odd, even), (even, odd) or (even, even). The following theorem states the result. Case(ii). If l is odd and k is even, then q = p − 1. In this case there exist infinitely many prime period two solutions of Eq.(2).
Case(iv). If l, k both are even then there does not exist any prime period two solutions of Eq.(2).
Proof. Let . . . , φ, ψ, φ, ψ, . . ., φ = ψ be a prime period two solution of Eq.(2). We establish the necessary conditions in the proof. The sufficient conditions are straight forward and are omitted in the proof. We divide the proof into the following four cases: Case(i). l and k both odd. Then φ = pφ + φ q + φ ψ = pψ + ψ q + ψ Note that, if φ and ψ are both positive or both zeroes, then there does not exist any prime period two solutions of Eq.(2). Otherwise, . . . , 0, p + 1 − q, 0, p + 1 − q, . . . is a prime period two solution of Eq.(2).% Case(ii). l is odd and k is even. Then Note that, in this case there exist infinite prime period two solutions . . . , φ, ψ, φ, . . . such that φ = ψ ψ − 1 Case(iii). l is even and k is odd. Then Case(iv). l and k both even. Then The proof is complete.  The value of l, k, p and q are given in each plot above in the Fig.1. Each of the above plot shows very clearly higher order periodicities. 5.1. Local stability of prime period 2 solutions. For simplicity, we show the local stability of prime period 2 solutions of the specialized equation where l = 0 and k = 1. Let . . . , φ, ψ, φ, ψ, . . ., φ = ψ be a prime period two solution of Eq.(2) where l = 0 and k = 1. We set u n = y n v n = y n−1 Then the equivalent form of the Eq.(2) with l = 0 and k = 1 is Then φ ψ is a fixed point of T 2 , the second iterate of T .
where g(u, v) = pu+v q+v and h(u, v) = pv+ pu+v q+v q+ pu+v q+v . Clearly the two cycle is locally asymptotically stable when the eigenvalues of the Jacobian matrix J T 2 , evaluated at φ ψ lie inside the unit disk. We have, Now, set Then it follows from the Linearized Stability that both the eigenvalues of the J T 2 φ ψ lie inside the unit disk if and only if |χ| < 1 + λ < 2. In other words, the equivalent inequalities are χ < 1 + λ, −1 − λ < χ and λ < 1.
In the present case, l is 0 and k is 1 and p + q < 1 then there exist infinitely many prime period two solutions of Eq.(2) with l = 0 and k = 1.
6. Semi-cycle analysis. Definition 6.1. A positive semi-cycle of {y n } ∞ n=−k consists of a "string" of terms {y s , y s+1 , . . . , y m } all greater than or equal toȳ, with s ≥ −k and m ≤ ∞ such that either s = −k or s > −k and y s−1 <ȳ, and either m = ∞ or m < ∞ and y m+1 <ȳ.
A negative semi-cycle of {y n } ∞ n=−k consists of a "string" of terms {y s , y s+1 , . . . , y m } all less thanȳ, with s ≥ −k and m ≤ ∞ such that either s = −k or s > −k and y s−1 ≥ȳ, and either m = ∞ or m < ∞ and y m+1 ≥ȳ.
The following two theorems given in [2] will be useful in establishing our main results in this section. Note that, when q > p − 1 we obtain the above result from Theorem(3.2)and Theorem(4.1). Thus the above conjecture needs to be shown only for q ≤ p − 1.
In the remaining case of l odd and k even, the existence of unbounded solutions have been established in [8]. This is part of a period-two trichotomy result. 8. Chaotic solutions. The method of Lyapunov characteristic exponents serves as a useful tool to quantify chaos. Specifically Lyapunov exponents measure the rates of convergence or divergence of nearby trajectories. Negative Lyapunov exponents indicate convergence, while positive Lyapunov exponents demonstrate divergence and chaos. The magnitude of the Lyapunov exponent is an indicator of the time scale on which chaotic behavior can be predicted or transients decay for the positive and negative exponent cases respectively. In this present study, the largest Lyapunov exponent is calculated for a given solution of finite length numerically [14].  The orbit (trajectory) plots corresponding to the 1st, 3rd and 6th cases in the Table-1 are given in the following Fig.2. In each row different number of iterations of orbit plots are shown for the same set of initial of values.
In each of the above cases the computed Lyapunov exponents happen to be positive which ensured the chaoticity of the solutions listed in the Table- Table-1.