# Global Dynamics of Higher-Order Exponential Systems of Difference Equations

Authors: A. Q. Khan; M. S. M. Noorani; H. S. Alayachi
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

In this paper, we study the global dynamics of three higher-order exponential systems of rational difference equations. This suggested work considerably extends and improves some existing results in the literature.


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## Body

## 1. Introduction

Global dynamical properties of difference equations or systems of difference equations involving exponential term have been widely investigated in recent years. For instance, Ozturk et al. [1] have explored the dynamical properties of the following exponential difference equation:(1)xn+1=α+βe-xnγ+xn-1,where α,β,γ, and xp(p=0,-1) are positive real numbers. Cömert et al. [2] have explored the dynamical properties of the following higher-order exponential difference equation:(2)xn+1=α+βe-xnγ+xn-k, where α,β,γ, and xp(p=0,-1,⋯,-k) are positive real numbers. Bozkurt [3] has explored the dynamical properties of the following exponential difference equation:(3)xn+1=αe-xn+βe-xn-1γ+αxn+βxn-1, where α,β,γ, and xp(p=0,-1) are positive real numbers. In 2009, Ozturk et al. [4] have explored the dynamical properties of the following higher-order exponential difference equation:(4)xn+1=αe-nxn+n-kxn-kβ+nxn+n-kxn-k, where α,β, and xp(p=0,-1,⋯,-k) are positive real numbers. Papaschinopoulos et al. [5] have explored the dynamical properties of the following exponential systems of difference equations:(5)xn+1=α+βe-ynγ+yn-1,yn+1=δ+ϵe-ynζ+xn-1xn+1=α+βe-ynγ+xn-1,yn+1=δ+ϵe-xnζ+yn-1xn+1=α+βe-xnγ+yn-1,yn+1=δ+ϵe-ynζ+xn-1, where α,β,γ,δ,ϵ,ζ and xp,yp(p=0,-1) are positive real numbers. Khan and Sharif [6] have explored the dynamical properties of the following 3×6 exponential systems of difference equations:(6)xn+1=α4+β4e-ynγ4+yn-1,yn+1=α5+β5e-znγ5+zn-1,zn+1=α6+β6e-xnγ6+xn-1xn+1=α7+β7e-znγ7+zn-1,yn+1=α8+β8e-xnγ8+xn-1,zn+1=α9+β9e-ynγ9+yn-1xn+1=α10+β10e-xnγ10+xn-1,yn+1=α11+β11e-ynγ11+zn-1,zn+1=α12+β12e-znγ12+yn-1xn+1=α13+β13e-xnγ13+zn-1,yn+1=α14+β14e-ynγ14+yn-1,zn+1=α15+β15e-znγ15+xn-1xn+1=α16+β16e-xnγ16+yn-1,yn+1=α17+β17e-ynγ17+xn-1,zn+1=α18+β18e-znγ18+zn-1, where αp,βp,γp(p=4,5,⋯,18) and xp,yp,zp(p=0,-1) are positive real numbers. Khan and Qureshi [7] have explored the dynamical properties of the following exponential system of difference equations:(7)xn+1=αe-yn+βe-yn-1γ+αxn+βxn-1,yn+1=α1e-xn+β1e-xn-1γ1+α1yn+β1yn-1, where α,β,γ,α1,β1,γ1 and xp,yp(p=0,-1) are positive real numbers. Psarros et al. [8] have explored the dynamical properties of the following six exponential systems of difference equations:(8)xn+1=a1ynb1+yn+c1xnek1-d1xn1+ek1-d1xn,yn+1=a2xnb2+xn+c2ynek2-d2yn1+ek2-d2ynxn+1=a1xnb1+xn+c1ynek1-d1yn1+ek1-d1yn,yn+1=a2ynb2+yn+c2xnek2-d2xn1+ek2-d2xnxn+1=a1yn2b1+yn+c1ed1-k1xn1+ed1-k1xn,yn+1=a2eb2-k2xn1+eb2-k2xn+c2ed2-k3yn1+ed2-k3ynxn+1=a1yn2b1+yn+c1ed1-k1xn1+ed1-k1xn,yn+1=a2xn2b2+xn+c2ed2-k2yn1+ed2-k2ynxn+1=a1xn2b1+xn+c1ed1-k1yn1+ed1-k1yn,yn+1=a2eb2-k2xn1+eb2-k2xn+c2ed2-k3yn1+ed2-k3ynxn+1=a1xn2b1+xn+c1ed1-k1yn1+ed1-k1yn,yn+1=a2yn2b2+yn+c2ed2-k2xn1+ed2-k2xn, where ap,bp,cp,dp,kq(p=1,2,q=1,2,3) and x0,y0 are positive real numbers. Mylona et al. [9] have explored the dynamical properties of the following two 3×3 close-to-cyclic systems of exponential difference equations:(9)xn+1=a1xn+b1yne-xn,yn+1=a2yn+b2zne-yn,zn+1=a3zn+b3xne-znxn+1=a1yn+b1xne-yn,yn+1=a2zn+b2yne-zn,zn+1=a3xn+b3zne-xn, where ap,bp(p=1,2,3) and x0,y0,z0 are positive real numbers. Mylona et al. [10] have explored the dynamical properties of the following systems of difference equations with exponential terms:(10)xn+1=a1xn+a2yne-xn,yn+1=a3yn+a4xne-ynxn+1=a1yn+a2xne-yn,yn+1=a3xn+a4yne-xn, where ap(p=1,⋯,4) and x0,y0 are positive real numbers, and for more other interesting results on difference equations as well as systems of difference equation, we refer the reader to [11–13] and the references cited therein. Motivated by the above systemic studies, in this paper we aim to explore the dynamical properties of the following higher-order exponential systems of difference equations, which are natural extension of the work studied by Ozturk et al. [4]:(11)xn+1=αe-nxn+n-kxn-kβ+nyn+n-kyn-k,yn+1=α1e-nyn+n-kyn-kβ1+nxn+n-kxn-k(12)xn+1=αe-nyn+n-kyn-kβ+nxn+n-kxn-k,yn+1=α1e-nxn+n-kxn-kβ1+nyn+n-kyn-k(13)xn+1=αe-nyn+n-kyn-kβ+nyn+n-kyn-k,yn+1=α1e-nxn+n-kxn-kβ1+nxn+n-kxn-k, where α,β,α1,β1 and xp,yp(p=0,-1,⋯,-k) are positive real numbers.The rest of this paper is organized as follows: Section2 is about the study of boundedness and persistence of systems (11), (12), and (13). This also includes construction of invariant rectangle of these systems. Section 3 is about the uniqueness and existence of +ve fixed point of systems (11), (12), and (13). Section 4 deals with the study of local stability, whereas in Section 5 we study the global dynamics of the unique +ve fixed point of these systems. Section 6 deals with the study of rate of convergence of systems (11), (12), and (13). Brief conclusion is given in Section 7.


## 2. Boundedness, Persistence, and Construction of Invariant Rectangle of Systems (11), (12), and (13)

Theorem 1.
If2n≥k, then every +ve solution {(xn,yn)}n=-k∞ of (i) system (11), (ii) system (12), and (iii) system (13) is bounded and persists.Proof.
( i ) If {(xn,yn)}n=-k∞ is a +ve solution of (11), then(14)xn≤αβ=U1,yn≤α1β1=U2,n=0,1,⋯.From (11) and (14) we get(15)xn≥αe--k+2nα/ββ+-k+2nα1/β1=L1,yn≥α1e--k+2nα1/β1β1+-k+2nα/β=L2,n=2,3,⋯.Finally from (14) and (15) we get(16)L1≤xn≤U1,L2≤yn≤U2,n=3,4,⋯.
( i i ) If {(xn,yn)}n=-k∞ is a +ve solution of (12), then(17)xn≤αβ=U3,yn≤α1β1=U4,n=0,1,⋯.From (12) and (17) we have(18)xn≥αe--k+2nα1/β1β+-k+2nα/β=L3,yn≥α1e--k+2nα/ββ1+-k+2nα1/β1=L4,n=2,3,⋯.Finally from (17) and (18) we get(19)L3≤xn≤U3,L4≤yn≤U4,n=3,4,⋯.
( i i i ) If {(xn,yn)}n=-k∞ is a +ve solution of (13), then(20)xn≤αβ=U5,yn≤α1β1=U6,n=0,1,⋯.From (13) and (20) we get(21)xn≥αe--k+2nα1/β1β+-k+2nα1/β1=L5,yn≥α1e--k+2nα/ββ1+-k+2nα/β=L6,n=2,3,⋯.Finally from (20) and (21) we get(22)L5≤xn≤U5,L6≤yn≤U6,n=3,4,⋯.Theorem 2.
If2n≥k, then following holds: (i)
If{(xn,yn)}n=-k∞ is a +ve solution of (11), then L1,U1×L2,U2 is invariant rectangle for (11).(ii)
If{(xn,yn)}n=-k∞ is a +ve solution of (12), then L3,U3×L4,U4 is invariant rectangle for (12).(iii)
If{(xn,yn)}n=-k∞ is a +ve solution of (13), then L5,U5×L6,U6 is invariant rectangle for (13).Proof.
The proof follows from mathematical induction.


