# Global Dynamics of Certain Homogeneous Second-Order Quadratic Fractional Difference Equation

Authors: M. Garić-Demirović; M. R. S. Kulenović; M. Nurkanović
Journal: The Scientific World Journal
 (2013)

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## Abstract

We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the formxn+1=xn-12/(axn2+bxnxn-1+cxn-12),n=0,1,2,…, where the parameters a,b, andc are positive numbers and the initial conditions x-1 and x0 are arbitrary nonnegative numbers. The unique feature of this equation is the coexistence of an equilibrium solution and the minimal period-two solution both of which are locally asymptotically stable.


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## Body

## 1. Introduction

We investigate global behavior of the equation(1)xn+1=xn-12axn2+bxnxn-1+cxn-12=f(xn,xn-1),
where the parametersa,b, andcare positive numbers and the initial conditionsx-1,andx0are arbitrary nonnegative numbers such thatx-1+x0>0. Equation (1) is a special case of equations
(2)xn+1=Axn2+Bxnxn-1+Cxn-12axn2+bxnxn-1+cxn-12,n=0,1,2,…,(3)xn+1=Axn2+Bxnxn-1+Cxn-12+Dxn+Exn-1+Faxn2+bxnxn-1+cxn-12+dxn+exn-1+f,mmmmmmmmmmmmmmmmmmnln=0,1,2,….
Some special cases of (3) have been considered in the series of papers [1–5]. Some special second-order quadratic fractional difference equations have appeared in analysis of competitive and anticompetitive systems of linear fractional difference equations in the plane; see [6–10]. Describing the global dynamics of (3) is a formidable task as this equation contains as a special case many equations with complicated dynamics, such as the linear fractional difference equation
(4)xn+1=Dxn+Exn-1+Fdxn+exn-1+f,n=0,1,2,…
which dynamics was investigated in great detail in [11] and in many papers which solved some conjectures and open problems posed in [11]. Equation (2) can be brought to the form
(5)xn+1=A(xn/xn-1)2+B(xn/xn-1)+Ca(xn/xn-1)2+b(xn/xn-1)+c,n=0,1,2,…,
and one can take the advantage of this auxiliary equation to describe the dynamics of (2). This approach was used in [1–4, 12]. In this paper, we take a different approach based on the theory of monotone maps developed in [13, 14] and use it to describe precisely the basins of attraction of all attractors of this equation. The special case of (1) whena=0is the linear fractional difference equation whose global dynamics is described in [11]. We show that (1) exhibits three types of global behavior characterized by the existence of a unique positive equilibrium solution and one or two minimal period-two solutions, one of which is locally stable and the other is a saddle point. The unique feature of (1) is the coexistence of an equilibrium solution and the minimal period-two solution both being locally asymptotically stable. This new phenomenon is caused by the presence of quadratic terms and did not exist in the case of (4).Our results will be based on the following theorem for a general second-order difference equation(6)xn+1=f(xn,xn-1),n=0,1,2,…;
see [15].Theorem 1.
LetIbe a set of real numbers and letf:I×I→Ibe a function which is nonincreasing in the first variable and nondecreasing in the second variable. Then, for every solution{xn}n=-1∞of the equation
(7)xn+1=f(xn,xn-1),x-1,x0∈I,n=0,1,2,…,
the subsequences{x2n}n=0∞and{x2n-1}n=0∞of even and odd terms of the solution do exactly one of the following:(i)
eventually, they are both monotonically increasing;(ii)
eventually, they are both monotonically decreasing;(iii)
one of them is monotonically increasing and the other is monotonically decreasing.The consequence of Theorem1 is that every bounded solution of (7) converges to either equilibrium or period-two solution or to the point on the boundary, and the most important question becomes determining the basins of attraction of these solutions as well as the unbounded solutions. The answer to this question follows from an application of theory of monotone maps in the plane which will be presented in Section 2.


