Dynamics of a System of Rational Higher-Order Difference Equation

Abstract

We focus on a system of a rational 𝑚-order difference equation 𝑥_𝑛+1=(𝑥_{𝑛−𝑚+1})/(𝐴+𝑦_𝑛𝑦_𝑛−1⋯𝑦_{𝑛−𝑚+1}), 𝑦_𝑛+1=(𝑦_{𝑛−𝑚+1})/(𝐵+𝑥_𝑛𝑥_𝑛−1⋯𝑥_{𝑛−𝑚+1}), 𝑛=0,1,…, where 𝐴,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 𝑥_𝑛+1=𝑥_𝑛−1𝑦_𝑛𝑥_𝑛−1,𝑦+1_𝑛+1=𝑦_𝑛−1𝑥_𝑛𝑦_𝑛−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 𝑥_𝑛+1=𝑥_𝑛−1𝑦_𝑛𝑥_𝑛−1,𝑦−1_𝑛+1=𝑦_𝑛−1𝑥_𝑛𝑦_𝑛−1,𝑧−1_𝑛+1=𝑧_𝑛−1𝑦_𝑛𝑧_𝑛−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 𝑥_𝑛+1=𝑥_𝑛−1𝑦_𝑛𝑥_𝑛−1,𝑦−1_𝑛+1=𝑦_𝑛−1𝑥_𝑛𝑦_𝑛−1,𝑧−1_𝑛+1=1𝑦_𝑛𝑧_𝑛, (3) where 𝑥₀,𝑥₋₁,𝑦₀,𝑦₋₁,𝑧₀,𝑧₋₁∈ℝ such that 𝑦₀𝑥₋₁≠1, 𝑥₀𝑦₋₁≠1 and 𝑦₀𝑧₀≠1.

In [4], Liu et al. gave more results of the solution of the system (2) including a new and simple expression of 𝑧_𝑛 and the asymptotical behavior of the solution.

In [5], Stević showed that the system of difference equations 𝑥_𝑛+1=𝑎𝑥_𝑛−1𝑏𝑦_𝑛𝑥_𝑛−1+𝑐,𝑦_𝑛+1=𝛼𝑦_𝑛−1𝛽𝑥_𝑛𝑦_𝑛−1,+𝛾𝑛=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 𝑥_𝑛+1𝑦=𝐴+^𝑛𝑥_𝑛−𝑝,𝑦_𝑛+1𝑥=𝐴+^𝑛𝑦_𝑛−𝑞,𝑛=0,1,…, (5) was studied by Papaschinopoulos and Schinas [7], where 𝑝,𝑞∈ℕ.

Moreover, the system of rational difference equations 𝑥_𝑛+1=𝑥_𝑛𝑎+𝑐𝑦_𝑛,𝑦_𝑛+1=𝑦_𝑛𝑏+𝑑𝑥_𝑛,𝑛=0,1,…, (6) was studied by Clark et al. [8, 9], where 𝑎,𝑏,𝑐,𝑑∈(0,∞) and 𝑥₀,𝑦₀∈[0,∞).

Liu et al. [10] studied the behavior of a system of rational difference equations 𝑥_𝑛+1=𝑥_𝑛−1𝑦_𝑛𝑥_𝑛−1−1,𝑦_𝑛+1=𝑦_𝑛−1𝑥_𝑛𝑦_𝑛−1,𝑧−1_𝑛+1=1𝑥_𝑛𝑧_𝑛−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 𝑥_𝑛+1=𝑥_𝑛−2𝐴+𝑦_𝑛𝑦_𝑛−1𝑦_𝑛−2,𝑦_𝑛+1=𝑦_𝑛−2𝐵+𝑥_𝑛𝑥_𝑛−1𝑥_𝑛−2,𝑛=0,1,…, (8) where 𝐴,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 rational 𝑚-order difference equation 𝑥_𝑛+1=𝑥_{𝑛−𝑚+1}𝐴+𝑦_𝑛𝑦_𝑛−1⋯𝑦_{𝑛−𝑚+1},𝑦_𝑛+1=𝑦_{𝑛−𝑚+1}𝐵+𝑥_𝑛𝑥_𝑛−1⋯𝑥_{𝑛−𝑚+1},𝑛=0,1,…, (9) where 𝐴,𝐵,𝑥₀,𝑥₋₁,…,𝑥_−𝑚+1,𝑦₀,𝑦₋₁,…,𝑦_−𝑚+1∈(0,∞).