## 3. Existence and Uniqueness of+ve Equilibrium of Systems (11), (12), and (13)

Theorem 3.
Assume that2n≥k, then the following statements hold: (i)
If0<β<1 and(23)αe--k+2nL1β1+-k+2nU1-k+2nU1+1αe--k+2nL1+1-βU1<-k+2nL12αe--k+2nU1-βU1,then (11) has a unique +ve equilibrium (x¯,y¯)∈L1,U1×L2,U2.(ii)
If(24)e-k+2nU3β+2-k+2nU3β1+2ln⁡αβ+-k+2nL3L3<α1-k+2n2β+-k+2nL3L3,then (12) has a unique +ve equilibrium (x¯,y¯)∈L3,U3×L4,U4.(iii)
If(25)α12n-k2e-2n-kL5ββ1+2n-kU5+2n-kα1e-2n-kL5U5Ω1<β1+2n-kL52Ω22,then (13) has a unique +ve equilibrium (x¯,y¯)∈L5,U5×L6,U6 where Ω1 and Ω2 are defined in (53) and (54).Proof.
( i ) Consider(26)x=αe-2n-kxβ+2n-ky,y=α1e-2n-kyβ1+2n-kx.From (26), we have(27)y=αe--k+2nx-k+2nx-β-k+2n,x=α1e--k+2ny-k+2ny-β1-k+2n.From (27), define(28)Qx=α1e--k+2nqx-k+2nqx-β1-k+2n-x,where(29)qx=αe--k+2nx-k+2nx-β-k+2n,and x∈L1,U1. Now our claim is Q(x)=0 has a single solution x∈L1,U1. From (28) and (29) one gets(30)Q′x=-α1e--k+2nqxq′x-k+2nqx+1-k+2nqx2-1,where(31)q′x=-αe--k+2nx-k+2nx+1-k+2nx2.If x¯∈L1,U1 is a solution of Q(x)=0, then from (28) and (29) one gets(32)α1e--k+2nqx¯qx¯=β1+-k+2nx¯,where(33)qx¯=αe--k+2nx¯-k+2nx¯-β-k+2n.In view of (31), (32), and (33), (30) becomes(34)Q′x=αe--k+2nx¯β1+-k+2nx¯-k+2nx¯+1αe--k+2nx¯+1-βx¯-k+2nx¯2αe--k+2nx¯-βx¯-1,≤αe--k+2nL1β1+-k+2nU1-k+2nU1+1αe--k+2nL1+1-βU1-k+2nL12αe--k+2nU1-βU1-1.Now assume that (23) holds, then from (34) one gets Q′(x)<0.
( i i ). From (12) one has(35)x=αe--k+2nyβ+-k+2nx,y=α1e--k+2nxβ1+-k+2ny.From (35) one gets(36)y=1-k+2nlnαβ+-k+2nxx,x=1-k+2nlnα1β1+-k+2nyy.From (36), define(37)Gx=1-k+2nlnα1β1+-k+2ngxgx-x,where(38)gx=1-k+2nln⁡αβ+-k+2nxx,and x∈L3,U3. We claim that G(x)=0 has a single solution x∈L3,U3. From (37) and (38) one gets(39)G′x=-g′xβ1+2-k+2ngx-k+2nβ1+-k+2ngxgx-1,where(40)g′x=-β+22n-kx2n-kβ+2n-kxx.If x¯∈L3,U3 is a solution of G(x)=0, then from (37) and (38) one gets(41)β1+-k+2ngx¯gx¯=α1e--k+2nx¯,where(42)gx¯=12n-klnαβ+2n-kx¯x¯.In view of (40), (41), and (42), (39) becomes(43)G′x=e2n-kx¯β+2-k+2nx¯β1+2lnα/β+-k+2nx¯x¯α1-k+2n2β+-k+2nx¯x¯-1,≤e-k+2nU3β+2-k+2nU3β1+2lnα/β+-k+2nL3L3α1-k+2n2β+-k+2nL3L3-1.Finally assume that (24) holds, then from (43) one gets G′(x)<0.
( i i i ) From system (13)(44)x=αe-2n-kyβ+2n-ky,y=α1e-2n-kxβ1+2n-kx.From (44), define(45)Jx=αe--k+2njxβ+-k+2njx-x,where(46)jx=α1e--k+2nxβ1+-k+2nx,and x∈L5,U5. We claim that J(x)=0 has a single solution x∈L5,U5. From (45) and (46) one gets(47)J′x=-α-k+2ne--k+2njxj′xβ+-k+2njx+1β+-k+2njx2-1,where(48)j′x=-α12n-ke-2n-kxβ1+2n-kx+1β1+2n-kx2.
Ifx¯∈L5,U5 is a solution of J(x)=0, then from (45) and (46) one gets(49)e--k+2njx¯=β+-k+2njx¯x¯α,where(50)jx¯=α1e--k+2nx¯β1+-k+2nx¯.In view of (48), (49), and (50), (47) becomes(51)J′x=α1-k+2n2e--k+2nx¯ββ1+2n-kx¯+-k+2nα1e--k+2nx¯x¯Ωβ1+-k+2nx¯2β1β1+-k+2nx¯+-k+2nα1e--k+2nx¯2-1,≤α1-k+2n2e--k+2nL5ββ1+-k+2nU5+-k+2nα1e--k+2nL5U5Ω1β1+-k+2nL52Ω22-1,where(52)Ω=β+1β1+2n-kx¯+-k+2nα1e-2n-kx¯,(53)Ω1=β+1β1+2n-kU5+-k+2nα1e-2n-kL5,and (54)Ω2=β1β1+2n-kL5+2n-kα1e-2n-kU5.Finally assume that (25) along with (53) and (54) holds, then from (51) one gets J′(x)<0.