## 2. Preliminaries

We now give some basic notions about monotone maps in the plane.Consider a partial ordering⪯onℝ2. Two pointsx,y∈ℝ2are said to be related ifx⪯yory⪯x. Also, a strict inequality between points may be defined asx≺yifx≺yandx≠y. A stronger inequality may be defined asx=(x1,x2)≪y=(y1,y2)ifx⪯ywithx1≠y1andx2≠y2.A mapT on a nonempty set ℛ⊂ℝ2 is a continuous function T:ℛ→ℛ. The map T is monotone if x⪯y implies that T(x)⪯T(y) for all x,y∈ℛ, and it is strongly monotone on ℛ if x≺y implies that T(x)≪T(y) for all x,y∈ℛ. The map is strictly monotone on ℛ if x≺y implies that T(x)≺T(y) for all x,y∈ℛ. Clearly, being related is invariant under iteration of a strongly monotone map.Throughout this paper, we will use theNorth-East ordering (NE) for which the positive cone is the first quadrant; that is, this partial ordering is defined by(x1,y1)⪯ne(x2,y2)ifx1≤x2andy1≤y2and the South-East (SE) ordering defined as(x1,y1)⪯se(x2,y2)ifx1≤x2andy1≥y2.A mapT on a nonempty set ℛ⊂ℝ2 which is monotone with respect to the North-East ordering is called cooperative and a map monotone with respect to the South-East ordering is called competitive.IfT is a differentiable map on a nonempty set ℛ, a sufficient condition for T to be strongly monotone with respect to the SE ordering is that the Jacobian matrix at all points x has the following sign configuration:
(8)sign(JT(x))=[+--+],
provided that ℛ is open and convex.Forx∈ℝ2, define Qℓ(x) for ℓ=1,…,4 to be the usual four quadrants based at x and numbered in a counterclockwise direction; for example, Q1(x)={y∈R2:x1≤y1,x2≤y2}. Basin of attraction of a fixed point (x-,y-) of a map T, denoted as ℬ((x-,y-)), is defined as the set of all initial points (x0,y0) for which the sequence of iterates Tn((x0,y0)) converges to (x-,y-). Similarly, we define a basin of attraction of a periodic point of period p. The next five results, from [13, 14], are useful for determining basins of attraction of fixed points of competitive maps. Related results have been obtained by Smith in [5, 16].Theorem 2.
LetTbe a competitive map on a rectangular region ℛ⊂ℝ2. Let x¯∈ℛ be a fixed point ofTsuch thatΔ:=ℛ∩int(Q1(x¯)∪Q3(x¯))is nonempty (i.e.,x¯is not the NW or SE vertex of ℛ), andTis strongly competitive onΔ. Suppose that the following statements are true.(a)
The mapThas aC1extension to a neighborhood of  x¯.(b)
The JacobianJT(x¯) of T at x¯ has real eigenvalues λ, μ such that 0<|λ|<μ, where |λ|<1, and the eigenspace Eλ associated with λ is not a coordinate axis.Then, there exists a curve 𝒞⊂ℛ through x¯ that is invariant and a subset of the basin of attraction of x¯, such that 𝒞 is tangential to the eigenspace Eλ at x¯, and 𝒞 is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of 𝒞 in the interior of ℛ are either fixed points or minimal period-two points. In the latter case, the set of endpoints of 𝒞 is a minimal period-two orbit of T.We will see in Theorem4 that the situation where the endpoints of 𝒞 are boundary points of ℛ is of interest. The following result gives a sufficient condition for this case.Theorem 3.
For the curve𝒞 of Theorem 2 to have endpoints in∂ℛ, it is sufficient that at least one of the following conditions is satisfied.(i)
The mapThas no fixed points nor periodic points of minimal period-two inΔ.(ii)
The mapThas no fixed points inΔ,detJT(x¯)>0, andT(x)=x¯has no solutionsx∈Δ.(iii)
The mapThas no points of minimal period-two inΔ,detJT(x¯)<0, andT(x)=x¯has no solutionsx∈Δ.For maps that are strongly competitive near the fixed point, hypothesis (b) of Theorem 2 reduces just to|λ|<1. This follows from a change of variables [5] that allows the Perron-Frobenius theorem to be applied. Also, one can show that in such case no associated eigenvector is aligned with a coordinate axis.The next result is useful for determining basins of attraction of fixed points of competitive maps.Theorem 4.