2. Preliminaries

Let 𝑚∈ℕ and let 𝑓∶𝐼^𝑚_𝑥×𝐼^𝑚_𝑦→𝐼_𝑥 and 𝑔∶𝐼^𝑚_𝑥×𝐼^𝑚_𝑦→𝐼_𝑦 be continuously differentiable functions, where 𝐼_𝑥 and 𝐼_𝑦 are intervals in ℝ.

For any (𝑥₀,𝑦₀),(𝑥₋₁,𝑦₋₁),…,(𝑥_−𝑚+1,𝑦_−𝑚+1)∈𝐼_𝑥×𝐼_𝑦, the system of difference equations 𝑥_𝑛+1=𝑓𝑥_𝑛,𝑥_𝑛−1,…,𝑥_{𝑛−𝑚+1},𝑦_𝑛,𝑦_𝑛−1,…,𝑦_{𝑛−𝑚+1},𝑦_𝑛+1=𝑔𝑥_𝑛,𝑥_𝑛−1,…,𝑥_{𝑛−𝑚+1},𝑦_𝑛,𝑦_𝑛−1,…,𝑦_{𝑛−𝑚+1},𝑛=0,1,…, (10) has a unique solution {(𝑥_𝑛,𝑦_𝑛)}^∞_{𝑛=−𝑚+1}.

Definition 1. A point (𝑥,𝑦)∈𝐼_𝑥×𝐼_𝑦 is called an equilibrium point of the system (10) if 𝑥=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 𝑥_𝑛+1=^𝑚−1_𝑖=0𝜕𝑓𝑥,𝑥,…,𝑥,𝑦,𝑦,…,𝑦𝜕𝑥_𝑛−𝑖𝑥_𝑛−𝑖+𝜕𝑓𝑥,𝑥,…,𝑥,𝑦,𝑦,…,𝑦𝜕𝑦_𝑛−𝑖𝑦_𝑛−𝑖,𝑦_𝑛+1=^𝑚−1_𝑖=0𝜕𝑔𝑥,𝑥,…,𝑥,𝑦,𝑦,…,𝑦𝜕𝑥_𝑛−𝑖𝑥_𝑛−𝑖+𝜕𝑔𝑥,𝑥,…,𝑥,𝑦,𝑦,…,𝑦𝜕𝑦_𝑛−𝑖𝑦_𝑛−𝑖. (11)

Definition 3. An equilibrium point (𝑥,𝑦) of the system (10) is said to be stable relative to 𝐼_𝑥×𝐼_𝑦 if for every 𝜖>0, there exists 𝛿>0 such that for any (𝑥₀,𝑦₀),(𝑥₋₁,𝑦₋₁),…,(𝑥_−𝑚+1,𝑦_−𝑚+1)∈𝐼_𝑥×𝐼_𝑦, with max⁰_{𝑖=−𝑚+1}𝑥_𝑖−𝑥,⁰_{𝑖=−𝑚+1}𝑦_𝑖−𝑦<𝛿. (12) One has max{|𝑥_𝑛−𝑥|,|𝑦_𝑛−𝑦|}<𝜖 for all 𝑛≥−𝑚+1.

Definition 4. An equilibrium point (𝑥,𝑦) of the system (10) is called an attractor relative to 𝐼_𝑥×𝐼_𝑦 if for all (𝑥₀,𝑦₀),(𝑥₋₁,𝑦₋₁),…,(𝑥_−𝑚+1,𝑦_−𝑚+1)∈𝐼_𝑥×𝐼_𝑦, one has lim_𝑛→∞𝑥_𝑛=𝑥 and lim_𝑛→∞𝑦_𝑛=𝑦.