## 4. Local Asymptotic Stability about Equilibrium of Systems (11), (12), and (13)

Theorem 4.
Assume that2n≥k, then the following statements hold:(i)
If(55)-k+nαe-2n-kL1β+2n-kU2+1+nαe-2n-kU1β+2n-kL2+1<β+2n-kL22,and(56)-k+nα1e--k+2nL2β1+-k+2nU1+1+nα1e--k+2nU2β1+-k+2nL1+1<β1+-k+2nL12,then(x¯,y¯)∈L1,U1×L2,U2 of (11) is a sink.(ii)
If(57)-k+nαe--k+2nL4β+-k+2nU3+1+nαe--k+2nU4β+-k+2nL3+1<β+-k+2nL32,and(58)-k+nα1e-2n-kL3β1+2n-kU4+1+nα1e-2n-kU3β1+2n-kL4+1<β1+2n-kL42,then(x¯,y¯)∈L3,U3×L4,U4 of (12) is a sink.(iii)
If(59)-k+nαe--k+2nL6β+-k+2nU6+1+nαe--k+2nU6β+-k+2nL6+1<β+-k+2nL62,and(60)-k+nα1e--k+2nL5β1+-k+2nU5+1+nα1e--k+2nU5β1+-k+2nL5+1<β1+-k+2nL52,then(x¯,y¯)∈L5,U5×L6,U6 of (13) is a sink.Proof.
( i ) If (x¯,y¯) is equilibrium of (11), then(61)x¯=αe--k+2nx¯β+-k+2ny¯,y¯=α1e--k+2ny¯β1+-k+2nx¯.Moreover, linearized equation of (11) about (x¯,y¯) is(62)Φn+1=EΦn,where(63)Φn=xn,⋯,xn-k,yn,⋯,yn-kT,(64)E=μ10…0μ2μ30…0μ410…0000…00⋮⋮…⋮⋮⋮⋱⋮⋮⋮00…1000…00μ50…0μ6μ70…0μ800…0010…00⋮⋮⋱⋮⋮⋮⋱⋮⋮⋮00…0000…10,and(65)μ1=-nαe--k+2nx¯β+-k+2ny¯,μ2=--k+nαe--k+2nx¯β+2n-ky¯,μ3=-nαe--k+2nx¯β+2n-ky¯2,μ4=--k+nαe--k+2nx¯β+2n-ky¯2,μ5=-nα1e--k+2ny¯β1+-k+2nx¯2,μ6=--k+nα1e--k+2ny¯β1+-k+2nx¯2,μ7=-nα1e--k+2ny¯β1+-k+2nx¯,μ8=--k+nα1e--k+2ny¯β1+-k+2nx¯.
Let eigenvalues ofE be λi,i=1,2,⋯,2k+2. Let D=diag(τ1,τ2,…,τ2k+2) be a diagonal matrix with τ1=τk+2=1, τ1+m=τk+2+m=1-mϵ(1≤m≤k), and(66)0<ϵ<min1k1--k+nαe--k+2nx¯β+-k+2ny¯+1β+-k+2ny¯2-nαe--k+2nx¯β+-k+2ny¯+1,1k1--k+nα1e--k+2ny¯β1+-k+2nx¯+1β1+-k+2nx¯2-nα1e--k+2ny¯β1+-k+2nx¯+1.Since D is invertible. So,(67)DED-1=τ1τ1-1μ10…0τ1τk+1-1μ2τ1τk+2-1μ30…0τ1τ2k+2-1μ4τ2τ1-10…0000…00⋮⋮…⋮⋮⋮⋱⋮⋮⋮00…τk+1τk-1000…00τk+2τ1-1μ50…0τk+2τk+1-1μ6τk+2τk+2-1μ70…0τk+2τ2k+2-1μ800…00τk+3τk+2-10…00⋮⋮⋱⋮⋮⋮⋱⋮⋮⋮00…0000…τ2k+2τ2k+1-10.
The following two inequalities,(68)τ1>τ2>⋯>τk+1>0,τk+2>τk+3>⋯>τ2k+2>0,imply that(69)τ2τ1-1<1,τ3τ2-1<1,⋮τk+1τk-1<1,τk+3τk+2-1<1,τk+4τk+3-1<1,⋮τ2k+2τ2k+1-1<1.Furthermore,(70)τ1τ1-1μ1+τ1τk+1-1μ2+τ1τk+2-1μ3+τ1τ2k+2-1μ4=αe--k+2nx¯β+-k+2ny¯+1β+-k+2ny¯2n+-k+n1-kϵ<1.(71)τk+2τ1-1μ5+τk+2τk+1-1μ6+μ7+τk+2τ2k+2-1μ8=α1e--k+2ny¯β1+-k+2nx¯+1β1+-k+2nx¯2n+-k+n1-kϵ<1.It is a well-known fact that E and DED-1 have the same eigenvalues. Thus(72)max1≤m≤2k+2λm≤DED-1∞=maxτ2τ1-1,⋯,τk+1τk-1,τk+3τk+2-1,⋯,τ2k+2τ2k+1-1,αe--k+2nx¯β+2n-ky¯+1β+-k+2ny¯2n+n-k1-kϵ,α1e--k+2ny¯β1+-k+2nx¯+1β1+2n-kx¯2n+2n-k1-kϵ<1.Hence (x¯,y¯)∈L1,U1×L2,U2 of (11) is a sink.
( i i ). If (x¯,y¯) is equilibrium of (12), then(73)x¯=αe--k+2ny¯β+2n-kx¯,y¯=α1e-2n-kx¯β1+2n-ky¯,and linearized equation of (12) about (x¯,y¯) is(74)Φn+1=EΦn,where Φn is defined in (63). Also(75)E=μ90…0μ10μ110…0μ1210…0000…00⋮⋮…⋮⋮⋮⋱⋮⋮⋮00…1000…00μ130…0μ14μ150…0μ1600…0010…00⋮⋮⋱⋮⋮⋮⋱⋮⋮⋮00…0000…10,
and(76)μ9=-nαe--k+2ny¯β+-k+2nx¯2,μ10=--k+nαe--k+2ny¯β+-k+2nx¯2,μ11=-nαe--k+2ny¯β+-k+2nx¯,μ12=--k+nαe--k+2ny¯β+2n-kx¯,μ13=-nα1e--k+2nx¯β1+-k+2ny¯,μ14=--k+nα1e--k+2nx¯β1+-k+2ny¯,μ15=-nα1e--k+2nx¯β1+-k+2ny¯2,μ16=--k+nα1e--k+2nx¯β1+-k+2ny¯2. Using arrangements as in the proof of (i) one gets(77)0<ϵ<min⁡1k1--k+nαe--k+2ny¯β+-k+2nx¯+1β+-k+2nx¯2-nαe--k+2ny¯β+-k+2nx¯+1,1k1--k+nα1e--k+2nx¯β1+-k+2ny¯+1β1+-k+2ny¯2-nα1e--k+2nx¯β1+-k+2ny¯+1,and(78)DED-1=τ1τ1-1μ90…0-τ1τk+1-1μ10-τ1τk+2-1μ110…0-τ1τ2k+2-1μ12τ2τ1-10…0000…00⋮⋮…⋮⋮⋮⋱⋮⋮⋮00…τk+1τk-1000…00-τk+2τ1-1μ130…0-τk+2τk+1-1μ14-τk+2τk+2-1μ150…0-τk+2τ2k+2-1μ1600…00τk+3τk+2-10…00⋮⋮⋱⋮⋮⋮⋱⋮⋮⋮00…0000…τ2k+2τ2k+1-10.Assume that (68) and (69) hold, then(79)μ9+τ1τk+1-1μ10+τ1τk+2-1μ11+τ1τ2k+2-1μ12=αe--k+2ny¯β+-k+2nx¯+1β+-k+2nx¯2n+-k+n1-kϵ<1.(80)μ13+τk+2τk+1-1μ14+μ15+τk+2τ2k+2-1μ16=α1e--k+2nx¯β1+-k+2ny¯+1β1+-k+2ny¯2n+-k+n1-kϵ<1. Finally,(81)max1≤m≤2k+2λm≤DED-1∞=maxτ2τ1-1,⋯,τk+1τk-1,τk+3τk+2-1,⋯,τ2k+2τ2k+1-1,αe--k+2ny¯β+-k+2nx¯+1β+-k+2nx¯2n+n-k1-kϵ,α1e--k+2nx¯β1+-k+2ny¯+1β1+-k+2ny¯2n+n-k1-kϵ<1.Hence equilibrium (x¯,y¯)∈L3,U3×L4,U4 (12) is a sink.( i i i ) If (x¯,y¯) is a fixed point of (13), then(82)x¯=αe--k+2ny¯β+-k+2ny¯,y¯=α1e--k+2nx¯β1+-k+2nx¯,and linearized equation of (13) about (x¯,y¯) is(83)Φn+1=EΦn,where Φn is again the same as (63). Moreover(84)E=00…00μ170…0μ1810…0000…00⋮⋮…⋮⋮⋮⋱⋮⋮⋮00…1000…00μ190…0μ2000…0000…0010…00⋮⋮⋱⋮⋮⋮⋱⋮⋮⋮00…0000…10, where(85)μ17=-nαe--k+2ny¯β+-k+2ny¯+1β+2n-ky¯2,μ18=--k+nαe--k+2ny¯β+2n-ky¯+1β+2n-ky¯2,μ19=-nα1e--k+2nx¯β1+-k+2nx¯+1β1+-k+2nx¯2,μ20=--k+nα1e--k+2nx¯β1+-k+2nx¯+1β1+-k+2nx¯2.Again using similar arrangements as in the proof of (i) one gets(86)0<ϵ<min⁡1k1--k+nαe--k+2ny¯β+-k+2ny¯+1β+-k+2ny¯2-nαe--k+2ny¯β+-k+2ny¯+1,1k1--k+nα1e--k+2nx¯β1+-k+2nx¯+1β1+-k+2ny¯2-nα1e--k+2nx¯β1+-k+2nx¯+1.Moreover,(87)DED-1=00…00-τ1τk+2-1μ170…0-τ1τ2k+2-1μ18τ2τ1-10…0000…00⋮⋮…⋮⋮⋮⋱⋮⋮⋮00…τk+1τk-1000…00-τk+2τ1-1μ190…0-τk+2τk+1-1μ2000…0000…00τk+3τk+2-10…00⋮⋮⋱⋮⋮⋮⋱⋮⋮⋮00…0000…τ2k+2τ2k+1-10.Assume that (68) and (69) hold true, then(88)τ1τk+2-1μ17+τ1τ2k+2-1μ18=αe--k+2ny¯β+2n-ky¯+1β+2n-ky¯2n+-k+n1-kϵ<1.(89)τk+2τ1-1μ19+τk+2τk+1-1μ20=α1e--k+2nx¯β1+-k+2nx¯+1β1+-k+2nx¯2n+-k+n1-kϵ<1.Finally,(90)max1≤m≤2k+2λm≤DED-1∞=maxτ2τ1-1,⋯,τk+1τk-1,τk+3τk+2-1,⋯,τ2k+2τ2k+1-1,αe--k+2ny¯β+-k+2ny¯+1β+-k+2ny¯2n+n--k1-kϵ,α1e--k+2nx¯β1+-k+2nx¯+1β1+-k+2nx¯2n+n-k1-kϵ<1.Hence equilibrium (x¯,y¯)∈L5,U5×L6,U6 of (13) is a sink.Hereafter we study the global dynamics about unique+ve equilibrium of systems (11), (12), and (13). Since investigating the global stability of difference equations or systems of difference equation for higher-order is a challenging task in recent year, here we will investigate the global dynamics about unique +ve equilibrium of systems (11), (12), and (13) by utilizing discrete-time Lyapunov function motivated by the work of [14–16].