(A) Assume the hypotheses of Theorem2, and let 𝒞 be the curve whose existence is guaranteed by Theorem 2. If the endpoints of 𝒞 belong to ∂ℛ, then 𝒞 separates ℛ into two connected components, namely,
(9)𝒲-:={x∈ℛ∖𝒞:∃y∈𝒞withx⪯sey},𝒲+:={x∈ℛ∖𝒞:∃y∈𝒞withy⪯sex},
such that the following statements are true.(i)
𝒲-is invariant, anddist(Tn(x),Q2(x¯))→0asn→∞for everyx∈𝒲-.(ii)
𝒲+is invariant, anddist(Tn(x),Q4(x¯))→0asn→∞for everyx∈𝒲+.(B) If, in addition to the hypotheses of part (A),x¯is an interior point of ℛ andTisC2and strongly competitive in a neighborhood ofx¯, thenThas no periodic points in the boundary ofQ1(x¯)∪Q3(x¯)except forx¯, and the following statements are true.(iii)
For everyx∈𝒲-, there existsn0∈ℕsuch thatTn(x)∈intQ2(x¯)forn≥n0.(iv)
For everyx∈𝒲+, there existsn0∈ℕsuch thatTn(x)∈intQ4(x¯)forn≥n0.IfTis a map on a set ℛ and ifx¯is a fixed point ofT, the stable set𝒲s(x¯)ofx¯is the set{x∈ℛ:Tn(x)→x¯}and unstable set𝒲u(x¯)ofx¯is the set
(10){x0=x,limn→-∞xn=x¯}x∈ℛ:thereexists{xn}n=-∞0⊂ℛs.t.T(xn)=xn+1,x0=x,limn→-∞xn=x¯}.WhenTis noninvertible, the set𝒲s(x¯)may not be connected and made up of infinitely many curves or𝒲u(x¯)may not be a manifold. The following result gives a description of the stable and unstable sets of a saddle point of a competitive map. If the map is a diffeomorphism on ℛ, the sets𝒲s(x¯)and𝒲u(x¯)are the stable and unstable manifolds ofx¯.Theorem 5.
In addition to the hypotheses of part (B) of Theorem4, suppose that μ>1 and that the eigenspace Eμ associated with μ is not a coordinate axis. If the curve 𝒞 of Theorem 2 has endpoints in ∂ℛ, then 𝒞 is the stable set 𝒲s(x¯) of x¯, and the unstable set 𝒲u(x¯) of x¯ is a curve in ℛ that is tangential to Eμ at x¯ and such that it is the graph of a strictly decreasing function of the first coordinate on an interval. Any endpoints of 𝒲u(x¯) in ℛ are fixed points of T.Remark 6.
We say thatf(u,v)is strongly decreasing in the first argument and strongly increasing in the second argument if it is differentiable and has first partial derivativeD1fnegative and first partial derivativeD2fpositive in a considered set. The connection between the theory of monotone maps and the asymptotic behavior of (7) follows from the fact that iffis strongly decreasing in the first argument and strongly increasing in the second argument, then the second iterate of a map associated to (7) is a strictly competitive map onI×I; see [14].
Setxn-1=unandxn=vnin (7) to obtain the equivalent system
(11)un+1=vnvn+1=f(vn,un),n=0,1,….
LetT(u,v)=(v,f(v,u)). The second iterateT2is given by
(12)T2(u,v)=(f(v,u),f(f(v,u),v))
and it is strictly competitive onI×I(see [14]).Remark 7.
The characteristic equation of (7) at an equilibrium point(x-,x-),
(13)λ2-D1f(x-,x-)λ-D2f(x-,x-)=0,
has two real rootsλ,μwhich satisfyλ<0<μand|λ|<μ, wheneverfis strictly decreasing in first and increasing in second variable. Thus, the applicability of Theorems 2–5 depends on the nonexistence of minimal period-two solution.There are several global attractivity results for (7). Some of these results give the sufficient conditions for all solutions to approach a unique equilibrium and they were used efficiently in [11]. The next result is from [17].Theorem 8 (see [17]).
Consider (7) wheref:I×I→Iis a continuous function andfis decreasing in the first argument and increasing in the second argument. Assume thatx¯is a unique equilibrium point which is locally asymptotically stable and assume that(φ,ψ)and(ψ,φ)are minimal period-two solutions which are saddle points such that
(14)(φ,ψ)⪯se(x¯,x¯)⪯se(ψ,φ).
Then, the basin of attraction ℬ((x¯,x¯)) of (x¯,x¯) is the region between the global stable sets 𝒲s((φ,ψ)) and 𝒲s((ψ,φ)). More precisely,
(15)ℬ((x¯,x¯))={(x,y):∃yu,yl:yu<y<yl,c(x,yl)∈𝒲s((φ,ψ)),(x,yu)∈𝒲s((ψ,φ))}.
The basins of attraction ℬ((φ,ψ))=𝒲s((φ,ψ)) and ℬ((ψ,φ))=𝒲s((ψ,φ)) are exactly the global stable sets of (φ,ψ) and (ψ,φ).
If(x-1,x0)∈𝒲+((ψ,φ))or(x-1,x0)∈𝒲-((φ,ψ)), thenTn((x-1,x0))converges to the other equilibrium point or to the other minimal period-two solutions or to the boundary of the regionI×I.