Definition 5. An equilibrium point (𝑥,𝑦) of the system (10) is said to be asymptotically stable relative to 𝐼_𝑥×𝐼_𝑦 if 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]). Let 𝑋(𝑛+1)=𝐹(𝑋(𝑛)), 𝑛=0,1,…, be a system of difference equations and let 𝑋 be the equilibrium point of the system. If all eigenvalues of the Jacobian matrix evaluated at 𝑋 lie inside the open unit disk, then 𝑋 is asymptotically stable. If one of them has a modulus greater than one, then 𝑋 is unstable.

Theorem 8 (see [13]). Let 𝑋(𝑛+1)=𝐹(𝑋(𝑛)), 𝑛=0,1,…, be a system of difference equations and let 𝑋 be the equilibrium point of the system. Assume that the characteristic polynomial of the system about 𝑋 is 𝑎₀𝜆^𝑛+𝑎₁𝜆^𝑛−1+⋯+𝑎_𝑛−1𝜆+𝑎_𝑛 where 𝑎_𝑖∈ℝ for all 𝑖 and 𝑎₀>0. Then all roots of the characteristic equation lie inside the open unit disk if and only if Δ_𝑘>0 for all positive integer 𝑘≤𝑛, where Δ_𝑘 is the principal minor of order 𝑘 of the 𝑛×𝑛 matrix Δ_𝑛=𝑎₁𝑎₃𝑎₅𝑎⋯0₀𝑎₂𝑎₄⋯00𝑎₁𝑎₃⋯0⋮⋮⋮⋱⋮000⋯𝑎_𝑛. (13)

3. Results

We note that(i)if 𝐴<1 and 𝐵<1 then the system (9) has equilibrium (0,0) and (√^𝑚1−𝐵,√^𝑚1−𝐴);(ii)if 𝐴=1 and 𝐵<1 then the system (9) has equilibrium (0,0) and (√^𝑚1−𝐵,0);(iii)if 𝐴<1 and 𝐵=1 then the system (9) has equilibrium (0,0) and (0,√^𝑚1−𝐴);(iv)if 𝐴>1 and 𝐵>1 then (0,0) is the unique equilibrium point of the system (9).

Theorem 9. Let (𝑥_𝑛,𝑦_𝑛) be positive solution of the system (9). For all nonnegative integer 𝑘, one has 0≤𝑥_𝑛≤𝑥_−𝑚+1𝑥𝐴^𝑘+1,𝑛=𝑚𝑘+1;_−𝑚+2𝐴^𝑘+1⋮𝑥,𝑛=𝑚𝑘+2;₀𝐴^𝑘+1,𝑛=𝑚𝑘+𝑚,0≤𝑦_𝑛≤𝑦_−𝑚+1𝑦𝐵^𝑘+1,𝑛=𝑚𝑘+1;_−𝑚+2𝐵^𝑘+1⋮𝑦,𝑛=𝑚𝑘+2;₀𝐵^𝑘+1,𝑛=𝑚𝑘+𝑚. (14)