## 5. Global Stability about Equilibrium of Systems (11), (12), and (13)

Theorem 5.
Assume that2n≥k, then the following statements hold:(i)
If(91)αe--k+2nL1<x¯β+-k+2nL2,and(92)α1e--k+2nL2<y¯β1+-k+2nL1,then(x¯,y¯)∈L1,U1×L2,U2 of (11) is globally asymptotically stable.(ii)
If(93)αe--k+2nL4<x¯β+2n-kL3,and(94)α1e--k+2nL3<y¯β1+-k+2nL4,then(x¯,y¯)∈L3,U3×L4,U4 of (12) is globally asymptotically stable.(iii)
If(95)αe--k+2nL6<x¯β+-k+2nL6,and(96)α1e--k+2nL5<y¯β1+-k+2nL5,then(x¯,y¯)∈L5,U5×L6,U6 of (13) is globally asymptotically stable.Proof.
( i ) Consider(97)Γn=x¯-1+xnx¯-lnxnx¯+y¯-1+yny¯-lnyny¯, where its positivity follows from the inequality below:(98)-1+x-ln⁡x≥0,∀x>0. Moreover,(99)-lnxn+1xn=ln-1-xnxn+1+1≤-xn+1-xnxn+1,-lnyn+1yn=ln-1-ynyn+1+1≤-yn+1-ynyn+1. Now(100)Γn+1-Γn=xn+1-xn-x¯lnxn+1xn+yn+1-yn-y¯lnyn+1yn. In view of (99), (100) takes the form(101)Γn+1-Γn≤xn+1-xn1-x¯xn+1+yn+1-yn1-y¯yn+1,=xn+1-xnαe-nxn+n-kxn-k-x¯β+nyn+n-kyn-kαe-nxn+n-kxn-k+yn+1-ynα1e-nyn+n-kyn-k-y¯β1+nxn+n-kxn-kα1e-nyn+n-kyn-k,≤U1-L1αe--k+2nL1-x¯β+-k+2nL2αe--k+2nL1+U2-L2α1e--k+2nL2-y¯β1+-k+2nL1α1e--k+2nL2.Assuming that (91) and (92) hold, then from (101) one gets Γn+1-Γn<0 for all n≥0. Hence, limn→∞(Γn+1-Γn)=0. Therefore, (x¯,y¯)∈L1,U1×L2,U2 of (11) is globally asymptotically stable.
( i i ). Using similar construction as in the proof of (i) one gets(102)Γn+1-Γn≤xn+1-xnαe-nyn+n-kyn-k-x¯β+nxn+n-kxn-kαe-nyn+n-kyn-k+yn+1-ynα1e-nxn+n-kxn-k-y¯β1+nyn+n-kyn-kα1e-nxn+n-kxn-k,≤U3-L3αe--k+2nL4-x¯β+-k+2nL3αe--k+2nL4+U4-L4α1e--k+2nL3-y¯β1+-k+2nL4α1e--k+2nL3.Assuming that (93) and (94) hold, then from (102) one gets Γn+1-Γn<0 for all n≥0. Hence, limn→∞(Γn+1-Γn)=0. Therefore, (x¯,y¯)∈L3,U3×L4,U4 of (12) is globally asymptotically stable.
( i i i ) Again using similar arrangements, one gets(103)Γn+1-Γn≤xn+1-xnαe-nyn+n-kyn-k-x¯β+nyn+n-kyn-kαe-nyn+n-kyn-k+yn+1-ynα1e-nxn+n-kxn-k-y¯β1+nxn+n-kxn-kα1e-nxn+n-kxn-k,≤U5-L5αe--k+2nL6-x¯β+-k+2nL6αe--k+2nL6+U6-L6α1e--k+2nL5-y¯β1+-k+2nL5α1e--k+2nL5.Assuming that (95) and (96) hold, then from (103) one gets Γn+1-Γn<0 for all n≥0. Hence, limn→∞(Γn+1-Γn)=0. Thus, (x¯,y¯)∈L5,U5×L6,U6 of (13) is globally asymptotically stable.Hereafter we will study rate of convergence of systems (11), (12), and (13) motivated by the work of [15, 17, 18].