## 3. Local Stability Analysis

Denote(16)f(u,v)=v2au2+buv+cv2,
and notice that the functionf(u,v)is decreasing in the first variable and increasing in the second variable. By Theorem 1, for every solutions{xn}n=-1∞of (1) the subsequences{x2n}n=0∞and{x2n-1}n=0∞are eventually monotonic.It is clear that (1) has a unique positive equilibrium solutionx¯=1/(a+b+c)and that a linearization of (1) is of the form
(17)yn+1=syn+tyn-1,
where
(18)s=-t=∂f∂u(x¯,x¯)=(-v2(2au+bv)(au2+buv+cv2)2)(x¯,x¯)=-x¯2(2ax¯+bx¯)(ax¯2+bx¯2+cx¯2)2=-1x¯·2a+b(a+b+c)2=-2a+ba+b+c.Lemma 9.
Equation (1) has a unique positive equilibrium solutionx¯=1/(a+b+c).(i)
If3a+b-c<0, then equilibrium solutionx¯is locally asymptotically stable.(ii)
If3a+b-c>0, then equilibrium solutionx¯is a saddle point.(iii)
If3a+b-c=0, then equilibrium solutionx¯is nonhyperbolic (with eigenvaluesλ1=-1andλ2=1/2).Proof.
By (17), a linearization of (1) is of the form
(19)yn+1+2a+ba+b+cyn-2a+ba+b+cyn-1=0.
Its characteristic equation is
(20)λ2+2a+ba+b+cλ-2a+ba+b+c=0,
with eigenvaluesλ±=(-(2a+b)±D)/2(a+b+c), whereD=(2a+b)(6a+5b+4c). It is clear thatλ-<0andλ+>0. Now, we prove thatλ+∈(0,1)and
(21)λ-{∈(-1,0)for3a+b-c<0<-1for3a+b-c>0=-1for3a+b-c=0.
Namely,(i)
λ+<1⇔-(2a+b)+D<2(a+b+c)⇔D<4a+3b+2c⇔4(a+b+c)2>0, which is always satisfied;(ii)
also,(22)λ-<-1⟺-(2a+b)-D<-2(a+b+c)⟺D>b+2c⟺(2a+b)2>(a-c)2⟺(a+b+c)(3a+b-c)>0⟺3a+b-c>0;(iii)
λ->-1⇔3a+b-c<0andλ-=-1⇔3a+b-c=0.Also, if3a+b-c=0, thenc=3a+b, and we haveλ+=1/2.


