Volume 2013, Article ID 179401,5pages http://dx.doi.org/10.1155/2013/179401
Research Article
Dynamics of a System of Rational Higher-Order Difference Equation
Banyat Sroysang
Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathumthani 12121, Thailand
Correspondence should be addressed to Banyat Sroysang; [email protected] Received 5 February 2013; Accepted 22 April 2013
Academic Editor: Cengiz C¸ inar
Copyright © 2013 Banyat Sroysang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We focus on a system of a rational𝑚-order difference equation𝑥𝑛+1 = (𝑥𝑛−𝑚+1)/(𝐴 + 𝑦𝑛𝑦𝑛−1⋅ ⋅ ⋅ 𝑦𝑛−𝑚+1),𝑦𝑛+1 = (𝑦𝑛−𝑚+1)/(𝐵 + 𝑥𝑛𝑥𝑛−1⋅ ⋅ ⋅ 𝑥𝑛−𝑚+1),𝑛 = 0, 1, . . ., where𝐴, 𝐵, 𝑥0, 𝑥−1, . . . , 𝑥−𝑚+1, 𝑦0, 𝑦−1, . . . , 𝑦−𝑚+1∈ (0, ∞). We investigate the dynamical behavior of positive solution for the system.
1. Introduction
In 2011, Kurbanli et al. [1] studied the behavior of positive solutions of the system of rational difference equations
𝑥𝑛+1= 𝑥𝑛−1 𝑦𝑛𝑥𝑛−1+ 1, 𝑦𝑛+1= 𝑦𝑛−1
𝑥𝑛𝑦𝑛−1+ 1,
(1)
where the initial conditions are arbitrary nonnegative real numbers.
In the same year, Kurbanli [2] studied the behavior of solutions of the system of rational difference equations
𝑥𝑛+1= 𝑥𝑛−1 𝑦𝑛𝑥𝑛−1− 1, 𝑦𝑛+1= 𝑦𝑛−1
𝑥𝑛𝑦𝑛−1− 1, 𝑧𝑛+1= 𝑧𝑛−1
𝑦𝑛𝑧𝑛−1− 1,
(2)
where the initial conditions are arbitrary real numbers.
Moreover, Kurbanli [3] studied the behavior of the solutions of the difference equation system
𝑥𝑛+1= 𝑥𝑛−1 𝑦𝑛𝑥𝑛−1− 1, 𝑦𝑛+1= 𝑦𝑛−1
𝑥𝑛𝑦𝑛−1− 1, 𝑧𝑛+1= 1
𝑦𝑛𝑧𝑛,
(3)
where 𝑥0, 𝑥−1, 𝑦0, 𝑦−1, 𝑧0, 𝑧−1 ∈ R such that 𝑦0𝑥−1 ̸= 1, 𝑥0𝑦−1 ̸= 1and𝑦0𝑧0 ̸= 1.
In [4], Liu et al. gave more results of the solution of the system (2) including a new and simple expression of𝑧𝑛 and the asymptotical behavior of the solution.
In [5], Stevi´c showed that the system of difference equa- tions
𝑥𝑛+1= 𝑎𝑥𝑛−1
𝑏𝑦𝑛𝑥𝑛−1+ 𝑐, 𝑦𝑛+1= 𝛼𝑦𝑛−1 𝛽𝑥𝑛𝑦𝑛−1+ 𝛾,
𝑛 = 0, 1, . . . , (4)
can be solved.
In 2012, Gu and Ding [6] derived two canonical state space forms from multiple-input multiple-output systems described by difference equations.
The system of two nonlinear difference equations 𝑥𝑛+1= 𝐴 + 𝑦𝑛
𝑥𝑛−𝑝, 𝑦𝑛+1= 𝐴 + 𝑥𝑛 𝑦𝑛−𝑞, 𝑛 = 0, 1, . . . ,
(5)
was studied by Papaschinopoulos and Schinas [7], where 𝑝, 𝑞 ∈N.
Moreover, the system of rational difference equations 𝑥𝑛+1= 𝑥𝑛
𝑎 + 𝑐𝑦𝑛, 𝑦𝑛+1= 𝑦𝑛 𝑏 + 𝑑𝑥𝑛, 𝑛 = 0, 1, . . . ,
(6) was studied by Clark et al. [8,9], where𝑎, 𝑏, 𝑐, 𝑑 ∈ (0, ∞)and 𝑥0, 𝑦0∈ [0, ∞).