## 6. Rate of Convergence of Systems (11), (12), and (13)

### 6.1. Rate of Convergence of System (11)

Let{(xn,yn)} be any solution of (11) such that the following hold:(104)limn→∞xn=x¯,limn→∞yn=y¯. Now computing error terms one gets(105)xn+1-x¯=αe-nxn+n-kxn-kβ+nyn+n-kyn-k-αe--k+2nx¯β+-k+2ny¯,=-αe-nxn+n-kxn-kenxn+n-kxn-k--k+2nx¯-1β+nyn+n-kyn-k-αne--k+2nx¯β+nyn+n-kyn-kβ+-k+2ny¯yn-y¯-αn-ke--k+2nx¯β+nyn+n-kyn-kβ+-k+2ny¯yn-k-y¯.After some tedious calculations, from (105) one gets(106)xn+1-x¯=-αne-nxn+n-kxn-kβ+nyn+n-kyn-kxn-x¯-αn-ke-nxn+n-kxn-kβ+nyn+n-kyn-kxn-k-x¯-αne--k+2nx¯β+nyn+n-kyn-kβ+-k+2ny¯yn-y¯-αn-ke--k+2nx¯β+nyn+n-kyn-kβ+-k+2ny¯yn-k-y¯+O1nxn+n-kxn-k--k+2nx¯2.Similarly,(107)yn+1-y¯=-α1ne--k+2ny¯β1+nxn+n-kxn-kβ1+-k+2nx¯xn-x¯-α1n-ke--k+2ny¯β1+nxn+n-kxn-kβ1+-k+2nx¯xn-k-x¯-α1ne-nyn+n-kyn-kβ1+nxn+n-kxn-kyn-x¯-α1n-ke-nyn+n-kyn-kβ1+nxn+n-kxn-kyn-k-y¯+O2nyn+n-kyn-k--k+2ny¯2.From (106) and (107), we have(108)xn+1-x-≈-αne-nxn+n-kxn-kβ+nyn+n-kyn-kxn-x--αn-ke-nxn+n-kxn-kβ+nyn+n-kyn-kxn-k-x--αne--k+2nx-β+nyn+n-kyn-kβ+-k+2ny-yn-y--αn-ke--k+2nx-β+nyn+n-kyn-kβ+-k+2ny-yn-k-y-,yn+1-y-≈-α1ne--k+2ny-β1+nxn+n-kxn-kβ1+-k+2nx-xn-x--α1n-ke--k+2ny-β1+nxn+n-kxn-kβ1+-k+2nx-xn-k-x--α1ne-nyn+n-kyn-kβ1+nxn+n-kxn-kyn-x--α1n-ke-nyn+n-kyn-kβ1+nxn+n-kxn-kyn-k-y-.Let(109)ϵn1=xn-x¯,ϵn2=yn-y¯,and then (108) becomes(110)ϵn+11≈An1ϵn1+An2ϵn-k1+An3ϵn2+An4ϵn-k2,ϵn+12≈An5ϵn1+An6ϵn-k1+An7ϵn2+An8ϵn-k2,where(111)An1=-αne-nxn+-k+nxn-kβ+nyn+-k+nyn-k,An2=-α-k+ne-nxn+n-kxn-kβ+nyn+n-kyn-k,An3=-αne--k+2nx¯β+nyn+n-ky-k+nβ+-k+2ny¯,An4=-α-k+ne--k+2nx¯β+nyn+n-kyn-kβ+-k+2ny¯,An5=-α1ne--k+2ny¯β1+nxn+n-kxn-kβ1+-k+2nx¯,An6=-α1-k+ne--k+2ny¯β1+nxn+n-kxn-kβ1+-k+2nx¯,An7=-α1ne-nyn+n-kyn-kβ1+nxn+n-kxn-k,An8=-α1-k+ne-nyn+n-kyn-kβ1+nxn+n-kxn-k.Moreover,(112)limn→∞An1=-nαe--k+2nx¯β+-k+2ny¯,limn→∞An2=--k+nαe--k+2nx¯β+-k+2ny¯,limn→∞An3=-nαe--k+2nx¯β+-k+2ny¯2,limn→∞An4=--k+nαe--k+2nx¯β+-k+2ny¯2,limn→∞An5=-nα1e--k+2ny¯β1+-k+2nx¯2,limn→∞An6=--k+nα1e--k+2ny¯β1+-k+2nx¯2,limn→∞An7=-nα1e--k+2ny¯β1+-k+2nx¯,limn→∞An8=--k+nα1e--k+2ny¯β1+-k+2nx¯.Hence limiting system of error terms becomes(113)En+1=EEn,where(114)E=-nαe--k+2nx¯β+-k+2ny¯0…0--k+nαe--k+2nx¯β+-k+2ny¯-nαe--k+2nx¯β+-k+2ny¯20…0--k+nαe--k+2nx¯β+-k+2ny¯210…0000…00⋮⋮…⋮⋮⋮⋱⋮⋮⋮00…1000…00-nα1e--k+2ny¯β1+-k+2nx¯20…0--k+nα1e--k+2ny¯β1+-k+2nx¯2-nα1e--k+2ny¯β1+-k+2nx¯0…0--k+nα1e--k+2ny¯β1+-k+2nx¯00…0010…00⋮⋮⋱⋮⋮⋮⋱⋮⋮⋮00…0000…10,which is similar to linearized system of (11). Finally we have the following theorem by utilizing Perron’s Theorems (see [19]).Theorem 6.
Assume that{(xn,yn)} is a positive solution of (11) such that (104) along with the following relation holds:(115)x¯∈L1,U1,y¯∈L2,U2.Then the error vector(116)En=ϵn1ϵn-11⋮ϵn-k1ϵn2ϵn-12⋮ϵn-k2of every solution of (11) satisfies(117)limn→∞En1/n=λE,limn→∞En+1En=λE,where λE are the characteristic roots of E about (x¯,y¯).


### 6.2. Rate of Convergence of System (12)

If{(xn,yn)} is any solution of (12) such that (104) holds, then(118)xn+1-x¯=αe-nyn+n-kyn-kβ+nxn+n-kxn-k-αe--k+2ny¯β+-k+2nx¯,=-αne--k+2ny¯β+nxn+n-kxn-kβ+-k+2nx¯xn-x¯-αn-ke--k+2ny¯β+nxn+n-kxn-kβ+-k+2nx¯xn-k-x¯-αe-nyn+n-kyn-kenyn+n-kyn-k--k+2ny¯-1β+nxn+n-kxn-k.After calculations, from (118) one gets(119)xn+1-x¯=-αne--k+2ny¯β+nxn+n-kxn-kβ+-k+2nx¯xn-x¯-αn-ke--k+2ny¯β+nxn+n-kxn-kβ+-k+2nx¯xn-k-x¯-αne-nyn+n-kyn-kβ+nxn+n-kxn-kyn-y¯-αn-ke-nyn+n-kyn-kβ+nxn+n-kxn-kyn-k-y¯+O3nyn+n-kyn-k--k+2ny¯2.Similarly,(120)yn+1-y¯=-α1ne-nxn+n-kxn-kβ1+nyn+n-kyn-kxn-x¯-α1n-ke-nxn+n-kxn-kβ1+nyn+n-kyn-kxn-k-x¯-α1ne--k+2nx¯β1+nyn+n-kyn-kβ1+-k+2ny¯yn-y¯-α1n-ke--k+2nx¯β1+nyn+n-kyn-kβ1+-k+2ny¯yn-k-y¯+O4nxn+n-kxn-k--k+2nx¯2.From (119) and (120), we have(121)xn+1-x-≈-αne--k+2ny-β+nxn+n-kxn-kβ+-k+2nx-xn-x--αn-ke--k+2ny-β+nxn+n-kxn-kβ+-k+2nx-xn-k-x--αne-nyn+n-kyn-kβ+nxn+n-kxn-kyn-y--αn-ke-nyn+n-kyn-kβ+nxn+n-kxn-kyn-k-y-,yn+1-y-≈-α1ne-nxn+n-kxn-kβ1+nyn+n-kyn-kxn-x--α1n-ke-nxn+n-kxn-kβ1+nyn+n-kyn-kxn-k-x--α1ne--k+2nx-β1+nyn+n-kyn-kβ1+-k+2ny-yn-y--α1n-ke--k+2nx-β1+nyn+n-kyn-kβ1+-k+2ny-yn-k-y-.Let (109) hold, then from (121) one gets(122)ϵn+11≈An9ϵn1+An10ϵn-k1+An11ϵn2+An12ϵn-k2,ϵn+12≈An13ϵn1+An14ϵn-k1+An15ϵn2+An10ϵn-k2,where(123)An9=-αne--k+2ny¯β+nxn+n-kxn-kβ+-k+2nx¯,An10=-α-k+ne--k+2ny¯β+nxn+n-kxn-kβ+-k+2nx¯,An11=-αne-nyn+n-kyn-kβ+nxn+n-kxn-k,An12=-α-k+ne-nyn+n-kyn-kβ+nxn+n-kxn-k,An13=-α1ne-nxn+n-kxn-kβ1+nyn+n-kyn-k,An14=-α1-k+ne-nxn+n-kxn-kβ1+nyn+n-kyn-k,An15=-α1ne--k+2nx¯β1+nyn+n-kyn-kβ1+-k+2ny¯,An16=-α1-k+ne--k+2nx¯β1+nyn+n-kyn-kβ1+-k+2ny¯.Moreover,(124)limn→∞An9=-nαe--k+2ny¯β+-k+2nx¯2,limn→∞An10=--k+nαe--k+2ny¯β+-k+2nx¯2,limn→∞An11=-nαe-2n-ky¯β+2n-kx¯,limn→∞An12=--k+nαe--k+2ny¯β+2n-kx¯,limn→∞An13=-nα1e--k+2nx¯β1+-k+2ny¯,limn→∞An14=--k+nα1e--k+2nx¯β1+-k+2ny¯,limn→∞An15=-nα1e--k+2nx¯β1+-k+2ny¯2,limn→∞An16=--k+nα1e--k+2nx¯β1+-k+2ny¯2.Hence limiting system of error terms becomes(125)En+1=EEn, where(126)E=-nαe--k+2ny¯β+-k+2nx¯20…0--k+nαe--k+2ny¯β+-k+2nx¯2-nαe--k+2ny¯β+-k+2nx¯0…0--k+nαe--k+2ny¯β+2n-kx¯10…0000…00⋮⋮…⋮⋮⋮⋱⋮⋮⋮00…1000…00-nα1e--k+2nx¯β1+-k+2ny¯0…0--k+nα1e--k+2nx¯β1+-k+2ny¯-nα1e--k+2nx¯β1+-k+2ny¯20…0--k+nα1e--k+2nx¯β1+-k+2ny¯200…0010…00⋮⋮⋱⋮⋮⋮⋱⋮⋮⋮00…0000…10,which is the same as linearized system of (12) about (x¯,y¯). Finally we have following theorem.Theorem 7.
Assume that{(xn,yn)} is a positive solution of (12) such that (104) along with the following relation holds:(127)x¯∈L3,U3,y¯∈L4,U4. Then the error vector En defined in (116) satisfies(128)limn→∞En1/n=λE,limn→∞En+1En=λE,where λE are the characteristic roots of E about (x¯,y¯).