## 4. Periodic Solutions

In this section, we present results for the existence of minimal period-two solutions of (7).Theorem 10.
(a) Equation (1) has the minimal period-two solution
(23){…,0,1c,0,1c,…},
for all positive values of parametersa,b, andc.
(b) If3a+b-c<0, then (1) has the minimal period-two solution:
(24){…,φ,ψ,φ,ψ,…},(φ≠ψ,φ>0,ψ>0),
where
(25)φ=1-D2(c-a),ψ=1+D2(c-a),D=3a+b-cb-a-c>0.Proof.
Suppose that there is a minimal period-two solution{φ,ψ,φ,ψ,…}of (1), whereφandψare distinct nonnegative real numbers such thatφ2+ψ2≠0. Then,φ,ψsatisfy
(26)φ=φ2aψ2+bψφ+cφ2,ψ=ψ2aφ2+bφψ+cψ2,
from which we obtain three cases:
(27)φ=0,aφ2+bφψ+cψ2=ψ,(28)aψ2+bφψ+cφ2=φ,ψ=0,(29)aψ2+bφψ+cφ2=φ,(30)aφ2+bφψ+cψ2=ψ.
Conclusion (a) follows from (27) and (28).
Subtracting (30) from (29), we have
(31)(ψ-φ)[(a-c)(φ-ψ)+1]=0,
that is,
(32)φ+ψ=1c-a
fora<c. Substituting (32) in (29), we obtain
(33)φ2-1c-aφ+a(a-b+c)(c-a)2=0,
from which
(34)φ±=1±D2(c-a).
Equation (32) implies that
(35)ψ±=1c-a-φ±=φ∓.
It is clear that
(36)D=3a+b-cb-a-c>0⟺{(3a+b-c>0∧b-a-c>0)cc×∨(3a+b-c<0∧b-a-c<0)}⟺(b-a>c∨3a+b<c).
Ifb-a>c, then
(37)D=1+4ab-a-c>1⟹(φ-<0,φ+>0)⟹(ψ->0,ψ+<0),
which is a contradiction.
If3a+b<c, then
(38)0<D=1+4ab-a-c<1⟹(φ±>0,ψ±>0).By using substitution(39)xn-1=un,xn=vn,
equation (1) becomes the system of equations
(40)un+1=vn,vn+1=un2avn2+bunvn+cun2.The mapTcorresponding to the system (40) is of the form
(41)T(uv)=(vg(u,v)),
whereg(u,v)=u2/(av2+buv+cu2).The second iteration of the mapTis
(42)T2(uv)=T(vg(u,v))=(g(u,v)g(v,g(u,v)))=(F(u,v)G(u,v)),
where
(43)F(u,v)=g(u,v),G(u,v)=v2aF2(u,v)+bvF(u,v)+cv2,
and the mapT2is competitive by Remark 7. The Jacobian matrix of the mapT2is
(44)JT2(uv)=(∂F∂u∂F∂v∂G∂u∂G∂v).Notice that periodic solutions(0,1/c),(1/c,0),(φ,ψ), and(ψ,φ)of (1) are equilibrium points of the mapT2.Now, we have(45)∂F∂u=2u(av2+buv+cu2)-u2(bv+2cu)(av2+buv+cu2)2=uv(2av+bu)(av2+buv+cu2)2,∂F∂v=-u2(2av+bu)(av2+buv+cu2)2,∂G∂u=-v2(aF2(u,v)+bvF(u,v)+cv2)2·(2aF(u,v)+bv)∂F∂u,∂G∂v=(∂F∂v2v(aF2(u,v)+bvF(u,v)+cv2)-v2(2aF(u,v)(∂F/∂v)+bF(u,v)+bv(∂F/∂v)+2cv)∂G∂v)×(aF2(u,v)+bvF(u,v)+cv2)-2=(∂F∂vv(2aF(u,v)+bv)F(u,v)-(2aF(u,v)+bv)v2(∂F/∂v)∂G∂v)×(aF2(u,v)+bvF(u,v)+cv2)-2.Theorem 11.