Proof. Obviously, they are true for 𝑘=0. Suppose that they are true for 𝑘=𝑙. Then𝑥_𝑛=𝑥_{𝑚(𝑙+1)+1}≤𝑥_{𝑚(𝑙+1)−𝑚+1}𝐴=1𝐴𝑥_𝑚𝑙+1≤1𝐴𝑥_−𝑚+1𝐴^𝑙+1()𝑥,𝑛=𝑚𝑙+1+1;_{𝑚(𝑚+1)+2}≤𝑥_{𝑚(𝑙+1)−𝑚+2}𝐴=1𝐴𝑥_𝑚𝑙+2≤1𝐴𝑥_−𝑚+2𝐴^𝑙+1()⋮𝑥,𝑛=𝑚𝑙+1+2;_{𝑚(𝑙+1)+𝑚}≤𝑥_𝑚(𝑙+1)𝐴=1𝐴𝑥_{𝑚𝑙+𝑚}≤1𝐴𝑥₀𝐴^𝑙+1()𝑦,𝑛=𝑚𝑙+1+𝑚,_𝑛=𝑦_{𝑚(𝑙+1)+1}≤𝑦_{𝑚(𝑙+1)−𝑚+1}𝐵=1𝐵𝑦_𝑚𝑙+1≤1𝐵𝑦_−𝑚+1𝐵^𝑙+1()𝑦,𝑛=𝑚𝑙+1+1;_4(𝑙+1)+2≤𝑦_{𝑚(𝑙+1)−𝑚+2}𝐵=1𝐵𝑦_𝑚𝑙+2≤1𝐵𝑦_−𝑚+2𝐵^𝑙+1()⋮𝑦,𝑛=4𝑚+1+2;_{𝑚(𝑙+1)+𝑚}≤𝑦_𝑚(𝑙+1)𝐵=1𝐵𝑦_{𝑚𝑙+𝑚}≤1𝐵𝑦₀𝐵^𝑙+1,𝑛=𝑚(𝑙+1)+𝑚. (15)
Thus, they are true for 𝑘=𝑙+1.
By the mathematical induction, this proof is completed.

Corollary 10. Let (𝑥_𝑛,𝑦_𝑛) be positive solution of the system (9). If 𝐴>1 and 𝐵>1, then the sequence {(𝑥_𝑛,𝑦_𝑛)} converges exponentially to the equilibrium point (0,0).

Theorem 11. Let 𝐴>1 and 𝐵>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 Φ_𝑛+1=𝐷Φ_𝑛, (16) where Φ_𝑛=𝑥_𝑛𝑥_𝑛−1𝑥_𝑛−2⋮𝑥_{𝑛−𝑚+1}𝑦_𝑛𝑦_𝑛−1𝑦_𝑛−2⋮𝑦_{𝑛−𝑚+1},1𝐷=00⋯0𝐴100⋯0010⋯0000⋯0001⋯0000⋯00⋮⋱⋱⋱⋱⋱⋱⋮⋮⋮00⋯100000000⋯000000𝐵.00⋯001000000⋯0001000⋮⋱⋱⋱⋱⋱⋱⋱⋱⋮00⋯0000010 (17)
The characteristic equation of the system (16) is 𝜆^𝑚−1𝐴𝜆^𝑚−1𝐵=0. (18) Thus, |𝜆|<1. By Theorem 7, the equilibrium point (0,0) is asymptotically stable.

Theorem 12. Let 𝐴<1 and 𝐵<1. Then both the equilibrium points (0,0) and (√^𝑚1−𝐵,√^𝑚1−𝐴) 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 (√^𝑚1−𝐵,√^𝑚1−𝐴). The linearized system of the system (9) about the equilibrium (√^𝑚1−𝐵,√^𝑚1−𝐴) is Φ_𝑛+1=𝐺Φ_𝑛, (19) where Φ_𝑛=𝑥_𝑛𝑥_𝑛−1𝑥_𝑛−2⋮𝑥_{𝑛−𝑚+1}𝑦_𝑛𝑦_𝑛−1𝑦_𝑛−2⋮𝑦_{𝑛−𝑚+1},𝐺=,00⋯01𝛼𝛼⋯𝛼𝛼10⋯0000⋯0001⋯0000⋯00⋮⋱⋱⋱⋱⋱⋱⋮⋮⋮00⋯1000000𝛽𝛽⋯𝛽𝛽0000100⋯001000000⋯0001000⋮⋱⋱⋱⋱⋱⋱⋱⋱⋮00⋯0000010 (20) in which 𝛼=−^𝑚(1−𝐴)^𝑚−1(1−𝐵),𝛽=−^𝑚(1−𝐴)(1−𝐵)^𝑚−1. (21)
The characteristic polynomial of the system (19) is 1−𝛼𝛽−^𝑚_𝑖=1𝑖𝛼𝛽𝜆^𝑖−1−2𝜆^𝑚−^𝑚−1_𝑖=1𝑖𝛼𝛽𝜆^{2𝑚−(𝑖+1)}+𝜆^2𝑚. (22)
We note the characteristic polynomial 𝑎₀𝜆^2𝑚+𝑎₁𝜆^2𝑚−1+⋯+𝑎_2𝑚−1𝜆+𝑎_2𝑚 that 𝑎₁=0. Thus, we obtain that not all of Δ_𝑘>0, 𝑘=1,2,…,2𝑚. By Theorems 7 and 8, the equilibrium point (√^𝑚1−𝐵,√^𝑚1−𝐴) is unstable.