### 6.3. Rate of Convergence of System (13)

If{(xn,yn)} is any solution of (13) such that (104) holds, then(129)xn+1-x¯=αe-nyn+n-kyn-kβ+nyn+n-kyn-k-αe--k+2ny¯β+-k+2ny¯,=-αe-nyn+n-kyn-kenyn+n-kyn-k--k+2ny¯-1β+nyn+n-kyn-k-αne--k+2ny¯β+nyn+n-kyn-kβ+-k+2ny¯yn-y¯-α-k+ne--k+2ny¯β+nyn+n-kyn-kβ+-k+2ny¯yn-k-y¯.After simplifying, one gets(130)xn+1-x¯=-αne-nyn+n-kyn-kβ+-k+2ny¯+e--k+2ny¯β+nyn+n-kyn-kβ+-k+2ny¯yn-y¯-α-k+ne-nyn+n-kyn-kβ+-k+2ny¯+e--k+2ny¯β+nyn+n-kyn-kβ+-k+2ny¯yn-k-y¯+O5nyn+n-kyn-k--k+2ny¯2.Similarly,(131)yn+1-y¯=-α1ne-nxn+n-kxn-kβ1+-k+2nx¯+e--k+2nx¯β1+nxn+n-kxn-kβ1+-k+2nx¯xn-x¯-α1n-ke-nxn+n-kxn-kβ1+-k+2nx¯+e--k+2nx¯β1+nxn+n-kxn-kβ1+-k+2nx¯xn-k-x¯+O6nxn+n-kxn-k--k+2nx¯2.From (130) and (131), we have(132)xn+1-x-≈-αne-nyn+n-kyn-kβ+-k+2ny-+e--k+2ny-β+nyn+n-kyn-kβ+-k+2ny-yn-y--α-k+ne-nyn+n-kyn-kβ+-k+2ny-+e--k+2ny-β+nyn+n-kyn-kβ+-k+2ny-yn-k-y-,yn+1-y-≈-α1ne-nxn+n-kxn-kβ1+-k+2nx-+e--k+2nx-β1+nxn+n-kxn-kβ1+-k+2nx-xn-x--α1n-ke-nxn+n-kxn-kβ1+-k+2nx-+e--k+2nx-β1+nxn+n-kxn-kβ1+-k+2nx-xn-k-x-.Let (109) hold, then (132) can be represented as(133)ϵn+11≈An17ϵn2+An18ϵn-k2,ϵn+12≈An19ϵn1+An20ϵn-k1,where(134)An17=-αne-nyn+n-kyn-kβ+-k+2ny¯+e--k+2ny¯β+nyn+n-kyn-kβ+-k+2ny¯,An18=-α-k+ne-nyn+n-kyn-kβ+-k+2ny¯+e--k+2ny¯β+nyn+n-kyn-kβ+-k+2ny¯,An19=-α1ne-nxn+n-kxn-kβ1+-k+2nx¯+e--k+2nx¯β1+nxn+n-kxn-kβ1+-k+2nx¯,An20=-α1-k+ne-nxn+n-kxn-kβ1+-k+2nx¯+e--k+2nx¯β1+nxn+n-kxn-kβ1+-k+2nx¯.Moreover,(135)limn→∞An17=-nαe--k+2ny¯β+-k+2ny¯+1β+-k+2ny¯2,limn→∞An18=--k+nαe--k+2ny¯β+-k+2ny¯+1β+-k+2ny¯2,limn→∞An19=-nα1e--k+2nx¯β1+-k+2nx¯+1β1+-k+2nx¯2,limn→∞An20=--k+nα1e--k+2nx¯β1+-k+2nx¯+1β1+-k+2nx¯2. Hence limiting system of error terms becomes(136)En+1=EEn, where(137)E=00…00-nαe--k+2ny¯β+-k+2ny¯+1β+-k+2ny¯20…0--k+nαe--k+2ny¯β+-k+2ny¯+1β+-k+2ny¯210…0000…00⋮⋮…⋮⋮⋮⋱⋮⋮⋮00…1000…00-nα1e--k+2nx¯β1+-k+2nx¯+1β1+-k+2nx¯20…0--k+nα1e--k+2nx¯β1+-k+2nx¯+1β1+-k+2nx¯200…0000…0010…00⋮⋮⋱⋮⋮⋮⋱⋮⋮⋮00…0000…10,which is the same as linearized system of (12) about (x¯,y¯). Finally we have following theorem.Theorem 8.
Assume that{(xn,yn)} is a positive solution of (13) such that (104) along with following relation holds:(138)x¯∈L5,U5,y¯∈L6,U6.Then the error vector En defined in (116) satisfies(139)limn→∞En1/n=λE,limn→∞En+1En=λE,where λE are the characteristic roots of E about (x¯,y¯).


## 6.1. Rate of Convergence of System (11)

Let{(xn,yn)} be any solution of (11) such that the following hold:(104)limn→∞xn=x¯,limn→∞yn=y¯. Now computing error terms one gets(105)xn+1-x¯=αe-nxn+n-kxn-kβ+nyn+n-kyn-k-αe--k+2nx¯β+-k+2ny¯,=-αe-nxn+n-kxn-kenxn+n-kxn-k--k+2nx¯-1β+nyn+n-kyn-k-αne--k+2nx¯β+nyn+n-kyn-kβ+-k+2ny¯yn-y¯-αn-ke--k+2nx¯β+nyn+n-kyn-kβ+-k+2ny¯yn-k-y¯.After some tedious calculations, from (105) one gets(106)xn+1-x¯=-αne-nxn+n-kxn-kβ+nyn+n-kyn-kxn-x¯-αn-ke-nxn+n-kxn-kβ+nyn+n-kyn-kxn-k-x¯-αne--k+2nx¯β+nyn+n-kyn-kβ+-k+2ny¯yn-y¯-αn-ke--k+2nx¯β+nyn+n-kyn-kβ+-k+2ny¯yn-k-y¯+O1nxn+n-kxn-k--k+2nx¯2.Similarly,(107)yn+1-y¯=-α1ne--k+2ny¯β1+nxn+n-kxn-kβ1+-k+2nx¯xn-x¯-α1n-ke--k+2ny¯β1+nxn+n-kxn-kβ1+-k+2nx¯xn-k-x¯-α1ne-nyn+n-kyn-kβ1+nxn+n-kxn-kyn-x¯-α1n-ke-nyn+n-kyn-kβ1+nxn+n-kxn-kyn-k-y¯+O2nyn+n-kyn-k--k+2ny¯2.From (106) and (107), we have(108)xn+1-x-≈-αne-nxn+n-kxn-kβ+nyn+n-kyn-kxn-x--αn-ke-nxn+n-kxn-kβ+nyn+n-kyn-kxn-k-x--αne--k+2nx-β+nyn+n-kyn-kβ+-k+2ny-yn-y--αn-ke--k+2nx-β+nyn+n-kyn-kβ+-k+2ny-yn-k-y-,yn+1-y-≈-α1ne--k+2ny-β1+nxn+n-kxn-kβ1+-k+2nx-xn-x--α1n-ke--k+2ny-β1+nxn+n-kxn-kβ1+-k+2nx-xn-k-x--α1ne-nyn+n-kyn-kβ1+nxn+n-kxn-kyn-x--α1n-ke-nyn+n-kyn-kβ1+nxn+n-kxn-kyn-k-y-.Let(109)ϵn1=xn-x¯,ϵn2=yn-y¯,and then (108) becomes(110)ϵn+11≈An1ϵn1+An2ϵn-k1+An3ϵn2+An4ϵn-k2,ϵn+12≈An5ϵn1+An6ϵn-k1+An7ϵn2+An8ϵn-k2,where(111)An1=-αne-nxn+-k+nxn-kβ+nyn+-k+nyn-k,An2=-α-k+ne-nxn+n-kxn-kβ+nyn+n-kyn-k,An3=-αne--k+2nx¯β+nyn+n-ky-k+nβ+-k+2ny¯,An4=-α-k+ne--k+2nx¯β+nyn+n-kyn-kβ+-k+2ny¯,An5=-α1ne--k+2ny¯β1+nxn+n-kxn-kβ1+-k+2nx¯,An6=-α1-k+ne--k+2ny¯β1+nxn+n-kxn-kβ1+-k+2nx¯,An7=-α1ne-nyn+n-kyn-kβ1+nxn+n-kxn-k,An8=-α1-k+ne-nyn+n-kyn-kβ1+nxn+n-kxn-k.Moreover,(112)limn→∞An1=-nαe--k+2nx¯β+-k+2ny¯,limn→∞An2=--k+nαe--k+2nx¯β+-k+2ny¯,limn→∞An3=-nαe--k+2nx¯β+-k+2ny¯2,limn→∞An4=--k+nαe--k+2nx¯β+-k+2ny¯2,limn→∞An5=-nα1e--k+2ny¯β1+-k+2nx¯2,limn→∞An6=--k+nα1e--k+2ny¯β1+-k+2nx¯2,limn→∞An7=-nα1e--k+2ny¯β1+-k+2nx¯,limn→∞An8=--k+nα1e--k+2ny¯β1+-k+2nx¯.Hence limiting system of error terms becomes(113)En+1=EEn,where(114)E=-nαe--k+2nx¯β+-k+2ny¯0…0--k+nαe--k+2nx¯β+-k+2ny¯-nαe--k+2nx¯β+-k+2ny¯20…0--k+nαe--k+2nx¯β+-k+2ny¯210…0000…00⋮⋮…⋮⋮⋮⋱⋮⋮⋮00…1000…00-nα1e--k+2ny¯β1+-k+2nx¯20…0--k+nα1e--k+2ny¯β1+-k+2nx¯2-nα1e--k+2ny¯β1+-k+2nx¯0…0--k+nα1e--k+2ny¯β1+-k+2nx¯00…0010…00⋮⋮⋱⋮⋮⋮⋱⋮⋮⋮00…0000…10,which is similar to linearized system of (11). Finally we have the following theorem by utilizing Perron’s Theorems (see [19]).Theorem 6.
Assume that{(xn,yn)} is a positive solution of (11) such that (104) along with the following relation holds:(115)x¯∈L1,U1,y¯∈L2,U2.Then the error vector(116)En=ϵn1ϵn-11⋮ϵn-k1ϵn2ϵn-12⋮ϵn-k2of every solution of (11) satisfies(117)limn→∞En1/n=λE,limn→∞En+1En=λE,where λE are the characteristic roots of E about (x¯,y¯).