(i) The minimal period-two points(46)(0,1c),(1c,0)
are locally asymptotically stable.
(ii) Ifa<c and 3a+b-c<0, then the minimal period-two points
(47)(φ,ψ),(ψ,φ)(φ≠ψ,φ>0,ψ>0),
whereφandψsatisfy (25), are saddle points.Proof.
(i) SinceF(0,1/c)=0, for periodic point(0,1/c), we have
(48)∂F∂u(01c)=∂F∂v(01c)=∂G∂u(01c)=∂G∂v(01c)=0,JT2(01c)=(0000)
with eigenvaluesλ1,2=0, which implies that(0,1/c)is locally asymptotically stable.
Similarly, sinceF(1/c,0)=1/c, for periodic solution(1/c,0), we have
(49)JT2(1c0)=(0-bc00)
with eigenvaluesλ1,2=0, which implies that(0,1/c)is locally asymptotically stable.
(ii) By (26), we have that
(50)F(φ,ψ)=φ,(51)∂F∂u(φψ)=ψ(2aψφ+b),(52)∂F∂v(φψ)=-φ(2aψφ+b).
By (50) and (51), we obtain
(53)∂G∂u(φψ)=-ψ2(aφ2+bψφ+cψ2)2·(2aφ+bψ)∂F∂u(φψ)=-ψ(2aφ+bψ)(2aψφ+b);
that is,
(54)∂G∂u(φψ)=-ψ2(2aφψ+b)(2aψφ+b).
Similarly, we have
(55)∂G∂v(φψ)=ψ(2aφ+bψ)φ-(2aφ+bψ)ψ2(∂F/∂v)(φψ)(aφ2+bψφ+cψ2)2=ψ(2aφ+bψ)φ+(2aφ+bψ)ψ2(2aψ+bφ)ψ2,
so that
(56)∂G∂v(φψ)=(2aφψ+b)φ+(2aφψ+b)(2aψφ+b)φψ.
Now, we obtain that Jacobian matrix of the mapT2at the point(φ,ψ)is of the form
(57)JT2(φψ)=(∂F∂u(φψ)∂F∂v(φψ)∂G∂u(φψ)∂G∂v(φψ)).
The corresponding characteristic equation is(58)λ2-pλ+q=0,
where
(59)p=∂F∂u(φψ)+∂G∂v(φψ)=ψ(2aψφ+b)+φ(2aφψ+bψ)+φψ(2aφψ+b)(2aψφ+b),
that is,
(60)p=m+n+mn,
where
(61)m=ψ(2aψφ+b),n=φ(2aφψ+b).
Notice that(62)∂F∂u(φψ)=m,∂F∂v(φψ)=-φψm,∂G∂u(φψ)=-ψφmn,∂G∂v(φψ)=n(1+m),
so that
(63)q=detJT2(φψ)=mn(1+m)-m2n=mn.
We need to show that(64)|p|>|1+q|p2-4q>0.(i)
Consider that(65)p2-4q>0⟺(m+n+mn)2-4mn>0⟺m2+n2+m2n2+2m2n+2mn2>2mn,
which is satisfied becausem2+n2≥2mn.(ii)
Notice that(66)m=ψ(2aψφ+b)=2aψ2φ+bψ=(26)2φ(φ-bφψ-cφ2)+bψ=2-bψ-2cφn=φ(2aφψ+b)=2aφ2ψ+bφ=(26)2ψ(ψ-bφψ-cψ2)+bφ=2-bφ-2cφ.
This implies that(67)|p|>|1+q|⟺p>1+q⟺m+n>1⟺3-(b+c)(φ+ψ)>0⇔(18)3-b+cc-a>0⟺3a+b-c<0,
which is satisfied.