Liu et al. [10] studied the behavior of a system of rational difference equations
𝑥𝑛+1= 𝑥𝑛−1
𝑦𝑛𝑥𝑛−1− 1, 𝑦𝑛+1= 𝑦𝑛−1 𝑥𝑛𝑦𝑛−1− 1, 𝑧𝑛+1= 1
𝑥𝑛𝑧𝑛−1, 𝑛 = 0, 1, . . . ,
(7)
where the initial conditions are nonzero real numbers.
In 2012, Zhang et al. [11] studied the solutions, stability character, and asymptotic behavior of the system of a rational third-order difference equation
𝑥𝑛+1= 𝑥𝑛−2
𝐴 + 𝑦𝑛𝑦𝑛−1𝑦𝑛−2, 𝑦𝑛+1= 𝑦𝑛−2 𝐵 + 𝑥𝑛𝑥𝑛−1𝑥𝑛−2,
𝑛 = 0, 1, . . . , (8) where𝐴, 𝐵, 𝑥0, 𝑥−1, 𝑥−2, 𝑦0, 𝑦−1, 𝑦−2∈ (0, ∞).
In this paper, we studied the solutions, stability character, and asymptotic behavior of the system of a rational𝑚-order difference equation
𝑥𝑛+1= 𝑥𝑛−𝑚+1 𝐴 + 𝑦𝑛𝑦𝑛−1⋅ ⋅ ⋅ 𝑦𝑛−𝑚+1, 𝑦𝑛+1= 𝑦𝑛−𝑚+1
𝐵 + 𝑥𝑛𝑥𝑛−1⋅ ⋅ ⋅ 𝑥𝑛−𝑚+1, 𝑛 = 0, 1, . . . , (9)
where𝐴, 𝐵, 𝑥0, 𝑥−1, . . . , 𝑥−𝑚+1, 𝑦0, 𝑦−1, . . . , 𝑦−𝑚+1∈ (0, ∞).
2. Preliminaries
Let𝑚 ∈Nand let𝑓 : 𝐼𝑥𝑚× 𝐼𝑦𝑚 → 𝐼𝑥and𝑔 : 𝐼𝑥𝑚× 𝐼𝑦𝑚 → 𝐼𝑦 be continuously differentiable functions, where𝐼𝑥and𝐼𝑦are intervals inR.
For any(𝑥0, 𝑦0), (𝑥−1, 𝑦−1), . . . , (𝑥−𝑚+1, 𝑦−𝑚+1) ∈ 𝐼𝑥× 𝐼𝑦, the system of difference equations
𝑥𝑛+1= 𝑓 (𝑥𝑛, 𝑥𝑛−1, . . . , 𝑥𝑛−𝑚+1, 𝑦𝑛, 𝑦𝑛−1, . . . , 𝑦𝑛−𝑚+1) , 𝑦𝑛+1= 𝑔 (𝑥𝑛, 𝑥𝑛−1, . . . , 𝑥𝑛−𝑚+1, 𝑦𝑛, 𝑦𝑛−1, . . . , 𝑦𝑛−𝑚+1) , 𝑛 = 0, 1, . . . ,
(10) has a unique solution{(𝑥𝑛, 𝑦𝑛)}∞𝑛=−𝑚+1.
Definition 1. A point(𝑥, 𝑦) ∈ 𝐼𝑥× 𝐼𝑦is called an equilibrium point of the system (10) if𝑥 = 𝑓(𝑥, 𝑥, . . . , 𝑥, 𝑦, 𝑦, . . . , 𝑦)and 𝑦 = 𝑔(𝑥, 𝑥, 𝑥, . . . , 𝑦, 𝑦, . . . , 𝑦).