Theorem 13. Let 𝐴,𝐵<1 and Ω₁=(0,√^𝑚1−𝐵)×(√^𝑚1−𝐴,∞), Ω₂=(√^𝑚1−𝐵,∞)×(0,√^𝑚1−𝐴). Assume that {(𝑥_𝑛,𝑦_𝑛)}^∞_{𝑛=−𝑚+1} satisfies the system (9). Then(i)if {(𝑥_𝑛,𝑦_𝑛)}⁰_{𝑛=−𝑚+1}⊆Ω₁, then {(𝑥_𝑛,𝑦_𝑛)}^∞_{𝑛=−𝑚+1}⊆Ω₁;(ii)if {(𝑥_𝑛,𝑦_𝑛)}⁰_{𝑛=−𝑚+1}⊆Ω₂, then {(𝑥_𝑛,𝑦_𝑛)}^∞_{𝑛=−𝑚+1}⊆Ω₂.

Proof. (i) Assume that {(𝑥_𝑛,𝑦_𝑛)}⁰_{𝑛=−𝑚+1}⊆Ω₁. Then, for any 𝑖∈{0,1,…,𝑚−1}, 𝑥_𝑖+1=𝑥_{𝑖−𝑚+1}𝐴+𝑦_𝑖𝑦_𝑖−1⋯𝑦_{𝑖−𝑚+1}<𝑥_{𝑖−𝑚+1}𝐴+√^𝑚1−𝐴^𝑚=𝑥_{𝑖−𝑚+1},𝑦_𝑖+1=𝑦_{𝑖−𝑚+1}𝐵+𝑥_𝑖𝑥_𝑖−1⋯𝑥_{𝑖−𝑚+1}>𝑦_{𝑖−𝑚+1}𝐵+√^𝑚1−𝐵^𝑚=𝑦_{𝑖−𝑚+1}. (23)
Then (𝑥₁,𝑦₁),(𝑥₂,𝑦₂),…,(𝑥_𝑚,𝑦_𝑚)∈Ω₁.
Next, we suppose that (𝑥_𝑘,𝑦_𝑘),(𝑥_𝑘−1,𝑦_𝑘−1),…,(𝑥_{𝑘−𝑚+1},𝑦_{𝑘−𝑚+1})∈Ω₁ where 𝑘 is a positive integer. Then 𝑥_𝑘+1=𝑥_{𝑘−𝑚+1}𝐴+𝑦_𝑘𝑦_𝑘−1⋯𝑦_{𝑘−𝑚+1}<𝑥_{𝑘−𝑚+1}𝐴+√^𝑚1−𝐴^𝑚=𝑥_{𝑘−𝑚+1},𝑦_𝑘+1=𝑦_{𝑘−𝑚+1}𝐵+𝑥_𝑘𝑥_𝑘−1⋯𝑥_{𝑘−𝑚+1}>𝑦_{𝑘−𝑚+1}𝐵+√^𝑚1−𝐵^𝑚=𝑦_{𝑘−𝑚+1}. (24)
Then (𝑥_𝑘+1,𝑦_𝑘+1)∈Ω₁.
By the mathematical induction, {(𝑥_𝑛,𝑦_𝑛)}^∞_{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.