## 5. Global Results and Basins of Attraction

In this section, we present global dynamics results for (1).Notice thatT(u,0)=(0,1/c),  u>0, andT(0,v)=(v,0),  v>0.Theorem 12.
If3a+b-c<0, then (7) has a unique equilibrium point x¯, which is locally asymptotically stable, and has the minimal period-two solution (φ,ψ), (ψ,φ), which is a saddle point and has the minimal period-two solution (0,1/c), (1/c,0) which is locally asymptotically stable. The basin of attraction ℬ((x¯,x¯)) of (x¯,x¯) is the region between the global stable sets 𝒲s((φ,ψ)) and 𝒲s((ψ,φ)). The basins of attraction ℬ((φ,ψ))=𝒲s((φ,ψ)) and ℬ((ψ,φ))=𝒲s((ψ,φ)) are exactly the global stable sets of (φ,ψ) and (ψ,φ). Furthermore, the basin of attraction of the minimal period-two solution (0,1/c), (1/c,0) is the union of the regions above 𝒲s((φ,ψ)) and below 𝒲s((ψ,φ)) in SE ordering; that is,(i)
if(x-1,x0)∈𝒲-((φ,ψ)), thenlimn→∞x2n=1/candlimn→∞x2n+1=0;(ii)
if(x-1,x0)∈𝒲+(ψ,φ), thenlimn→∞x2n=0andlimn→∞x2n+1=1/c.Proof.
Using assumption3a+b-c<0(and its consequences:c-a>0,b-c-a<0) and (25), it is easy to check that(0,1/c)⪯se(φ,ψ)⪯se(x¯,x¯)⪯se(ψ,φ)⪯se(1/c,0). It is easy to check that the equilibrium point(x¯,x¯)is locally asymptotically stable for the strictly competitive mapT2as well. Equation (1) is equivalent to the system of difference equations (40) which can be decomposed into the system of the even-indexed and odd-indexed terms as follows:
(68)u2n=v2n-1,u2n+1=v2n,v2n=v2n-12au2n-12+bu2n-1v2n-1+cv2n-12,v2n+1=v2n2au2n2+bu2nv2n+cv2n2.
The conclusion follows from Lemma9 and from Theorems 10, 11, and 8 and using the facts that(i)
if(x-1,x0)∈𝒲-((φ,ψ)), then
(69)(u2n,v2n)=T2n((u0,v0))⟶(0,1c),(u2n+1,v2n+1)=T2n+1((u0,v0))⟶(1c,0);(ii)
if(u0,v0)∈𝒲+((x¯,x¯)), then
(70)(u2n,v2n)=T2n((u0,v0))⟶(1c,0),(u2n+1,v2n+1)=T2n+1((u0,v0))⟶(0,1c).
It means that(i)
if(x-1,x0)∈𝒲-((x¯,x¯)), thenT2n((x-1,x0))→(0,1/c)andT2n+1((x-1,x0))→(1/c,0), that is,
(71)limn→∞x2n=1climn→∞x2n+1=0;(ii)
if(x-1,x0)∈𝒲+(ψ,φ), thenT2n+1((x-1,x0))→(0,1/c)andT2n((x-1,x0))→(1/c,0), that is,
(72)limn→∞x2n=0limn→∞x2n+1=1c(see Figure 1).Figure 1
Global dynamics of (1) fora=1,b=1, andc=5—an illustration of Theorem 12.Theorem 13.
If3a+b-c>0, then (1) has a unique equilibrium point x¯ which is saddle point and has the minimal period-two solution (0,1/c), (1/c,0) which is locally asymptotically stable. There exists a set 𝒞 which is an invariant subset of the basin of attraction of (x¯,x¯). The set 𝒞 is a graph of a strictly increasing continuous function of the first variable on an interval and separates ℛ∖(0,0), where ℛ=[0,∞)×[0,∞), into two connected and invariant components 𝒲-((x¯,x¯)) and 𝒲+((x¯,x¯)) which satisfy that(i)
if(x-1,x0)∈𝒲-((x¯,x¯)), thenlimn→∞x2n=1/candlimn→∞x2n+1=0;(ii)
if(x-1,x0)∈𝒲+((x¯,x¯)), thenlimn→∞x2n=0andlimn→∞x2n+1=1/c.Proof.
It is easy to check that(x¯,x¯)is a saddle point for the strictly competitive mapT2as well. The existence of the set 𝒞 with stated properties follows from Lemma 9 and Theorems 2, 4, and 10. Therefore, using (68), we obtain that(i)