Definition 2. The linearized system of the system (10) about the equilibrium (𝑥, 𝑦) is the system of linear difference equations
𝑥𝑛+1= 𝑚−1∑
𝑖=0
(𝜕𝑓 (𝑥, 𝑥, . . . , 𝑥, 𝑦, 𝑦, . . . , 𝑦)
𝜕𝑥𝑛−𝑖 𝑥𝑛−𝑖 +𝜕𝑓 (𝑥, 𝑥, . . . , 𝑥, 𝑦, 𝑦, . . . , 𝑦)
𝜕𝑦𝑛−𝑖 𝑦𝑛−𝑖) , 𝑦𝑛+1= 𝑚−1∑
𝑖=0
(𝜕𝑔 (𝑥, 𝑥, . . . , 𝑥, 𝑦, 𝑦, . . . , 𝑦)
𝜕𝑥𝑛−𝑖 𝑥𝑛−𝑖 +𝜕𝑔 (𝑥, 𝑥, . . . , 𝑥, 𝑦, 𝑦, . . . , 𝑦)
𝜕𝑦𝑛−𝑖 𝑦𝑛−𝑖) . (11)
Definition 3. An equilibrium point(𝑥, 𝑦)of the system (10) is said to be stable relative to𝐼𝑥 × 𝐼𝑦 if for every 𝜖 > 0, there exists𝛿 > 0such that for any(𝑥0, 𝑦0), (𝑥−1, 𝑦−1), . . . , (𝑥−𝑚+1, 𝑦−𝑚+1) ∈ 𝐼𝑥× 𝐼𝑦, with
max{ ∑0
𝑖=−𝑚+1𝑥𝑖− 𝑥 , ∑0
𝑖=−𝑚+1𝑦𝑖− 𝑦} < 𝛿. (12) One has max{|𝑥𝑛− 𝑥|, |𝑦𝑛− 𝑦|} < 𝜖for all𝑛 ≥ −𝑚 + 1.
Definition 4. An equilibrium point (𝑥, 𝑦) of the system (10) is called an attractor relative to 𝐼𝑥 × 𝐼𝑦 if for all (𝑥0, 𝑦0), (𝑥−1, 𝑦−1), . . . , (𝑥−𝑚+1, 𝑦−𝑚+1) ∈ 𝐼𝑥 × 𝐼𝑦, one has lim𝑛 → ∞𝑥𝑛= 𝑥and lim𝑛 → ∞𝑦𝑛= 𝑦.
Definition 5. An equilibrium point(𝑥, 𝑦)of the system (10) is said to be asymptotically stable relative to𝐼𝑥× 𝐼𝑦if it is stable, and it is also an attractor.
Definition 6. An equilibrium point(𝑥, 𝑦)of the system (10) is said to be unstable if it is not stable.
Theorem 7 (see [12]). Let𝑋(𝑛 + 1) = 𝐹(𝑋(𝑛)),𝑛 = 0, 1, . . ., be a system of difference equations and let𝑋be the equilibrium point of the system. If all eigenvalues of the Jacobian matrix evaluated at𝑋lie inside the open unit disk, then𝑋is asymp- totically stable. If one of them has a modulus greater than one, then𝑋is unstable.
Theorem 8 (see [13]). Let𝑋(𝑛 + 1) = 𝐹(𝑋(𝑛)),𝑛 = 0, 1, . . ., be a system of difference equations and let𝑋be the equilibrium point of the system. Assume that the characteristic polynomial of the system about𝑋is𝑎0𝜆𝑛+ 𝑎1𝜆𝑛−1+ ⋅ ⋅ ⋅ + 𝑎𝑛−1𝜆 + 𝑎𝑛where 𝑎𝑖 ∈Rfor all𝑖and𝑎0 > 0. Then all roots of the characteristic equation lie inside the open unit disk if and only ifΔ𝑘 > 0for
all positive integer𝑘 ≤ 𝑛, whereΔ𝑘 is the principal minor of order𝑘of the𝑛 × 𝑛matrix
Δ𝑛= (
𝑎1 𝑎3 𝑎5 ⋅ ⋅ ⋅ 0 𝑎0 𝑎2 𝑎4 ⋅ ⋅ ⋅ 0 0 𝑎1 𝑎3 ⋅ ⋅ ⋅ 0 ... ... ... d ... 0 0 0 ⋅ ⋅ ⋅ 𝑎𝑛
) . (13)
3. Results
We note that
(i) if𝐴 < 1and𝐵 < 1then the system (9) has equilibrium (0, 0)and(√1 − 𝐵,𝑚 √1 − 𝐴);𝑚
(ii) if𝐴 = 1and𝐵 < 1then the system (9) has equilibrium (0, 0)and(√1 − 𝐵, 0);𝑚
(iii) if𝐴 < 1and𝐵 = 1then the system (9) has equilibrium (0, 0)and(0,√1 − 𝐴);𝑚
(iv) if𝐴 > 1and𝐵 > 1then(0, 0)is the unique equilib- rium point of the system (9).