## 6.2. Rate of Convergence of System (12)

If{(xn,yn)} is any solution of (12) such that (104) holds, then(118)xn+1-x¯=αe-nyn+n-kyn-kβ+nxn+n-kxn-k-αe--k+2ny¯β+-k+2nx¯,=-αne--k+2ny¯β+nxn+n-kxn-kβ+-k+2nx¯xn-x¯-αn-ke--k+2ny¯β+nxn+n-kxn-kβ+-k+2nx¯xn-k-x¯-αe-nyn+n-kyn-kenyn+n-kyn-k--k+2ny¯-1β+nxn+n-kxn-k.After calculations, from (118) one gets(119)xn+1-x¯=-αne--k+2ny¯β+nxn+n-kxn-kβ+-k+2nx¯xn-x¯-αn-ke--k+2ny¯β+nxn+n-kxn-kβ+-k+2nx¯xn-k-x¯-αne-nyn+n-kyn-kβ+nxn+n-kxn-kyn-y¯-αn-ke-nyn+n-kyn-kβ+nxn+n-kxn-kyn-k-y¯+O3nyn+n-kyn-k--k+2ny¯2.Similarly,(120)yn+1-y¯=-α1ne-nxn+n-kxn-kβ1+nyn+n-kyn-kxn-x¯-α1n-ke-nxn+n-kxn-kβ1+nyn+n-kyn-kxn-k-x¯-α1ne--k+2nx¯β1+nyn+n-kyn-kβ1+-k+2ny¯yn-y¯-α1n-ke--k+2nx¯β1+nyn+n-kyn-kβ1+-k+2ny¯yn-k-y¯+O4nxn+n-kxn-k--k+2nx¯2.From (119) and (120), we have(121)xn+1-x-≈-αne--k+2ny-β+nxn+n-kxn-kβ+-k+2nx-xn-x--αn-ke--k+2ny-β+nxn+n-kxn-kβ+-k+2nx-xn-k-x--αne-nyn+n-kyn-kβ+nxn+n-kxn-kyn-y--αn-ke-nyn+n-kyn-kβ+nxn+n-kxn-kyn-k-y-,yn+1-y-≈-α1ne-nxn+n-kxn-kβ1+nyn+n-kyn-kxn-x--α1n-ke-nxn+n-kxn-kβ1+nyn+n-kyn-kxn-k-x--α1ne--k+2nx-β1+nyn+n-kyn-kβ1+-k+2ny-yn-y--α1n-ke--k+2nx-β1+nyn+n-kyn-kβ1+-k+2ny-yn-k-y-.Let (109) hold, then from (121) one gets(122)ϵn+11≈An9ϵn1+An10ϵn-k1+An11ϵn2+An12ϵn-k2,ϵn+12≈An13ϵn1+An14ϵn-k1+An15ϵn2+An10ϵn-k2,where(123)An9=-αne--k+2ny¯β+nxn+n-kxn-kβ+-k+2nx¯,An10=-α-k+ne--k+2ny¯β+nxn+n-kxn-kβ+-k+2nx¯,An11=-αne-nyn+n-kyn-kβ+nxn+n-kxn-k,An12=-α-k+ne-nyn+n-kyn-kβ+nxn+n-kxn-k,An13=-α1ne-nxn+n-kxn-kβ1+nyn+n-kyn-k,An14=-α1-k+ne-nxn+n-kxn-kβ1+nyn+n-kyn-k,An15=-α1ne--k+2nx¯β1+nyn+n-kyn-kβ1+-k+2ny¯,An16=-α1-k+ne--k+2nx¯β1+nyn+n-kyn-kβ1+-k+2ny¯.Moreover,(124)limn→∞An9=-nαe--k+2ny¯β+-k+2nx¯2,limn→∞An10=--k+nαe--k+2ny¯β+-k+2nx¯2,limn→∞An11=-nαe-2n-ky¯β+2n-kx¯,limn→∞An12=--k+nαe--k+2ny¯β+2n-kx¯,limn→∞An13=-nα1e--k+2nx¯β1+-k+2ny¯,limn→∞An14=--k+nα1e--k+2nx¯β1+-k+2ny¯,limn→∞An15=-nα1e--k+2nx¯β1+-k+2ny¯2,limn→∞An16=--k+nα1e--k+2nx¯β1+-k+2ny¯2.Hence limiting system of error terms becomes(125)En+1=EEn, where(126)E=-nαe--k+2ny¯β+-k+2nx¯20…0--k+nαe--k+2ny¯β+-k+2nx¯2-nαe--k+2ny¯β+-k+2nx¯0…0--k+nαe--k+2ny¯β+2n-kx¯10…0000…00⋮⋮…⋮⋮⋮⋱⋮⋮⋮00…1000…00-nα1e--k+2nx¯β1+-k+2ny¯0…0--k+nα1e--k+2nx¯β1+-k+2ny¯-nα1e--k+2nx¯β1+-k+2ny¯20…0--k+nα1e--k+2nx¯β1+-k+2ny¯200…0010…00⋮⋮⋱⋮⋮⋮⋱⋮⋮⋮00…0000…10,which is the same as linearized system of (12) about (x¯,y¯). Finally we have following theorem.Theorem 7.
Assume that{(xn,yn)} is a positive solution of (12) such that (104) along with the following relation holds:(127)x¯∈L3,U3,y¯∈L4,U4. Then the error vector En defined in (116) satisfies(128)limn→∞En1/n=λE,limn→∞En+1En=λE,where λE are the characteristic roots of E about (x¯,y¯).