if(u0,v0)∈𝒲-((x¯,x¯)),then
(73)(u2n,v2n)=T2n((u0,v0))⟶(0,1c),(u2n+1,v2n+1)=T2n+1((u0,v0))⟶(1c,0);(ii)
if(u0,v0)∈𝒲+((x¯,x¯)), then
(74)(u2n,v2n)=T2n((u0,v0))⟶(1c,0),(u2n+1,v2n+1)=T2n+1((u0,v0))⟶(0,1c).Consequently,(i)
if(x-1,x0)∈𝒲-((x¯,x¯)), thenT2n((x-1,x0))→(0,1/c)andT2n+1((x-1,x0))→(1/c,0); that is,
(75)limn→∞x2n=1climn→∞x2n+1=0;(ii)
if(x-1,x0)∈𝒲+((x¯,x¯)), thenT2n+1((x-1,x0))→(0,1/c)andT2n((x-1,x0))→(1/c,0); that is,
(76)limn→∞x2n=0limn→∞x2n+1=1c(see Figure 2).Figure 2
Global dynamics of (1) fora=2,b=2, andc=1—an illustration of Theorem 13.Theorem 14.
If3a+b-c=0, then (1) has a unique equilibrium point x¯ which is nonhyperbolic and has two minimal period-two points (0,1/c), (1/c,0) which are locally asymptotically stable points. There exists a set 𝒞 which is an invariant subset of the basin of attraction of (x¯,x¯). The set 𝒞 is a graph of a strictly increasing continuous function of the first variable on an interval and separates ℛ∖(0,0), where ℛ=[0,∞)×[0,∞), into two connected and invariant components 𝒲-((x¯,x¯)) and 𝒲+((x¯,x¯)) which satisfy that(i)
if(x-1,x0)∈𝒲-((x¯,x¯)), thenlimn→∞x2n=1/candlimn→∞x2n+1=0;(ii)
if(x-1,x0)∈𝒲+((x¯,x¯)),limn→∞x2n=0andlimn→∞x2n+1=1/c.Proof.
In view of Lemma9, the eigenvalues of the mapTat the equilibrium point(x¯,x¯)areλ1=-1  and  λ2=1/2, which means thatμ1=λ12=1andμ2=λ22=1/4are the eigenvalues of the mapT2.Using (51), (52), (54), and (56), we obtain
(77)JT2(x¯,x¯)=(Q-Q-Q2(Q-1)(Q-2)),
whereQ=(2a+b)/(a+b+c).A straightforward calculation yields that the eigenvector corresponding to the eigenvalueμ2=1/4is of the form
(78)v2=(2αα),α∈ℝ∖{0}.
We see that eigenvectorv2is not parallel to coordinate axes. Therefore, all conditions of Theorem 2 are satisfied for the mapT2with ℛ=(0,∞)×(0,∞). As a consequence of this and using (68), we have that(i)
if(u0,v0)∈𝒲-((x¯,x¯)),then(u2n,v2n)→(0,1/c)and(u2n+1,v2n+1)→(1/c,0);(ii)
if(u0,v0)∈𝒲+((x¯,x¯)), then(u2n,v2n)→(1/c,0)and(u2n+1,v2n+1)→(0,1/c).
It means that(i)
if(x-1,x0)∈𝒲-((x¯,x¯)), thenT2n((x-1,x0))→(0,1/c)andT2n+1((x-1,x0))→(1/c,0); that is,
(79)limn→∞x2n=1climn→∞x2n+1=0;(ii)
if(x-1,x0)∈𝒲+((x¯,x¯)), thenT2n+1((x-1,x0))→(0,1/c)andT2n((x-1,x0))→(1/c,0); that is,
(80)limn→∞x2n=0limn→∞x2n+1=1c(see Figure 3).Figure 3
Global dynamics of (1) fora=1/9,b=1/3, andc=2/3—an illustration of Theorem 14.Remark 15.
As one may notice from the figures all stable manifolds of either saddle point equilibrium points or saddle period-two solutions are asymptotic to the origin, which is the point where (1) is not defined. These manifolds cannot end in any other point on the axes since the union of axes without the origin is an invariant set. Thus, the limiting points of the global stable manifolds of either saddle point equilibrium points or saddle period-two solutions have endpoints at(0,0)and(∞,∞).



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*Source: 210846-2013-12-04.xml*