Theorem 9. Let(𝑥𝑛, 𝑦𝑛)be positive solution of the system(9).
For all nonnegative integer𝑘, one has
0 ≤ 𝑥𝑛≤ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {
𝑥−𝑚+1
𝐴𝑘+1 , 𝑛 = 𝑚𝑘 + 1;
𝑥−𝑚+2
𝐴𝑘+1 , 𝑛 = 𝑚𝑘 + 2;
... 𝑥0
𝐴𝑘+1, 𝑛 = 𝑚𝑘 + 𝑚,
0 ≤ 𝑦𝑛≤ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {
𝑦−𝑚+1
𝐵𝑘+1 , 𝑛 = 𝑚𝑘 + 1;
𝑦−𝑚+2
𝐵𝑘+1 , 𝑛 = 𝑚𝑘 + 2;
... 𝑦0
𝐵𝑘+1, 𝑛 = 𝑚𝑘 + 𝑚.
(14)
Proof. Obviously, they are true for𝑘 = 0. Suppose that they are true for𝑘 = 𝑙. Then
𝑥𝑛= {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {
𝑥𝑚(𝑙+1)+1≤𝑥𝑚(𝑙+1)−𝑚+1
𝐴 = 1
𝐴𝑥𝑚𝑙+1≤ 1
𝐴(𝑥−𝑚+1
𝐴𝑙+1 ) , 𝑛 = 𝑚 (𝑙 + 1) + 1;
𝑥𝑚(𝑚+1)+2≤ 𝑥𝑚(𝑙+1)−𝑚+2
𝐴 = 1
𝐴𝑥𝑚𝑙+2≤ 1 𝐴(𝑥−𝑚+2
𝐴𝑙+1 ) , 𝑛 = 𝑚 (𝑙 + 1) + 2;
...
𝑥𝑚(𝑙+1)+𝑚≤ 𝑥𝑚(𝑙+1) 𝐴 = 1
𝐴𝑥𝑚𝑙+𝑚≤ 1 𝐴( 𝑥0
𝐴𝑙+1) , 𝑛 = 𝑚 (𝑙 + 1) + 𝑚,
𝑦𝑛= {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {
𝑦𝑚(𝑙+1)+1≤ 𝑦𝑚(𝑙+1)−𝑚+1
𝐵 = 1
𝐵𝑦𝑚𝑙+1≤ 1 𝐵(𝑦−𝑚+1
𝐵𝑙+1 ) , 𝑛 = 𝑚 (𝑙 + 1) + 1;
𝑦4(𝑙+1)+2≤ 𝑦𝑚(𝑙+1)−𝑚+2
𝐵 = 1
𝐵𝑦𝑚𝑙+2≤ 1 𝐵(𝑦−𝑚+2
𝐵𝑙+1 ) , 𝑛 = 4 (𝑚 + 1) + 2;
...
𝑦𝑚(𝑙+1)+𝑚≤𝑦𝑚(𝑙+1) 𝐵 = 1
𝐵𝑦𝑚𝑙+𝑚≤ 1 𝐵( 𝑦0
𝐵𝑙+1) , 𝑛 = 𝑚 (𝑙 + 1) + 𝑚.
(15)
Thus, they are true for𝑘 = 𝑙 + 1.
By the mathematical induction, this proof is completed.
Corollary 10. Let(𝑥𝑛, 𝑦𝑛)be positive solution of the system (9). If𝐴 > 1and𝐵 > 1, then the sequence{(𝑥𝑛, 𝑦𝑛)}converges exponentially to the equilibrium point(0, 0).
Theorem 11. Let𝐴 > 1and𝐵 > 1. Then the equilibrium point (0, 0)of the system(9)is asymptotically stable.
