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

We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the form x n+1 = x n−1 2/(ax n 2 + bx n x n−1 + cx n−1 2), n = 0,1, 2,…, where the parameters a,  b, and  c are positive numbers and the initial conditions x −1 and x 0 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.


Introduction
We investigate global behavior of the equation where the parameters , , and are positive numbers and the initial conditions −1 , and 0 are arbitrary nonnegative numbers such that −1 + 0 > 0. Equation (1) is a special case of equations Some special cases of (3) have been considered in the series of papers [1][2][3][4][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][7][8][9][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 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 +1 = ( / −1 ) 2 + ( / −1 ) + ( / −1 ) 2 + ( / −1 ) + , = 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][2][3][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) when 2 The Scientific World Journal = 0 is 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 see [15].
the subsequences { 2 } ∞ =0 and { 2 −1 } ∞ =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 Theorem 1 is that every bounded solution of (7) converges to either equilibrium or periodtwo 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.

Preliminaries
We now give some basic notions about monotone maps in the plane.
Consider a partial ordering ⪯ on R 2 . Two points , ∈ R 2 are said to be related if ⪯ or ⪯ . Also, a strict inequality between points may be defined as ≺ if ≺ and ̸ = . A stronger inequality may be defined as The map is monotone if ⪯ implies that ( ) ⪯ ( ) for all , ∈ R, and it is strongly monotone on R if ≺ implies that ( ) ≪ ( ) for all , ∈ R. The map is strictly monotone on R if ≺ implies that ( ) ≺ ( ) for all , ∈ R. Clearly, being related is invariant under iteration of a strongly monotone map.
A map on a nonempty set R ⊂ R 2 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.
If is a differentiable map on a nonempty set R, a sufficient condition for to be strongly monotone with respect to the SE ordering is that the Jacobian matrix at all points has the following sign configuration: provided that R is open and convex. For ∈ R 2 , define ℓ ( ) for ℓ = 1, . . . , 4 to be the usual four quadrants based at and numbered in a counterclockwise direction; for example, 1 ( ) = { ∈ 2 : Basin of attraction of a fixed point ( , ) of a map , denoted as B(( , )), is defined as the set of all initial points ( 0 , 0 ) for which the sequence of iterates (( 0 , 0 )) converges to ( , ). Similarly, we define a basin of attraction of a periodic point of period . 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.
Let be a competitive map on a rectangular region R ⊂ R 2 . Let x ∈ R be a fixed point of such that Δ := R ∩ int( 1 (x) ∪ 3 (x)) is nonempty (i.e., x is not the NW or SE vertex of R), and is strongly competitive on Δ. Suppose that the following statements are true.
(a) The map has a 1 extension to a neighborhood of x.
(b) The Jacobian (x) of at x has real eigenvalues , such that 0 < | | < , where | | < 1, and the eigenspace associated with 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 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 .
We will see in Theorem 4 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. For maps that are strongly competitive near the fixed point, hypothesis (b) of Theorem 2 reduces just to | | < 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 Theorem 2, 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, such that the following statements are true. 2 and strongly competitive in a neighborhood of x, then has no periodic points in the boundary of 1 (x) ∪ 3 (x) except for x, and the following statements are true.

(B) If, in addition to the hypotheses of part (A), x is an interior point of R and is
If is a map on a set R and if x is a fixed point of , the stable set W (x) of x is the set { ∈ R : ( ) → x} and unstable set W (x) of x is the set When is noninvertible, the set W (x) may not be connected and made up of infinitely many curves or W (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 sets W (x) and W (x) are the stable and unstable manifolds of .

Theorem 5. In addition to the hypotheses of part (B) of Theorem 4, suppose that
> 1 and that the eigenspace associated with is not a coordinate axis. If the curve C of Theorem 2 has endpoints in R, then C is the stable set W (x) of x, and the unstable set W (x) of is a curve in R that is tangential to at x and such that it is the graph of a strictly decreasing function of the first coordinate on an interval. Any endpoints of W (x) in R are fixed points of . Remark 6. We say that ( , V) is strongly decreasing in the first argument and strongly increasing in the second argument if it is differentiable and has first partial derivative 1 negative and first partial derivative 2 positive in a considered set. The connection between the theory of monotone maps and the asymptotic behavior of (7) follows from the fact that if is 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 on × ; see [14].
Set −1 = and = V in (7) to obtain the equivalent system Let ( , V) = (V, (V, )). The second iterate 2 is given by and it is strictly competitive on × (see [14]).

Remark 7.
The characteristic equation of (7) at an equilibrium point ( , ), has two real roots , which satisfy < 0 < and | | < , whenever is 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) where : × → is a continuous function and is decreasing in the first argument and increasing in the second argument. Assume that is a unique equilibrium point which is locally asymptotically stable and assume that ( , ) and ( , ) are minimal period-two solutions which are saddle points such that Then, the basin of attraction B(( , )) of ( , ) is the region between the global stable sets W (( , )) and W (( , )). 4 The Scientific World Journal The basins of attraction B(( , )) = W (( , )) and B(( , )) = W (( , )) are exactly the global stable sets of ( , ) and ( , ). If ( −1 , 0 ) ∈ W + (( , )) or ( −1 , 0 ) ∈ W − (( , )), then (( −1 , 0 )) converges to the other equilibrium point or to the other minimal period-two solutions or to the boundary of the region × .

Periodic Solutions
In this section, we present results for the existence of minimal period-two solutions of (7).
Proof. Suppose that there is a minimal period-two solution { , , , , . . .} of (1), where and are distinct nonnegative real numbers such that 2 + 2 ̸ = 0. Then, , satisfy The Scientific World Journal 5 from which we obtain three cases: Conclusion ( ) follows from (27) and (28). Subtracting (30) from (29), we have that is, for < . Substituting (32) in (29), we obtain from which Equation (32) implies that It is clear that If − > , then which is a contradiction. If 3 + < , then By using substitution equation (1) becomes the system of equations The map corresponding to the system (40) is of the form where ( , V) = 2 /( V 2 + V + 2 ). The second iteration of the map is where and the map 2 is competitive by Remark 7. The Jacobian matrix of the map 2 is Notice that periodic solutions (0, 1/ ), (1/ , 0), ( , ), and ( , ) of (1) are equilibrium points of the map 2 .
(ii) By (26), we have that By (50) and (51), we obtain that is, Similarly, we have so that Now, we obtain that Jacobian matrix of the map 2 at the point ( , ) is of the form ) .
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 (∞, ∞).