## 6.3. Rate of Convergence of System (13)

If{(xn,yn)} is any solution of (13) such that (104) holds, then(129)xn+1-x¯=αe-nyn+n-kyn-kβ+nyn+n-kyn-k-αe--k+2ny¯β+-k+2ny¯,=-αe-nyn+n-kyn-kenyn+n-kyn-k--k+2ny¯-1β+nyn+n-kyn-k-αne--k+2ny¯β+nyn+n-kyn-kβ+-k+2ny¯yn-y¯-α-k+ne--k+2ny¯β+nyn+n-kyn-kβ+-k+2ny¯yn-k-y¯.After simplifying, one gets(130)xn+1-x¯=-αne-nyn+n-kyn-kβ+-k+2ny¯+e--k+2ny¯β+nyn+n-kyn-kβ+-k+2ny¯yn-y¯-α-k+ne-nyn+n-kyn-kβ+-k+2ny¯+e--k+2ny¯β+nyn+n-kyn-kβ+-k+2ny¯yn-k-y¯+O5nyn+n-kyn-k--k+2ny¯2.Similarly,(131)yn+1-y¯=-α1ne-nxn+n-kxn-kβ1+-k+2nx¯+e--k+2nx¯β1+nxn+n-kxn-kβ1+-k+2nx¯xn-x¯-α1n-ke-nxn+n-kxn-kβ1+-k+2nx¯+e--k+2nx¯β1+nxn+n-kxn-kβ1+-k+2nx¯xn-k-x¯+O6nxn+n-kxn-k--k+2nx¯2.From (130) and (131), we have(132)xn+1-x-≈-αne-nyn+n-kyn-kβ+-k+2ny-+e--k+2ny-β+nyn+n-kyn-kβ+-k+2ny-yn-y--α-k+ne-nyn+n-kyn-kβ+-k+2ny-+e--k+2ny-β+nyn+n-kyn-kβ+-k+2ny-yn-k-y-,yn+1-y-≈-α1ne-nxn+n-kxn-kβ1+-k+2nx-+e--k+2nx-β1+nxn+n-kxn-kβ1+-k+2nx-xn-x--α1n-ke-nxn+n-kxn-kβ1+-k+2nx-+e--k+2nx-β1+nxn+n-kxn-kβ1+-k+2nx-xn-k-x-.Let (109) hold, then (132) can be represented as(133)ϵn+11≈An17ϵn2+An18ϵn-k2,ϵn+12≈An19ϵn1+An20ϵn-k1,where(134)An17=-αne-nyn+n-kyn-kβ+-k+2ny¯+e--k+2ny¯β+nyn+n-kyn-kβ+-k+2ny¯,An18=-α-k+ne-nyn+n-kyn-kβ+-k+2ny¯+e--k+2ny¯β+nyn+n-kyn-kβ+-k+2ny¯,An19=-α1ne-nxn+n-kxn-kβ1+-k+2nx¯+e--k+2nx¯β1+nxn+n-kxn-kβ1+-k+2nx¯,An20=-α1-k+ne-nxn+n-kxn-kβ1+-k+2nx¯+e--k+2nx¯β1+nxn+n-kxn-kβ1+-k+2nx¯.Moreover,(135)limn→∞An17=-nαe--k+2ny¯β+-k+2ny¯+1β+-k+2ny¯2,limn→∞An18=--k+nαe--k+2ny¯β+-k+2ny¯+1β+-k+2ny¯2,limn→∞An19=-nα1e--k+2nx¯β1+-k+2nx¯+1β1+-k+2nx¯2,limn→∞An20=--k+nα1e--k+2nx¯β1+-k+2nx¯+1β1+-k+2nx¯2. Hence limiting system of error terms becomes(136)En+1=EEn, where(137)E=00…00-nαe--k+2ny¯β+-k+2ny¯+1β+-k+2ny¯20…0--k+nαe--k+2ny¯β+-k+2ny¯+1β+-k+2ny¯210…0000…00⋮⋮…⋮⋮⋮⋱⋮⋮⋮00…1000…00-nα1e--k+2nx¯β1+-k+2nx¯+1β1+-k+2nx¯20…0--k+nα1e--k+2nx¯β1+-k+2nx¯+1β1+-k+2nx¯200…0000…0010…00⋮⋮⋱⋮⋮⋮⋱⋮⋮⋮00…0000…10,which is the same as linearized system of (12) about (x¯,y¯). Finally we have following theorem.Theorem 8.
Assume that{(xn,yn)} is a positive solution of (13) such that (104) along with following relation holds:(138)x¯∈L5,U5,y¯∈L6,U6.Then the error vector En defined in (116) satisfies(139)limn→∞En1/n=λE,limn→∞En+1En=λE,where λE are the characteristic roots of E about (x¯,y¯).


## 7. Conclusion

This work is about the global dynamics of three higher-order exponential systems of difference equations. We proved that every positive solution is bounded and persistent, and furtherL1,U1×L2,U2, L3,U3×L4,U4, and L5,U5×L6,U6, respectively, are invariant rectangle for systems (11), (12), and (13). We studied existence and uniqueness of +ve equilibrium and global stability, and conclusions are presented in Table 1. Finally rate of convergence for (11), (12), and (13) is also investigated.Table 1
Systems with corresponding qualitative behavior.
Systems Corresponding behavior (11) Unique positive equilibrium point(x-,y-)∈L1,U1×L2,U2 if 0<β<1 and αe-(-k+2n)L1β1+-k+2nU1(-k+2n)U1+1αe-(-k+2n)L1+(1-β)U1<(-k+2n)2L12αe-(-k+2n)L1-βU1; ( x - , y - ) ∈ L 1 , U 1 × L 2 , U 2 is locally asymptotically stable if (-k+n)αe-(-k+2n)L1(β+(-k+2n)U2+1)+nαe-(-k+2n)U1(β+(-k+2n)L2+1)<(β+(-k+2n)L2)2, and (-k+n)α1e-(-k+2n)L2(β1+(-k+2n)U1+1)+nα1e-(-k+2n)U2(β1+(-k+2n)L1+1)<(β1+(-k+2n)L1)2; ( x - , y - ) ∈ L 1 , U 1 × L 2 , U 2 is globally asymptotically stable if αe-(-k+2n)L1<x-β+(-k+2n)L2andα1e-(-k+2n)L2<y-β1+(-k+2n)L1. (12) Unique positive equilibrium point(x-,y-)∈L3,U3×L4,U4 if e ( - k + 2 n ) U 3 β + 2 ( - k + 2 n ) U 3 β 1 + 2  l n  α / β + ( - k + 2 n ) L 3 L 3 < α 1 ( - k + 2 n ) 2 β + ( - k + 2 n ) L 3 L 3; ( x - , y - ) ∈ L 3 , U 3 × L 4 , U 4 is locally asymptotically stable if (-k+n)αe-(-k+2n)L4(β+(-k+2n)U3+1)+nαe-(-k+2n)U4(β+(-k+2n)L3+1)<(β+(-k+2n)L3)2, and (-k+n)α1e-(-k+2n)L3(β1+(-k+2n)U4+1)+nα1e-(-k+2n)U3(β1+(-k+2n)L4+1)<(β1+(-k+2n)L4)2; ( x - , y - ) ∈ L 3 , U 3 × L 4 , U 4 is globally asymptotically stable if αe-(-k+2n)L4<x-β+(-k+2n)L3andα1e-(-k+2n)L3<y-β1+(-k+2n)L4. (13) Unique positive equilibrium point(x-,y-)∈L5,U5×L6,U6 if α1(-k+2n)2e-(-k+2n)L5ββ1+(-k+2n)U5+(-k+2n)α1e-(-k+2n)L5U5Ω1<β1+(-k+2n)L52Ω22; ( x - , y - ) ∈ L 5 , U 5 × L 6 , U 6 is locally asymptotically stable if (-k+n)αe-(-k+2n)L6(β+(-k+2n)U6+1)+nαe-(-k+2n)U6(β+(-k+2n)L6+1)<(β+(-k+2n)L6)2, and (-k+n)α1e-(-k+2n)L5(β1+(-k+2n)U5+1)+nα1e-(-k+2n)U5(β1+(-k+2n)L5+1)<(β1+(-k+2n)L5)2; ( x - , y - ) ∈ L 5 , U 5 × L 6 , U 6 is globally asymptotically stable if αe-(-k+2n)L6<x-β+(-k+2n)L6andα1e-(-k+2n)L5<y-β1+(-k+2n)L5.



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*Source: 3825927-2019-06-02.xml*