Proof. The linearized system of the system (9) about the equilibrium(0, 0)is
Φ𝑛+1= 𝐷Φ𝑛, (16)
where
Φ𝑛= (( (( (( (( (( (
( 𝑥𝑛 𝑥𝑛−1 𝑥𝑛−2 ... 𝑥𝑛−𝑚+1
𝑦𝑛 𝑦𝑛−1 𝑦𝑛−2 ... 𝑦𝑛−𝑚+1
)) )) )) )) )) )
) ,
𝐷 = (( (( (( (( (( (( (( (( (
(
0 0 ⋅ ⋅ ⋅ 0 1
𝐴 0 0 ⋅ ⋅ ⋅ 0 0 1 0 ⋅ ⋅ ⋅ 0 0 0 0 ⋅ ⋅ ⋅ 0 0 0 1 ⋅ ⋅ ⋅ 0 0 0 0 ⋅ ⋅ ⋅ 0 0 ... d d d d d d ... ... ... 0 0 ⋅ ⋅ ⋅ 1 0 0 0 0 0 0 0 0 ⋅ ⋅ ⋅ 0 0 0 0 0 0 1 0 0 ⋅ ⋅ ⋅ 0 0 1 0 0 0 0𝐵 0 0 ⋅ ⋅ ⋅ 0 0 0 1 0 0 0 ... d d d d d d d d ... 0 0 ⋅ ⋅ ⋅ 0 0 0 0 0 1 0
)) )) )) )) )) )) )) )) )
) .
(17)
The characteristic equation of the system (16) is
(𝜆𝑚− 1
𝐴) (𝜆𝑚− 1
𝐵) = 0. (18)
Thus,|𝜆| < 1. ByTheorem 7, the equilibrium point(0, 0)is asymptotically stable.
Theorem 12. Let𝐴 < 1and𝐵 < 1. Then both the equilibrium points (0, 0) and (√1 − 𝐵,𝑚 √1 − 𝐴)𝑚 of the system (9) are unstable.
Proof. We note by the characteristic equation (18) that|𝜆| >
1 and then, by Theorem 7, the equilibrium point (0, 0) is unstable.
Next, we consider the equilibrium point (√1 − 𝐵,𝑚
√1 − 𝐴). The linearized system of the system (9) about the𝑚
equilibrium(√1 − 𝐵,𝑚 √1 − 𝐴)𝑚 is
Φ𝑛+1= 𝐺Φ𝑛, (19)
where
Φ𝑛= (( (( (( (( (( (
( 𝑥𝑛 𝑥𝑛−1 𝑥𝑛−2 ... 𝑥𝑛−𝑚+1
𝑦𝑛 𝑦𝑛−1 𝑦𝑛−2 ... 𝑦𝑛−𝑚+1
)) )) )) )) )) )
) ,
𝐺 = (( (( (( (( (( (( ((
(
0 0 ⋅ ⋅ ⋅ 0 1 𝛼 𝛼 ⋅ ⋅ ⋅ 𝛼 𝛼 1 0 ⋅ ⋅ ⋅ 0 0 0 0 ⋅ ⋅ ⋅ 0 0 0 1 ⋅ ⋅ ⋅ 0 0 0 0 ⋅ ⋅ ⋅ 0 0 ... d d d d d d ... ... ... 0 0 ⋅ ⋅ ⋅ 1 0 0 0 0 0 0 𝛽 𝛽 ⋅ ⋅ ⋅ 𝛽 𝛽 0 0 0 0 1 0 0 ⋅ ⋅ ⋅ 0 0 1 0 0 0 0 0 0 ⋅ ⋅ ⋅ 0 0 0 1 0 0 0 ... d d d d d d d d ... 0 0 ⋅ ⋅ ⋅ 0 0 0 0 0 1 0
)) )) )) )) )) )) ))
) ,
(20) in which
𝛼 = −√(1 − 𝐴)𝑚 𝑚−1(1 − 𝐵), 𝛽 = −√(1 − 𝐴) (1 − 𝐵)𝑚 𝑚−1.
(21)
The characteristic polynomial of the system (19) is 1 − 𝛼𝛽 −∑𝑚
𝑖=1
(𝑖𝛼𝛽𝜆𝑖−1) − 2𝜆𝑚
−𝑚−1∑
𝑖=1
(𝑖𝛼𝛽𝜆2𝑚−(𝑖+1)) + 𝜆2𝑚.
(22)
We note the characteristic polynomial𝑎0𝜆2𝑚+ 𝑎1𝜆2𝑚−1+
⋅ ⋅ ⋅ + 𝑎2𝑚−1𝜆 + 𝑎2𝑚 that 𝑎1 = 0. Thus, we obtain that not all ofΔ𝑘 > 0,𝑘 = 1, 2, . . . , 2𝑚. By Theorems7and 8, the equilibrium point(√1 − 𝐵,𝑚 √1 − 𝐴)𝑚 is unstable.
Theorem 13. Let𝐴, 𝐵 < 1andΩ1= (0,√1 − 𝐵) × (𝑚 √1 − 𝐴,𝑚
∞), Ω2 = (√1 − 𝐵, ∞) × (0,𝑚 √1 − 𝐴). Assume that𝑚 {(𝑥𝑛, 𝑦𝑛)}∞𝑛=−𝑚+1satisfies the system(9). Then
(i)if{(𝑥𝑛, 𝑦𝑛)}0𝑛=−𝑚+1⊆ Ω1, then{(𝑥𝑛, 𝑦𝑛)}∞𝑛=−𝑚+1⊆ Ω1; (ii)if{(𝑥𝑛, 𝑦𝑛)}0𝑛=−𝑚+1⊆ Ω2, then{(𝑥𝑛, 𝑦𝑛)}∞𝑛=−𝑚+1⊆ Ω2. Proof. (i) Assume that{(𝑥𝑛, 𝑦𝑛)}0𝑛=−𝑚+1 ⊆ Ω1. Then, for any 𝑖 ∈ {0, 1, . . . , 𝑚 − 1},
𝑥𝑖+1= 𝑥𝑖−𝑚+1
𝐴 + 𝑦𝑖𝑦𝑖−1⋅ ⋅ ⋅ 𝑦𝑖−𝑚+1 < 𝑥𝑖−𝑚+1
𝐴 + (√1 − 𝐴)𝑚 𝑚 = 𝑥𝑖−𝑚+1, 𝑦𝑖+1= 𝑦𝑖−𝑚+1
𝐵 + 𝑥𝑖𝑥𝑖−1⋅ ⋅ ⋅ 𝑥𝑖−𝑚+1 > 𝑦𝑖−𝑚+1
𝐵 + (√1 − 𝐵)𝑚 𝑚 = 𝑦𝑖−𝑚+1. (23) Then(𝑥1, 𝑦1), (𝑥2, 𝑦2), . . . , (𝑥𝑚, 𝑦𝑚) ∈ Ω1.
Next, we suppose that(𝑥𝑘, 𝑦𝑘), (𝑥𝑘−1, 𝑦𝑘−1), . . . , (𝑥𝑘−𝑚+1, 𝑦𝑘−𝑚+1) ∈ Ω1where𝑘is a positive integer. Then
𝑥𝑘+1= 𝑥𝑘−𝑚+1
𝐴 + 𝑦𝑘𝑦𝑘−1⋅ ⋅ ⋅ 𝑦𝑘−𝑚+1 < 𝑥𝑘−𝑚+1
𝐴 + (√1 − 𝐴)𝑚 𝑚 = 𝑥𝑘−𝑚+1, 𝑦𝑘+1= 𝑦𝑘−𝑚+1
𝐵 + 𝑥𝑘𝑥𝑘−1⋅ ⋅ ⋅ 𝑥𝑘−𝑚+1 > 𝑦𝑘−𝑚+1
𝐵 + (√1 − 𝐵)𝑚 𝑚 = 𝑦𝑘−𝑚+1. (24) Then(𝑥𝑘+1, 𝑦𝑘+1) ∈ Ω1.
By the mathematical induction,{(𝑥𝑛, 𝑦𝑛)}∞𝑛=−𝑚+1⊆ Ω1. (ii) This is similar to the proof of (i).
Acknowledgment
The author would like to thank the referees for their useful comments and suggestions.
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